text:
\begin{document}

\title{On Good Universality and the Riemann Hypothesis}

\maketitle

\begin{abstract}
We use subsequence and moving average ergodic
theorems applied to Boole's transformation and its
variants and their invariant measures on the real
line to give new characterisations of the
{\color{black} Lindel\"of} Hypothesis and the
Riemann {\color{black} Hypothesis}.  These ideas are then used
to study the value distribution of  Dirichlet $L$
{\color{black} -functions}, and the zeta functions of  Dedekind,
Hurwitz and Riemann  and their  derivatives.  This
builds on earlier work of R. L. Adler and B. Weiss,
M. Lifshits
and M. Weber, J. Steuding,
J. Lee and A. I. {\color{black}Suriajaya} using
Birkhoff's ergodic theorem  and probability theory.
\end{abstract}

\vspace{0.7cm}

\noindent
Keywords:
\keywords{Ergodic Averages, Boole's Transformation,
Dedekind Zeta Function, \textcolor{black}{Dirichlet $L$-function,
$L$-Series},
Hurwitz Zeta Function, Riemann Zeta Function,
The {\color{black} Lindel\"of} Hypothesis,
The Riemann Hypothesis.}
\vspace{0.5cm}

\noindent
2020 Mathematics Subject Classification:
11M06, 28D05, 37A44, 11M26, 11M35, 30D35.

\tableofcontents

\numberwithin{equation}{section} \numberwithin{figure}{section}
\newpage
\section{Introduction}
\label{S1}

Equivalent statements
of the Riemann {\color{black}Hypothesis}  are well-known
and can be found e.g. in
Titchmarsh \cite{titchmarsh} or more recently
e.g. in Mazur and Stein \cite{mazurstein}.
The Lindel\"of Hypothesis asserts that for every
$\epsilon > 0$ we have
\begin{equation}
\label{LH}
|\zeta(1/2 + it)| = O(|t|^{\epsilon})
\qquad \quad
\mbox{as}~ |t| \to \infty,
\end{equation}
where $\zeta (z)$ is the Riemann zeta funtion,
and Landau's notation ``$O$" means
that there exists a neighbourhood $\mathcal{V}$
of $\infty$ and a constant $c \geq 0$ such that:
$|\zeta(1/2 + it)| \leq c |t|^{\epsilon}$
for all $t \in \mathcal{V}$.

Before stating the main Theorems
\ref{theorem1}--\ref{theorem1.5},

and
Theorem \ref{theorem3} in the context of
moving average ergodic theorems,
let us make precise the context in probability theory.

Let $(X_i)_{i\geq 1}$ be a sequence of
independent Cauchy random variables,
with characteristic function
$\phi (t)=e^{|t|}$ and consider
the partial sums
$S_n= X_1+ \ldots + X_n$ $(n=1,2, \ldots )$.

M. Lifshits and M. Weber
\cite{lifshitsweber3}

studied the
value distribution of the Riemann zeta function
$\zeta ( s)$ sampled along the  random walk
$(S_n)_{n\geq 1}$ showing, for $b> 2$, that
\begin{equation}
\label{limLW2}
\lim _{N\to \infty}{1\over N} \sum _{n=1}^N
\zeta \left  ({1\over 2} +iS_n \right )
= 1+ o\left ({(\log N)^b \over N^{1\over 2}}
\right ).
\end{equation}
They also showed that
\begin{equation}
\label{supIneqLW2}
\left \| {\sup_{n\geq 1}}{|\sum _{q=1}^n
\zeta({1\over 2}+iS_q)-n|\over n^{1\over 2}
( \log n)^b}\right \| _2 < \infty .
\end{equation}
Here of course $f(x) =o(g(x))$ means
$\lim _{x\to \infty}{f(x) \over g(x)} = 0$.
This result was extended to $L${\color{black} -functions} and Hurwitz
zeta functions by T. Srichan
\cite{srichan}.

In \textcolor{black}{\cite{steuding2}}

J. Steuding replaced
$(S_n)_{n\geq 1}$ in
\cite{lifshitsweber3}

by
$(T^nx)_{n\geq 1}$ for almost all $x$ on
$\mathbb{ R}$ with respect the Lebesgue measure, for
the Boolean dynamical system by $Tx:= x-{1\over x}$.
This result has its roots in the observation, due
to G. Boole
\cite{boole},

subsequently developed by
J.W.L. Glashier
\cite{glashier}
\cite{glashier2},

that if $f$
is integrable on the real numbers, then
$$
\int _{-\infty}^{\infty} f(x)dx =
\int _{-\infty}^{\infty} f\Bigl(x-{1\over x}\Bigr) dx.
$$
See
\cite{adlerweiss}

for a proof of the ergodicity of
this dynamical system.
In Theorem 3.1 in
\cite{leeetal},

the maps
\begin{equation}
\label{Talphabeta}
T_{\alpha , \beta} (x) \ = \

\left\{
\begin{array}{cc}
{ \alpha \over 2} \left (
{x+ \beta \over \alpha } - {\alpha \over x- \beta }
\right )
& \mbox{\quad if}~ x \ \not= \beta, \\%;\cr
\beta
& \mbox{\quad if}~ x\ = \beta,
\end{array}
\right.

\end{equation}
for $\alpha > 0$ and real $\beta$,
are shown to be measure preserving and ergodic with
respect to the probability measure
\begin{equation}
\label{mualphabeta}
\mu _{\alpha , \beta } (A)
= {\alpha \over \pi}
\int _A {dt \over \alpha ^2 + (t-\beta )^2},
\end{equation}
for any Lebesgue measureable  subset $A$ of the
real numbers.

As noted in
\cite{leeetal},

if $\lambda$ denotes
Lebesgue measure then
$\mu _{\alpha , \beta }(A) \leq {1\over \alpha \pi}
\lambda (A)$ for all $A \in \cal{B}$,
where $\mathcal{B}$ denotes the Lebesgue
$\sigma$-algebra.
Also if
$\phi _{\alpha , \beta} (x) = \alpha x + \beta$,
with $T= T_{1,0}$ and $\mu = \mu _{1,0}$
we have
\begin{equation}
\label{TalphabetaT}
T_{\alpha , \beta} = \phi  _{\alpha , \beta} \circ
T \circ \phi ^{-1} _{\alpha , \beta},
\end{equation}
which implies the $\mu _{\alpha , \beta }$-measure
preservation and ergodicity of
$T_{\alpha , \beta}$. {\color{black} The referee of this paper has brought to the authors' attention
the paper
\cite{maugmaisrichan},

which considers a family of
rational maps more general than that
in
\cite{leeetal},

which we consider in this paper.
The suggestion is that our methods in this paper will
adapt to the more general framework in
\cite{maugmaisrichan}

--
at least to a degree.  We will
not however explore this possibility in this paper,
as it would be too much of a digression.}

\bigskip

We consider a number of general definitions.

We say $(k_n)_{n\geq 1} \in \mathbb{ N}$ is
{\it $L^p$-good
universal} if for each dynamical system
$(X,\beta , \mu , T)$ and  for each
$g \in L^p(X, \beta , \mu )$  the limit
\begin{equation}
\label{Lp_gooduniversal}
\ell _{T,g}(x)=\lim _{N\to \infty}
{1 \over N}\sum _{n=0}^{N-1}g(T^{k_n}x)
\end{equation}
exists $\mu$ almost everywhere.

Also throughout the rest of this paper $\cal{B}$
will be the Lebesgue $\sigma$- algebra on
$\mathbb{ R}$ and when a $\sigma$-algebra on
$\mathbb{R}$ is not explicitly mentioned it is in
fact $\cal{B}$.

We say that
a  sequence
$ x_1, \ldots , x_N, \ldots $ is
{\it uniformly
distributed modulo one} if
\begin{equation}
\label{unifdistrmod1}
 \lim_{N\to \infty} {1 \over N} \# \{
 1 \leq n \leq N : \{ x_n \} \in I \} =|I|
\end{equation}
for every interval $I \subseteq [0,1)$. For a real
number $y$ we have used $\{ y \}$ to denote its
fractional part and let $[y]= y-\{ y \}$ denote its
integer part.
We say a  sequence of integers is
{\it uniformly distributed on $\mathbb{ Z}$}
if it is
uniformly distributed among the residue classes
modulo $m$, for each natural number $m>1$.
We say a sequence of natural numbers
$(k_n)_{n\geq 1}$ is {\it Hartman uniformly distributed
(on $\mathbb{ Z}$)}
if it is uniformly distributed on
$\mathbb{ Z}$, and for each irrational number
$\alpha$, the sequence
$(\{ k_n \alpha \})_{n\geq 1}$ is uniformly
distributed modulo one .
This condition coincides
with $(k_n)_{n\geq 1}$ being
uniformly distributed
on the Bohr compactification of $\mathbb{ Z}$.
Some basics on Hartman uniform distribution
can be found in the book of Kuipers and
Niederreiter \cite{kuipersniederreiter}.
Note that if $(k_n)_{n\geq 1}$ is
Hartman uniformly
distributed on $\mathbb{ Z}$,
and if for $z$ with
$|z|=1$ we set
\begin{equation}
\label{FNz_definition}
\mbox{} \qquad
\qquad \qquad
F(N, z ) := {1\over N} \sum _{n=0}^{N-1} z^{k_n}

\qquad \qquad \qquad(N=1,2,\cdots)
\end{equation}
then
\begin{equation}
\label{FNz_limite}
\lim _{N\to \infty} F(N,z) =

\left\{
\begin{array}{cc}
1
&  \mbox{{\rm if}} \ z =1, \\%\cr
0 &
\ {\rm otherwise.}
\end{array}
\right.
\end{equation}

\medskip

For a $c \in \mathbb{ R}$ we use $\mathbb{ H}_c$ to
denote the half plane
$\{ z \in \mathbb{ C} : \Re (z) > c \}$ and  use
$\mathbb{ L}_c$  to denote
the line $\{ z \in \mathbb{ C} : \Re (z) = c \}$.
We establish the following theorem.

\begin{theorem}
\label{theorem1}
Let $f$ be a meromorphic  function on
$\mathbb{ H}_c$ satisfying the following conditions:

(1)  there exists {\color{black}a} $K>0$ and a $c'>c$
such that for any $t\in \mathbb{ R}$, we have
\begin{equation}
\label{majoration_f}
|f(  \{ \sigma + it | \sigma > c' \} ) | \leq K;
\end{equation}

(2)  there  exists a non-increasing
$\nu :(c,  \infty ) \to \mathbb{ R}$ such that if $\sigma$ is sufficiently near $c$ then $\nu (\sigma ) \leq
1+c- \sigma$
and that for any $\epsilon > 0$,
we have $f(\sigma + it ) \ll _{f, \epsilon}
|t|^{\nu (\sigma )+\epsilon}$ as
$|t| \to \infty $;

(3) $f$ has  at most one pole of
order $m$
in $\mathbb{ H}_c$ at $s_0= \sigma_0 +it_0$,
that is, we can write its Laurent expansion
near $s = s_0$ as
$$
{a_{-m} \over (s-s_0)^m} + {a_{-(m-1)} \over (s-s_0)^{(m-1)}} + \ldots   {a_{-1} \over (s-s_0)^{-1 }}
+ a _0 + \sum _{n=1}^{\infty} a_n(s-s_0)^n
$$
for $m \geq 0$ where we set $m=0$ if
$f$ has no pole in $\mathbb{ H}_c$.

Then if
$(k_n)_{n\geq 1}$ is $L^p$-good universal and
Hartman uniformly distributed, for any
$s \in \mathbb{ H}_{c} \backslash \mathbb{ L}_{\sigma_0}$,
we have
\begin{equation}
\label{lim_Lpgood_Hartman}
\lim _{N \to \infty} {1\over N}
\sum _{n=0}^{N-1} f(s+iT^{k_n}_{\alpha , \beta }
( x))
= {\alpha \over \pi } \int _{\mathbb{ R}}
{f(s+ i \tau ) \over \alpha ^2 +
(\tau - \beta )^2}  d\tau ,
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

In the case $k_n= n \ (n=1,2, \ldots )$

Theorem \ref{theorem1}
appears in
\cite{leeetal},

where it is
shown,
using (1), (2) and (3)
and contour integration,  for
$$
l_{\alpha , \beta }(s) :=  {\alpha \over \pi}
\int _{\mathbb{ R}} {f(s+ i \tau ) \over \alpha ^2 +
(\tau - \beta )^2}  d\tau ,
$$
that if $f$ has a pole at $s_0
=
\sigma_0 + i t_0$,
\begin{equation}
\label{l_alphabeta_classicalzeta}
l_{\alpha , \beta} (s)
=
\left\{
\begin{array}{cc}

{\color{black} f(s+ \alpha +i \beta)} +
\widetilde{B_m}(s_o) ,
& \mbox{\quad if}~  c < \Re (s) < \sigma _o ,
~ s \not=
s_o- \alpha - i \beta ,\\%\cr
\displaystyle
\sum _{n=0}^m{a_{-n} \over (-2\alpha )^n}  ,
& \mbox{\quad if}~  c < \Re (s) < \sigma _o ,
~ s = s_o- \alpha - i \beta ,\\% \cr
f(s+ \alpha + i \beta ),
&  \mbox{\quad if}~ \Re (s) > \sigma _o ,
\end{array}
\right.
\end{equation}
or
$l_{\alpha, \beta}(s) =
f(s-\alpha + i \beta)$
if $s \in \mathbb{H}_c$ and
$f$ has no pole.
Here $\Re(s)$ denotes the real part of $s$
and
$$\widetilde{B_m}(s_0)
=
\sum  _{n=1}^m
\Bigl\{
{{\color{black}a_{-n}} \over i^n(\beta + i\alpha  -i(s-s_0) )^n }
-{ {\color{black}a_{-n}} \over i^n(\beta - i\alpha  -i(s-s_0) )^n }
\Bigr\}.$$
Moreover, when $m=1$,
we can extend Theorem \ref{theorem1}
to the line ${\mathbb{L}}_{\sigma_0}$
by setting
$$
l_{\alpha , \beta} (\sigma _0 + it)
=
f(\sigma _0 +\alpha  +i(t+ \beta ) )
- { a_{-1} \alpha \over \alpha ^2 +(t_0-t-\beta )^2 }.
$$
Applications will be given in the next section.

As noted earlier Steuding
\textcolor{black}{\cite{steuding2}}

showed that if $\zeta$ denotes the Riemann zeta function

\begin{equation}
\label{limit_Steuding_zeta}
\lim _{N\to \infty} {1 \over N}\sum _{n=0}^{N-1}
\zeta (s+ iT^{n}x)
= {1\over \pi } \int _{\mathbb{ R}}
{\zeta (s+i \tau ) \over 1+ \tau ^2} d\tau
\end{equation}
exists $\mu$ almost everywhere
(for $x$).  We can measure the
stability of these averages another way.
Henceforth  $C$, possibly with subscripts,
will denote a positive constant,
though non-necessarily the same on each occasion.

\begin{theorem}
\label{theorem4}
Suppose $s=\sigma + i t$
with ${1\over 2} <  \sigma < 1$.  Let

$$
Y_N (\sigma ) = Y_N (\sigma , x )
:= {1 \over N}\sum _{n=0}^{N-1}\zeta (s+ iT^{n}x).
\eqno (N=1,2, \ldots )
$$
Suppose $(N_k)_{k \geq 1}$ is a strictly
increasing sequence
of positive integers.  Then there exists an absolute $C>0$
such that
\begin{equation}
\label{zeta_supaverages_upperbound2}
\sum _{k=1}^{\infty} \|  \sup _{N_k \leq N < N_{k+1}}|Y_N(s ) - Y_{N_k}(s )| \  \|_2^2 \leq
{ C \over \pi \sigma}
\Bigl| {\zeta (2\sigma ) \over 2}+ {\zeta ( {2\sigma -1}) \over 2\sigma -1}
\Bigr|.
\end{equation}
\end{theorem}

There are other ergodic theoretic measures of
the stability of the Riemann zeta function.
In this direction we
consider the frequencies  associated to the
sequences
$$
\Bigl(
{\zeta (s+ iT^{n}x)  \over n}
\Bigr)_{n \geq 1}.
$$

Evidently
$$
\lim _{n\to \infty}{\zeta (s+ iT^{n}x)  \over n} = 0,
$$
for almost all $x$.  It is possible to make this more precise as follows.
Suppose $0< p < \infty $.
For a sequence  of real numbers
$\underline{x}=\{ x_n :  n\geq 1 \}$
set
\begin{equation}
\label{underline_x}
\|\underline {x} \| _{p, \infty}
:= \bigl(
\sup _{t>0}\, \{t^p \, \# \{n: |x_n|>t \}\}
\bigr)^{1\over p}.
\end{equation}

\begin{theorem}
\label{theorem5}
(i) Suppose  ${1\over 2} <  \sigma < 1$.  Then
\begin{equation}
\label{limzeta_average}
\lim _{m\to \infty}{\# \{ n :{ |\zeta (\sigma + iT^nx )| \over n} \geq {1\over m } \} \over m } = \zeta ( \sigma + 1) - {2 \over \sigma (2- \sigma )},
\end{equation}
for almost all $x \in \mathbb{ R}$. and in $L^1( \mu )$ norm.

(ii) Further
\begin{equation}
\label{zeta_average_upperbound1}
\left \| \left \| \left \{ { |\zeta (\sigma + iT^nx )| \over n} : n \geq 1  \right \} \right \| _{1, \infty }  \right \| _1< \infty .
\end{equation}
\end{theorem}

Moreover,
\begin{theorem}
\label{theorem1.4}

Suppose $s=\sigma + it $ with
${1\over 2} <  \sigma < 1$.
Suppose $\kappa = (k_n)_{n\geq 1}$
is  $L^p$- good universal for $p > 2$
and let
\begin{equation}
\label{}
R_N(s, x , \kappa )
:= {1\over N} \sum _{n=0}^{N-1}\zeta (s+ iT^{k_n}x).

\qquad \qquad (N=1,2, \ldots ).
\end{equation}
Then

\noindent
(i) there exists a
constant $C>0$
such that
\begin{equation}
\label{}
\sum _{N=1}^{\infty}
\bigl\|  R_{N+1}(s,.,\kappa ) - R_{N}(s,.,\kappa ) \
\bigr\|_2^2
\leq
{ C \over \pi \sigma} \
\Bigl| {\zeta (2\sigma ) \over 2}+
{\zeta ( {2\sigma -1}) \over 2\sigma -1}
\Bigr| ,
\end{equation}

\noindent
(ii) there exists $C>0$
\begin{equation}
\label{}
\mu _{\alpha , \beta} \Bigl(
\Bigl\{ x : \sum _{N=1}^{\infty}
|  R_{N+1}(s,x,\kappa ) - R_{N}(s,x,\kappa ) |^2
\geq \lambda
\Bigr\}
\Bigr)
\leq
{C \over \lambda ^2}
\Bigl| {\zeta (2\sigma ) \over 2}
+ {\zeta ( {2\sigma -1}) \over 2\sigma -1}
\Bigr|.
\end{equation}
\end{theorem}

We will  provide additional information
about various special families of sequences
in {\color{black}S}ection {\color{black}\ref{S4}} and {\color{black}S}ection \ref{S9}.
On the other hand if we consider
the $L^1$-norm we have the
following theorem in the opposite direction.

\begin{theorem}
\label{theorem1.5}

Suppose $\kappa = (k_n)_{n\geq 1}$ is Hartman uniformly distributed and $L^p$- good universal for fixed
$p  \geq \ 1$ and $\sigma \in ({1\over 2}, 1)$.  Then
\begin{equation}
\label{equation1.5}
\sum _{N\geq 1}
\bigl|R_{N+1}(\sigma, x, \kappa )
- R_{N}(\sigma, x , \kappa )
\bigr|
 \ = \ + \infty ,
\end{equation}
almost everywhere in $x$ with respect to
Lebesgue measure.
\end{theorem}

It would be interesting to get good bounds almost everywhere in $x$ for
$$
\sum _{N=1} ^J
\bigl|R_{N+1}( \sigma , x , \kappa )-
R_{N}( \sigma , x , \kappa )
\bigr|,
\qquad \qquad (J=1,2 \ldots )
$$
even for specific cases of  $\kappa$.

\bigskip

The paper is organized as follows: in {\color{black}S}ection
\ref{S2} new characterisations of the
(extended)
Lindel\"of Hypothesis are obtained for the Riemann zeta function, and
for Dirichlet $L${\color{black} -functions},
Dedekind zeta functions of number fields,
Hurwitz zeta functions,
as applications of Theorem
\ref{theorem1}. The sequences  {\color{black} here are}
assumed to be Hartman uniformly distributed and
$L^p$-good universal. Replacing this assumption by
being Stoltz leads to similar limit
theorems, which are reported
in {\color{black}S}ection \ref{S8}.
The role played by sublinear sequences
having controlled growth is investigated
in {\color{black}S}ection \ref{S9} for the Riemann  zeta function.
Section \ref{S7} contains some examples of
$L^p$-good universal sequences

and Hartman uniformly distributed sequences.
Comparison between the dynamical and probabilistic
models of Weber  \cite{weber}, resp. Srichan \cite{srichan}, is done
in {\color{black}S}ection \ref{S6}.

\section{Applications of Theorem \ref{theorem1}}
\label{S2}

From now,
we assume that
$(k_n)_{n\geq 1}$ satisfies the  hypothesis
in Theorem \ref{theorem1}.

In the sequel, for a function $f$
denote $f^{(0)} = f$ and $f^{(k)}$
the $k$-th derivative of $f$ and set
$$
P_k(s) := {(-1)^kk!\over i^{k+1}}\left ({1\over (\beta +i\alpha -i(s-1))^{k+1}} {\color{black}-} {1\over (\beta - i\alpha -i(s-1))^{k+1}} \right )
$$
for any non-negative integer $k$.

\subsection{The Riemann zeta function}
\label{S2.1}

\begin{theorem}
\label{theorem9}

Suppose $(k_n)_{n\geq 1}$ is
$L^p$-good universal for some $p\in [1, \infty ]$
and Hartman uniformly distributed.
Then for any $k \geq 0$ and

$s \in \mathbb{ H}_{-{1 \over 2}} \backslash \mathbb{ L}_1$
we have
\begin{equation}
\label{zetak_Lpgoodunivhartman}
\lim _{N \to \infty} {1\over N} \sum _{n=0}^{N-1} \zeta ^{(k)} (s+iT^{k_n}_{\alpha , \beta }( x))
= {\alpha \over \pi } \int _{\mathbb{ R}}
{\zeta ^{(k)} (s+ i \tau ) \over \alpha ^2 +(\tau - \beta )^2}  d\tau
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

Denote the right hand side of this limit by {\color{black}$l^{(k)}_{\alpha , \beta }$.  Then}
\begin{equation}
\label{lk_alphabeta_zeta}
l^{(k)}_{\alpha , \beta }
=
\left\{
\begin{array}{cc}
\zeta ^{(k)}(s+ \alpha +i \beta) + P_k(s) ,
& \mbox{if}~  -{1\over 2} < \Re (s) < 1 ,
~ s \not= 1- \alpha - i \beta ,\\%\cr
(-1)^k\gamma _k- {{k!} \over (2\alpha )^{k+1}}  ,
& \mbox{if}~  -{1\over 2} < \Re (s) < 1 , ~
s = 1- \alpha - i \beta , \\%\cr
\zeta ^{(k)} (s+ \alpha + i \beta ),
&  \mbox{if}~ \Re (s) > 1,
\end{array}
\right.
\end{equation}
where
\begin{equation}
\label{gammak_alphabeta_zeta}
\gamma _k =
\lim _{N\to \infty}
\left (  \sum _{n=1}^N {\log ^k n \over n} -
{\log ^{k+1} N \over k+1} \right ).
\end{equation}
In the case $k=0$
we can extend the result
to the line $\mathbb{ L}_1$ by setting
\begin{equation}
\label{lzero_alphabeta_zeta}
l^{(0)}_{\alpha , \beta }
=
l^{(0)}_{\alpha , \beta } (1+it) =
\zeta ^{(0)} (1+\alpha +i(t+ \beta )) - {\alpha \over \alpha ^2 +(t^2+\beta ^2)}.
\end{equation}

If we set $k_n=n \ (n=1,2, \ldots )$
{\color{black}
Theorem} \ref{theorem9}
appears in \cite{leeetal}

and this is
in the case $\alpha =1, \beta = 0$
and $k=1$
in
\textcolor{black}{\cite{steuding2}}.

\textcolor{black}
{We also mention here that the case
$k_n = n \,\,(n = 1,2, \ldots )$
and $T=T_{{1,0}}$ of
Theorems \ref{theorem11}

and
\ref{theorem12}

to follow,
appeared in
\textcolor{black}{\cite{steuding2}},

of Theorem
\ref{theorem12}

appeared first in
\cite{balazardsaiasyor}

and of Theorems
\ref{theorem14}

and
\ref{theorem15}

appeared in
\cite{leeetal}.}

\begin{proof}
Note that for
$k\geq 0$, we know that $\zeta ^{(k)}$ is
meromorphic and has an absolutely convergent
Dirichlet {\color{black} series} for $\Re (s) > 1$.
Thus the
condition (1) of Theorem \ref{theorem1}
is satisfied for $c' > 1$.
This is because the Laurent expansion of
$\zeta$ near the pole at $s=1$ is known
\cite{briggs}

and then that the Laurent expansion of
$\zeta ^{(k)}$
has the form
$$
\zeta ^{(k)}(s) =
{ (-1)^k \, k! \over (s-1)^{k+1}} +
(-1)^k \gamma _k +
\sum _{n=k}^{\infty }{(-1)^{n+1} \gamma _n \over (n-k+1)!} (s-1)^{n-k+1}.
$$
Thus if $k\geq 0$ the function
$\zeta ^{(k)}$ has a pole of order $k+1$ at $s=1$.
We can in addition show  (Titchmarsh
\cite{titchmarsh},

pp. 95--96)
that given $\epsilon > 0$ we have
$$
\zeta ^{(k)}(\sigma + it)  \ll  |t|^{\mu (\sigma ) + \epsilon} ,
$$
where

$$
\mu  (\sigma )
=
\left\{
\begin{array}{cc}

0 ,
& \mbox{if}~  \sigma > 1 ,\\%\cr
{(1-\sigma ) \over 2} ,
& \quad \quad \mbox{if}~  0 \leq  \sigma \leq 1 , \\%\cr
{1\over 2}- \sigma,
&  \mbox{if}~ \sigma <0,
\end{array}
\right.
$$
if $k\geq 0$.
Therefore Theorem \ref{theorem9} follows from
Theorem \ref{theorem1}
with $c= -{1\over 2}$, $s_0=1$ and
$m=k+1$ to $\zeta ^{(k)}(s)$ as required.

\end{proof}

We now state a  formulation of the
{\color{black} Lindel\"of} {\color{black} Hypothesis} in terms of
$\zeta ^{(k)}$ from \cite{leeetal}.

\begin{lemma}
\label{lemma10}

The {\color{black} Lindel\"of} {\color{black} Hypothesis}
\eqref{LH}

is equivalent, {\color{black}for every $\epsilon > 0$, to}
\begin{equation}
\label{LH_k}
{\color{black} \left |
\zeta ^{(k)} ({1\over 2}+it)
\right | }
\ll |t|^{\epsilon}
\end{equation}
as $|t|$ tends to $\infty$, for any $k \geq 0$.
\end{lemma}

 We now give a new characterization of the {\color{black} Lindel\"of} {\color{black} Hypothesis} for the Riemann zeta function, which generalizes the classical one. The classical case $k_n = n \,
 (n= 0, 1, 2, \ldots)$
 appears in \cite{leeetal}.

\begin{theorem}
\label{theorem11}

Suppose
$(k_n)_{n\geq 1}$ is $L^p$-good universal
for some $p\in [1, \infty ]$ and
Hartman uniformly distributed.
Suppose $k$ is any non-negative integer.
Then the statement, for any natural number
$l$,
\begin{equation}
\label{limitzetak_Talphabeta_iterates}
\lim _{N \to \infty} {1\over N} \sum _{n=0}^{N-1} {\color{black} \left | \zeta ^{(k)} \left  ({1\over 2}+iT^{k_n}_{\alpha , \beta }( x) \right ) \right  |}^{\color{black}2l}
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{|\zeta ^{(k)} ({\color{black}{1\over 2}}+ i \tau )|^{\color{black}2l}\over \alpha ^2 +(\tau - \beta )^2}  d\tau
\end{equation}
for $\mu _{\alpha , \beta }$ -almost all $x$
in $\mathbb{ R}$, is equivalent
to the {\color{black} Lindel\"of} {\color{black} Hypothesis}.
\end{theorem}

\begin{proof}
Via Lemma \ref{lemma10}
the {\color{black} Lindel\"of}
{\color{black} Hypothesis} implies that given $\epsilon > 0$
we have

$$
\left |\zeta ^{(k)}{\color{black}\left ({1\over 2}+.\right )} \right |^{\color{black}2m} \in L^p (\mathbb{ R}, {\cal B},  \mu _{\alpha , \beta })
$$
for each pair of natural numbers $k,m$.  {\color{black}Here ${.}$ represents the variable $t \in \mathbb{ R}$.}
Theorem \ref{theorem1}
and the ergodicity of $T_{\alpha , \beta }$
implies that

$$
\lim _{N \to \infty} {1\over N} \sum _{n=0}^{N-1}{\color{black} \left | \zeta ^{(k)}
 \left ( {1\over 2}+iT^{k_n}_{\alpha , \beta }( x) \right ) \right |^{2l}}
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{|\zeta ^{(k)} ({\color{black}{1\over 2}}+ i \tau )|^{\color{black}2l} \over \alpha ^2 +(\tau - \beta )^2}  d\tau.
$$

{\color{black} We now prove the converse.  First}

$$
{\color{black} \zeta ^{(k)} (s) = (-1)^{k-1} \int _1^{\infty}
{[x] -x +{1\over 2} \over x^{s+1}}(\log x)^{k-1}
(-s\log x +k)dx + {(-1)^kk! \over (s-1)^{k+1}}.}
$$

Thus there exists $C_k>0${\color{black}, dependent only on
$k, \alpha$ and $\beta$,} such that for $|t|\geq 1$ we have
$$
|\zeta ^{(k)} ({1\over 2} + it) | <C_k|t|.
$$
Also evidently there exists $c_{\alpha , \beta } > 0$ such that if $|
\tau | \geq 1$
$$
{1\over  \alpha ^2 + (\tau - \beta )^2} \geq c_{\alpha , \beta } {1\over 1+ \tau ^2}.
$$
{\color{black}Assuming that the Lindel\"of Hypothesis is false, implies there exists}
$\eta >0$ and $\tau _m \to \infty$  and
$C^1_{\alpha , \beta }$ such that
$$
{\color{black} \left| \zeta ^{(m)}({1\over 2} + it)\right|} >
C^1_{\alpha , \beta } \tau _m^{\eta}.
$$
Now{\color{black}, $|\zeta ^{(k)} ({1\over 2} + it) | <C_k|t|$}
for any $|t| \geq 1$ with $C_k>0$, and
$$
{\color{black} \left|\zeta ^{(k)} ({1\over 2}+i\tau ) - \zeta ({1\over 2}+i\tau _m ) \right|} = \left | \int _{\tau _m}^{\tau}
|\zeta ^{(k+1)} ({1\over 2}+it )dt \right | < C^2_{\alpha , \beta }| \tau - \tau _m|  \tau.
$$
So $|\zeta ({1\over 2}+i\tau )| \geq {1\over 2}  C^1_{\alpha , \beta } \tau _m^{\eta}$ for $\tau$ with $| \tau  - \tau _m| \leq \tau _m ^{-1}$ with $m$ large enough. {\color{black}  Let $L:= {2\over 3}\tau _m$;
then the interval $I := (\tau _m - \tau _m ^{-1} , \tau _m + \tau _m  ^{-1} )$  contains the interval  $(L, 2L)$ for large $m$.}  Hence
$$
\int _L^{2L}
{\color{black} \Bigl|
{\zeta ^{(k)}}
\left ( {1\over 2} + i \tau \right )}
\Bigr|^{2 l} \left ({1\over 2} {d\tau \over 1+\tau ^2} \right )
\geq
\left ( {C_1 \over 2} \right )^{2 l} \int _I \tau _m ^{2 l \eta -2}d\tau
=
2. \left ( {C_1 \over 2} \right )^{2 l} . \tau _m ^{2 l \eta -3},
$$
which is  $\gg T^{2 l \eta -3}$, and this  is
impossible as $l \to \infty$.
So our theorem is proved.

\end{proof}

We now specialize to the case $T=T_{1,0}$
and give a condition in terms of ergodic averages
equivalent to the Riemann {\color{black} Hypothesis}.
\textcolor{black}{We denote by $\rho= \b +i\gamma$,  a representative complex zeros of the Riemann zeta function.
See Titschmarsh \cite{titchmarsh}, p. 30 for instance for details.}

\begin{theorem}
\label{theorem12}

Suppose $(k_n)_{n\geq 1}$ is both
Hartman uniformly distributed and $L^p$-good universal for some $p>1$.  Then
for almost all $x$ in $\mathbb{ R}$ with respect to Lebesgue measure we have
\begin{equation}
\label{zeta_Titerates_zeroes}
\lim _{N\to \infty} {1\over N}
\sum _{n=1}^N \log
{\color{black}\left | \zeta \left ({1\over 2} +
{1\over 2}i T^{k_n}x \right )
\right |}
=
 \sum _{\rho: \Re (\rho ) {\color{black}>} {1\over 2}}
\log \left | {\rho \over 1- \rho } \right | .
\end{equation}
\end{theorem}

Evidently, the Riemann {\color{black} Hypothesis} follows
if either side is zero.

To prove Theorem \ref{theorem12},
we need the following lemma due to  M. Balazard, E.
Saias and M. Yor \cite{balazardsaiasyor}.

\begin{lemma}
\label{lemma13}

We have
\begin{equation}
\label{logzeta_balazard_zeroes}
{1\over 2 \pi } \int _{\Re (s)
=
 {1\over 2} } {\log |\zeta (s) | \over |s|^2} |ds| =
\sum _{ \rho : \Re (\rho) {\color{black}>} {1\over 2}}
 \log \left | {\rho \over 1- \rho } \right |  .
\end{equation}
\end{lemma}

\bigskip \bigskip

\begin{proof}
We wish to use
Theorem \ref{theorem1}
to deduce Theorem \ref{theorem12}
using Lemma \ref{lemma13}.
To show Theorem \ref{theorem1}
is relevant we need to show that
$$
\log {\color{black} \left | \zeta\left  ({1\over 2} + i. \right ) \right |} \in L^ p ({\mathbb{ R}} ,
\mu _{1,0}),
$$
{\color{black} (where $.$ denotes $t \in \mathbb{R}$)}, i.e. that
$$
\int _{\Re (s) =
{1\over 2} }\left  ({| \log |\zeta (s))|^ p | \over |s|^2} \right ) |ds| < \infty .
$$
We mentioned earlier that there exists $C>0$ such that if $|t|>1${\color{black},}
$$
{\color{black} \Bigl |\zeta \left ({1\over 2} + i t  \right ) \Bigr |  }
\leq C|t|.
$$
Also notice that
${(\log |\zeta (s) |)^p \over |s|^2}$ is
continuous on an interval on
$\Re (s) = {1\over 2}$ centred on
$s={1\over 2}$.
Away from that interval on $\Re (s) = {1\over 2}$
we use the observation that
$\left |\zeta ({1\over 2} + i t) \right |
\leq C|t|$.
Hence {\color{black}(for $s ={1\over 2} + it$),} given $\delta > 0$ we have
$$
{(\log |\zeta (s) |)^p \over |s|^2}
\ll
{1\over  |s|^{2-\delta}}.

$$
This means
$$
{\color{black} \log \left | \zeta \left  ({1\over 2} + i.  \right ) \right | }\in L^ p (\mathbb{ R} , \mu _{1,0}),
$$
as required.  Using the fact that $T$ preserves the measure $\mu _{1,0}$ and is ergodic with respect to this measure, we have
$$
\lim _{N\to \infty} {1\over N} \sum _{n=1}^N
\log
{\color{black} \left| \zeta \left ({1\over 2} + {1\over 2}i T^{k_n}x \right )
\right | }
= {1\over 2 \pi}  \int _{\Re(s) {\color{black} >}
{1\over 2 }} {\log |\zeta (s) | \over |s|^2}
|ds|.
$$
$\mu _{1,0}$ almost everywhere.
Theorem \ref{theorem12}
now follows from Lemma \ref{lemma13}.

\end{proof}

\subsection{Dirichlet $L${\color{black} -functions}}
\label{S2.2}

\begin{theorem}
\label{theorem14}

Let $L(s, \chi )$ denote the $L${\color{black} -functions}
associated to the character $\chi$.
Suppose $(k_n)_{n\geq 1}$ is
Hartman uniformly distributed and $L^p$-good universal for some $p>1$.
Then, for $k \geq 1$,

(i) if $\chi$ is non-principal,
for $s \in \mathbb{ H}_{-{ 1\over 2}}$

we have
$$
\lim _{N \to \infty} {1\over N}
\sum _{n=0}^{N-1}
L ^{(k)} (s+iT^{k_n}_{\alpha , \beta }( x), \chi ) =
{\alpha \over \pi } \int _{\mathbb{ R}}
{L ^{(k)} (s+ i \tau, \chi  ) \over \alpha ^2 +(\tau - \beta )^2}  d\tau
\hspace{3cm}\mbox{}
$$
\begin{equation}
\label{lseries_chinonprincipal}
\mbox{} \hspace{2cm}
=
  ~{L ^{(k)} (s +\alpha +i\beta  ,  \chi  )}
\qquad \quad \mbox{for almost all $x$ in $\mathbb{ R}$},
\end{equation}
(ii) if $\chi (= \chi_0)$ is principal, for
$s \in \mathbb{ H}_{-{1\over 2}}\backslash
\mathbb{ L}_{1}$ we have
\begin{equation}
\label{lseries_chiprincipal}
\lim _{N \to \infty} {1\over N}
\sum _{n=0}^{N-1}
L ^{(k)} (s+iT^{k_n}_{\alpha , \beta }( x), \chi ) =
{\alpha \over \pi } \int _{\mathbb{ R}}
{L ^{(k)} (s+ i \tau, \chi  ) \over \alpha ^2 +
(\tau - \beta )^2}  d\tau ,
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

Denote this limit, i.e. the
right hand side of

\eqref{lseries_chiprincipal}, by
{\color{black}$l^{(k)}_{\alpha , \beta } (s, \chi_0)$.  Then}
\begin{equation}
\label{lk_alphabeta_dirichlet}
l^{(k)}_{\alpha , \beta } (s, \chi_0)
=
\left\{
\begin{array}{cc}

L ^{(k)}(s+ \alpha +i \beta , \chi _0)
+ \gamma _{-1} (\chi _0) P_k(s) ,
& \mbox{if}~  -{1\over 2} < \Re (s) < 1 , ~
s \not= 1- \alpha - i \beta ,\\%\cr

{\color{black}\gamma _k (\chi _0)} -
{{k!} \gamma_{-1} (\chi _{0})  \over
(2\alpha )^{k+1}}  ,
& \mbox{if}~  -{1\over 2} < \Re (s) < 1 , ~
s = 1- \alpha - i \beta , \\%\cr
 L^{(k)} (s+ \alpha + i \beta , \chi _0 ),
 &  \mbox{if}~ \Re (s) > 1,
 \end{array}
 \right.
\end{equation}
where $\gamma _{-1} (\chi _0),
\gamma _k (\gamma _0)$, are constants
that depend on $\chi _0$.
These are  the coefficients of the Laurent
expansion of $L^{(k)}(s, \chi _0)$ near $s=1$.
If $k=0$ we can extend the result to the line

$\mathbb{ L}_1$
by defining
$$l^{(0)}_{\alpha , \beta }
=
l^{(0)}_{\alpha , \beta }(1+ it, \chi _0)
=
L(1+ \alpha +i(t+\beta ), \chi _0) -
{\alpha \gamma _{-1}(\chi _0) \over
\alpha ^2 +(t^2 + \beta ^2)}.
$$

\begin{proof}
We know that $L^{(k)}{\color{black}(s, \chi )}$ has a Dirichlet {\color{black}series}
expansion for $\Re (s) >1$,
for each non-negative integer $k$.
From this we can show that
$$
\bigl|L^{(k)} (s, \chi )\bigr| \ll _k, _{\epsilon}
|t|^{\mu ( \sigma) + \epsilon }
$$
where
$$
\mu  (\sigma )
=  \left\{
\begin{array}{cc}

0 ,
& \mbox{if}~  \sigma > 1 ,\\%\cr
{(1-\sigma ) \over 2} ,
& \mbox{if}~  0 \leq  \sigma \leq 1 ,\\% \cr
{1\over 2}- \sigma,
&  \mbox{if}~ \sigma <0.
\end{array}
\right.
$$
If $\chi$ is non-principal
then $L^{(k)}(s, \chi )$ is entire
for all $k\geq 0$, so $L^{(k)}(s, \chi )$
satisfies Theorem \ref{theorem1}
for all $s \in \mathbb{ H}_{-{1\over 2}}$.
If $\chi = \chi _0$ is principal,
$L^{(k)}(s, \chi _0 ) \ (k\geq 1)$
has a pole of order $k+1$ at $s=1$.
We can therefore apply
Theorem \ref{theorem1}
with $c= -{1\over 2}$ , $s_0 = 1$ and
$m=k+1$, to $L^{(k)} (s, \chi _0)$ with Laurent
coefficients coming from
\cite{ishibashikanemitsu}.

\end{proof}

\subsection{The Dedekind zeta function of a number field}
\label{S2.3}

We now consider the
Dedekind $\zeta _{\mathbb{ K}}(s)$ function of a
number field $\mathbb{ K}$ of degree
$d_{\mathbb{ K}}$ over the rationals
$\mathbb{ Q}$ which is defined by as follows.
For $s \in \mathbb{C}$ such that
$\Re (s) > 1$, let
$$
\zeta _{K}(s) =
\sum _{I \subset \mathcal{O}_{\mathbb{K}}}
{1\over (N_{\mathbb{K}/\mathbb{ Q}}(I))^s}.
$$
Here $I$ runs over the ideals contained in
the ring of
integers of $K$ denoted ${{\mathcal{O}}_K} $
and $N_{K/{\mathbb{Q}}}(I)$ denotes the
absolute norm of $I$ in $K$, which is
the cardinality of the quotient
${{{\cal O}_K}/ I}$.
In the case $\mathbb{K} = {\mathbb{Q}}$,
the function $\zeta _{\mathbb{K}} (s)$ reduces to
the Riemann zeta function $ \zeta (s)$.
The complex function $\zeta _{\mathbb{K}} (s)$ can
be extended meromorphically to the
entire complex plane with a simple pole
at $s=1$.  See \cite{cohen}  p. 216, for more details.

\begin{theorem}
\label{theorem15}

Suppose $(k_n)_{n\geq 1}$ is
$L^p$-good universal for some
$p\in [1, \infty ]$ and Hartman uniformly
distributed. For any
$k\geq 0$ and
any $s \in \mathbb{ H}_{{1\over 2}-{1\over d_{\mathbb{ K}}} }\backslash \mathbb{ L}_1$
we have
\begin{equation}
\label{limDedekindk_iteratesTalphabeta}
\lim _{N \to \infty} {1\over N} \sum _{n=0}^{N-1} \zeta ^{(k)}_{\mathbb{ K}}
(s+iT^{k_n}_{\alpha , \beta }( x))
=
 {\alpha \over \pi } \int _{\mathbb{ R}}
{\zeta ^{(k)}_{\mathbb{ K}} (s+ i \tau ) \over \alpha ^2 +(\tau - \beta )^2}  d\tau
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

Denote the right hand side of this limit by
$ l^{(k)}_{\mathbb{K}_{\alpha , \beta }}$.
We have
\begin{equation}
\label{lk_alphabeta_dedekind}
l^{(k)}_{\mathbb{ K} _{\alpha , \beta }}(s)
=
\left\{
\begin{array}{cc}

\zeta ^{(k)}_{\mathbb{ K}} (s+ \alpha +i \beta)
+\gamma _{-1}({\mathbb{ K}})  P_k(s) ,
& \mbox{if}~  {1\over 2} -{1\over d_{\mathbb{ K}}}
< \Re (s) < 1 , ~
s \not = 1- \alpha - i \beta ,\\%\cr
k!\gamma _k( \mathbb{ K})- {k! \gamma_{-1}(\mathbb{ K} ) \over {(2 \alpha )^{k+1}}},
& \mbox{if}~  {1\over 2}-{1\over d_{\mathbb{ K}}}
< \Re (s) < 1 , ~
s = 1- \alpha - i \beta ,\\% \cr
\zeta ^{(k)} (s+ \alpha + i \beta ),
&  \mbox{if}~ \Re (s) > 1.
\end{array}
\right.
\end{equation}
Here $\gamma _{-1}( \mathbb{ K} )$ and
$\gamma _{k}( \mathbb{ K} )$  are constants
{\color{black}dependent} only on $\mathbb{ K}$,
which are coefficients of the Laurent expansion
of $\zeta _{\mathbb{ K}}^{(k)}$ near $s=1$.
Also  in the case $k=0$ we can extend
the result to the line
$\mathbb{ L}_1$ by setting
\begin{equation}
\label{lzero_alphabeta_dedekind}

l^{(0)}_{\mathbb{ K}_{\alpha , \beta }} (1+it)
=
 \zeta ^{(0)} (1+\alpha +i(t+ \beta )) -
 {\alpha \gamma _{-1}(\mathbb{ K}) \over \alpha ^2 +(t^2+\beta ^2)}.
\end{equation}

\begin{proof}
We refer to
\cite{steuding}

for a bound
for $L$ on the half line
and to \cite{hashimotoetal}

for the {\color{black}coefficients} of the Laurent
expansion $\zeta _{\mathbb{ K}}(s)$
near the pole $s=1$.
We then proceed {\color{black} as earlier} with
Theorem \ref{theorem9}
{\color{black},}  $c={1\over 2}-{1\over d_{\mathbb{ K}}}$,
$s_0=1$ and $m=k+1$.

\end{proof}

\begin{remark}
Results analogous to the ergodic
characterisation of the {\color{black} Lindel\"of}
{\color{black} Hypothesis} just given,
can be proved
for
$L${\color{black} -functions} {\color{black}associated to primitive}
characters and Dedekind  $\zeta$ function.
In both cases, the statements
of ergodic characterisation are of the type:

there exist constants $k$, $\varepsilon >0 $ such that

$$
 f{\color{black} \left ({1\over 2} + it \right )} \ll _{f,k, \varepsilon  } |t|^{\epsilon},
$$
as $|t| \to \infty$, for suitable {\color{black} $L$} {\color{black} -functions}  $f$.
This inequality when $f$ is an $L$-{\color{black} function}
\textcolor{black}{arising from Dirichlet series}
is called the
{\color{black} Generalized} {\color{black} Lindel\"of} {\color{black} Hypothesis}.
\end{remark}

\subsection{The Hurwitz zeta function}
\label{S2.4}

Recall that the Hurwitz zeta function is
defined for $a>0$ and $s \in {\mathbb{C}}$
with $\Re (s) >1 $ by
$$\zeta (s, a) =
\sum _{n=1}^{\infty} {1\over (n+a)^s}.$$
It is continued meromorphically to the
whole of ${\mathbb{ C}}$ with a single pole at $s=1$.

\begin{theorem}
\label{theorem16}

Suppose $(k_n)_{n\geq 1}$ is
$L^p$-good universal for some $p\in [1, \infty ]$
and Hartman uniformly distributed.
Then for any $s$ such that  $ \Re (s) > -{1\over 2},
s\not= 1$, with  $0 \leq a < 1$
and $k$ a  non-negative integer,  we have
\begin{equation}
\label{limHurwitzk_iteratesTalphabeta}
\lim _{N \to \infty}
{1\over N} \sum _{n=0}^{N-1}
\zeta ^{(k)} (s+iT^{k_n}_{\alpha , \beta}( x), a)
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{\zeta ^{(k)} (s+ i \tau , a ) \over \alpha ^2 +
(\tau - \beta )^2}  d\tau  ,
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

\bigskip

Denote the right hand side of this limit by {\color{black}$l^{(k)}_{\alpha , \beta  }(s,a)$.  Then}
\begin{equation}
\label{lk_alphabeta_hurwitz}
l^{(k)}_{\alpha , \beta  }(s,a)
=
\left\{
\begin{array}{cc}

\zeta ^{(k)}(s+ \alpha +i \beta , a) + P_k(s) ,
& \mbox{if}~  -{1\over 2} < \Re (s) < 1 , ~
s \not= 1- \alpha - i \beta ,\\%\cr
(-1)^k\gamma _k(a) - {{k!} \over (2\alpha )^{k+1}}  ,
& \mbox{if}~  -{1\over 2} < \Re (s) < 1 , ~
s = 1- \alpha - i \beta , \\%\cr
\zeta ^{(k)} (s+ \alpha + i \beta ,a ),
&  \mbox{if}~ \Re (s) > 1,
\end{array}
\right.
\end{equation}
where
\begin{equation}
\label{gammak_alphabeta_hurwitz}
\gamma _k(a) =
\lim _{N\to \infty} \left (
\sum _{n=1}^N {\log ^k (n+a) \over n+a} -
{\log ^{k+1} (N+a) \over k+1} \right ).
\end{equation}
In the case $k=0$
we can extend the result to the
line $\mathbb{ L}_1$ by setting
\begin{equation}
\label{lzero_alphabeta_hurwitz}
l^{(0)}_{\alpha , \beta } (1+it,a)
=
\zeta ^{(0)} (1+\alpha +i(t+ \beta ),a) -
{\alpha \over \alpha ^2 +(t^2+\beta ^2)}.
\end{equation}
If we set $k_n=n \ (n=1,2, \ldots )$
Theorem \ref{theorem9}
appears in \cite{leeetal}

and this is the case
$\alpha =1, \beta = 0$, and $k=1$
in \textcolor{black}{\cite{steuding2}}.

\begin{proof}
Following the proof of Theorem
\ref{theorem9}
choose $c=-{1\over 2}, s_0 =1$ and $m=k+1$.  For the bound on
$|\zeta ^{(k)}(s, a)|$ on the half line
and the
coefficients of the Laurent series {\color{black} of}
$\zeta ^{(k)}(s,a)$ near $s=1$
we
refer to \cite{steuding}.

\end{proof}

\section{Proof of Theorem \ref{theorem1}}
\label{S3}

The proof is in the continuation,
and {\color{black} a generalization},
of a Theorem of \cite{leeetal} in which {\color{black} conditions}
(1), (2) and (3) are used.
We first recall a special case of a Theorem of S. Sawyer \cite{sawyer}:

Suppose for a dynamical system
$(\mathbb{ R} , \mathcal{B}, \mu _{\alpha , \beta }  ,
T _{\alpha , \beta})$ that
$g \in L^p(\mathbb{ R}, \mathcal{B}, \mu _{\alpha , \beta })$ and let
$\|g\|= (\int _X |f|^p
d\mu _{\alpha , \beta })^{1\over p}$.
Set
$$
Mg(x) = \sup _{N\geq 1}
\Bigl|
{1\over N} \sum _{n=1}^Ng(s+iT_{\alpha , \beta } ^{k_n}(x) )
\Bigr|.  \eqno (N=1,2, \dots )
$$
If $(k_n)_{n\geq 1}$ is
$L^p$-good universal for $p > 1$,
then there exists $C > 0$ such that
\begin{equation}
\label{(3.1)}
\bigl\|Mg\bigr\|_p  \leq C \|g\|_p.
\end{equation}

Because $\left  |{1\over N}
\sum _{n=1}^Ng( s+
i T_{\alpha , \beta} ^{k_n}(x) ) \right |
\leq Mg(x)$ $(N =1,2, \ldots)$
and $Mg \in L^p$, the dominated convergence
theorem implies
$$
h(x) = \lim _{N\to \infty} {1\over N}
\sum _{n=1}^Nh( s+ iT^{k_n}_{\alpha , \beta} (x) )
$$
exists in  $L^p$-norm.
Our next order of business is to show
that $h(s+ i T_{\alpha , \beta } (x)) =h(s+ ix)$.
Let $U_{\alpha  , \beta } g(s+ i x) =
g(s+ i T_{\alpha , \beta } ( x))$.
This is a norm preserving operator on $L^p$
as $T_{\alpha , \beta }$ is $\mu _{\alpha , \beta}$
measure preserving.
Also let $U^{-1}_{\alpha , \beta }$ denote the
$L^2$ adjoint of $U_{\alpha , \beta }$.
Recall that we say any sequence
$(c_n)_{n \in \mathbb{ Z}}$ is positive
definite if given a bi-sequence of complex numbers
$(z_n)_{n\in  \mathbb{  Z}}$,
only finitely many of whose terms are
non-zero, we have
$\sum _{n,m \in \mathbb{ Z}}c_{n-m}z_n
{\overline {z_m}} \geq 0$.
Here $\overline z$ is the conjugate of the complex
number $z$.
Let $\left< a,b\right > = \int _{\mathbb{ R}}
a{ \overline b} d\mu _{\alpha , \beta }$
(i.e. the standard inner product on $L^2$).
Notice that
$\bigl(\left< U_{\alpha , \beta }^ng,
g \right> \bigr)_{n\in \mathbb{ Z}}$
is positive definite.
Recall that the Bochner-Herglotz theorem
\cite{katznelson}
says that there is a measure
$\omega _g$ on ${\mathbb{ T}}$,
such that
$$
\left< U_{\alpha , \beta } ^ng, g \right>
=
\int _{\mathbb{ T}}z^n d\omega _g (z). \eqno
(n \in \mathbb{ Z})
$$
This tells us that
$$
\Bigl\|  {1\over N}
\sum _{n=1}^Ng(s+ iT^{k_n+1}_{\alpha , \beta }(x) )
-
{1\over N}
\sum _{n=1}^Ng(s+ iT^{k_n}_{\alpha , \beta }(x) )
\Bigr\|_2
$$
$$
=\int _{\mathbb{ T}}(2-z-z^{-1})
\Bigl| {1\over N}
\sum _{n=1}^Nz^{k_n}
\Bigr| ^2d\omega _g(z)
$$
using the parametrization
$z = e^{2\pi i \theta }$
for $\theta \in [0,1)$, this is
$$
=
4\int _{\mathbb{ T}} \sin ^2\left (
{\theta \over 2 } \right )
\Bigl|
{1\over N} \sum _{n=1}^Nz^{k_n}
\Bigr|^2
d\omega _g (z).
$$
Using the fact that
$\sin {\theta \over 2} = 0$ if
$ \theta =0$ and the fact that
$(k_n)_{n\geq 1}$ is Hartman
uniformly distributed,
by
\eqref{FNz_definition} and \eqref{FNz_limite},
 we see that
$g(s+ iT_{\alpha , \beta }(x)) = g(x)$.
A standard fact from ergodic theory is that if
$T_{\alpha , \beta }$ is ergodic and
$g(s+ iT_{\alpha , \beta } (x)) =g(x)$ for
measurable $g$ then $g(x)$ is constant,
which must be
$\int _{\mathbb{ R}} g d\mu _{\alpha , \beta }$.
This extends to the $L^p$-norm  for all $p>1$.
\par
All we have to do now is to show that the
pointwise limit is
the same as the norm limit,
i.e. that
${\overline g(x)} = g(x) =
\int _{\mathbb{  R}} gd\mu _{\alpha , \beta }$.
We consider the sequence of natural numbers
$(N_t)_{t\geq 1}$ such that
$$
\Bigl|{1\over N_t}
\sum _ {n=1}^{N_t}g(s+ iT_{\alpha , \beta }^{k_n}(x)) -
\int _{\mathbb{ R}}  g(x) d \mu _{\alpha , \beta }
\Bigr| _p \le {1\over t}.
$$
Thus
$$
\sum _{t=1}^{\infty}\int _X
\Bigl|{1\over N_t}
\sum _ {n=1}^{N_t}g(s+ iT_{\alpha , \beta }^{k_n}(x)) -
\int _{\mathbb{ R}}  g(x) d \mu_{\alpha , \beta }
\Bigr|^p d\mu  < \infty .
$$
Fatou's lemma gives
$$
\int _{\mathbb{ R}}
\Bigl( \sum _{t=1}^{\infty}
\Bigl|{1\over N_t}
\sum _ {n=1}^{N_t}g(s+ i T_{\alpha , \beta }^{k_n}(x)) -
\int _{\mathbb{ R}} g(x)  d \mu _{\alpha , \beta }
\Bigr|^p
\Bigr) d\mu  < \infty .
$$
This implies
$$
 \sum _{t=1}^{\infty}
 \Bigl|{1\over N_t}
 \sum _ {n=1}^{N_t}g(s+ i T_{\alpha , \beta }^{k_n}(x)) -
 \int _{\mathbb{ R}}  g(x) d \mu _{\alpha , \beta }
 \Bigr|^p   < \infty .
$$
almost everywhere.
This means

$$
\Bigl|{1\over N_t}
\sum _ {n=1}^{N_t}g (s+ i T_{\alpha , \beta }^{k_n}(x)) -
\int _X g(x) d \mu _{\alpha , \beta }
\Bigr|
= o(1) .
$$
$\mu  _{\alpha , \beta }$ almost everywhere.
As $(k_n)_{n\geq 1}$ is $L^p$-good
universal we must have
$$\lim _{N \to \infty} {1\over N}
\sum _{n=0}^{N-1} g(s+iT^{k_n}_{\alpha , \beta } x) =
\int _{\mathbb{ R}}  g(s+ix) d\mu _{\alpha , \beta }$$
$\mu _{\alpha , \beta }$ almost everywhere as
required to prove Theorem \ref{theorem1}.

\bigskip

\section{Proof of Theorems \ref{theorem4} to \ref{theorem1.5}}
\label{S4}

We begin by recalling the spectral regularization
method of Lifshits and Weber
\cite{lifshitsweber}
\cite{lifshitsweber2},

and
some technical preliminaries.
Assume $(X, \beta , \nu )$ is a measure space with
finite measure $\nu$ and that
$S$ is a $\nu$ measure preserving
transformation of  $(X, \beta , \nu )$.
For $f \in L^1(X, \beta , \nu )$ let
\begin{equation}
\label{A_N_definition}
B_N(f) = {1\over N}\sum _{k=0}^{N-1}f \circ T^k.
\qquad \qquad (N=1,2, \ldots)
\end{equation}

We need the following lemma, due to R. Jones,
R. Kaufman, J. Rosenblatt and M. Wierdl
\cite{jonesetal}.

\begin{lemma}
\label{lemma6}

Suppose $(N_p)_{p\geq }$ is a strictly
increasing sequence of positive integers.
Then there exists an absolute
constant $C>0$ such that
\begin{equation}
\label{equation6}
\sum _{p=1}^{\infty}
\bigl\|
\sup _{N_p \leq N < N_{p+1}}|B_N(f ) - B_{N_p}(f)| \  \bigr\|_2^2
\, \leq
\,C \| f\| _2.
\end{equation}
\end{lemma}

Write $\log_{+}(u)= \max (1, \log (u ))$
for $u\geq 1$.
Let $\omega$ denote
the spectral measure associated
to the element $f$ and define its regularised
measure $\hat {\omega }$
via its Radon-Nykodim
derivative by
$$
{d \hat{\omega } \over dx}(x) =
\int _{-\pi}^{ \pi} Q (\theta , x)
\omega ( d \theta ),
$$
where
$$
Q( \theta , x) \ = \
\left\{
\begin{array}{cc}

|\theta |^{-1} \log_{+}^ 2
(\left |{ \theta \over x } \right | )
& \mbox{if}~ |x| < | \theta |,\\% \cr
\theta^2 |x|^{-3}
& \mbox{if}~ |\theta | \leq |x| \leq \pi .
\end{array}
\right.
$$
The following is a theorem of
Lifshits and Weber
\cite{lifshitsweber}.

\begin{lemma}
\label{lemma7}

Suppose $N_0$ and $N_1$ with $N_0< N_1$  are positive integers.
Then there exists an absolute
constant $C>0$ such that
\begin{equation}
\label{equation4.2}
\bigl\|  \sup _{N_0 \leq N < N_{1}}
|B_N(f ) - B_{N_0}(f )| \
\bigr\|_2^2
~\leq~
\, C \hat {\omega}
\bigl(
[{1\over N_1}{1\over N_0} )
\bigr).
\end{equation}
\end{lemma}

Suppose $0< p < \infty $.
For a sequence  of real numbers
$\underline{x}=\{ x_n :  n\geq 1 \}$
the quantity
$
\|\underline {x} \| _{p, \infty}$
is defined
in \eqref{underline_x}.
Also if $r< p$ we have
$$
\| \underline {x} \| _{p, \infty} \leq \| \underline {x} \| _{p}
\leq
\left ( {p\over p-r} \right )^{1\over p} \| \underline {x} \| _{p, \infty}.
$$

We also consider
$$
N^*_{\underline{x}}=\sup_{m\geq 1} {\# \{ n: {x_n \over n}>{1\over m} \}\over m}.
$$

In our considerations,
$x_n = f(\tau^n(x)) \ (n=1,2, \ldots )$
and \,$\underline{x} =O_f(x)=
(f(\tau ^n(x))_{n\geq 1}$  which is the value of $f$
along the orbit of a point $x \in X$.
Also let $N_f(x) = N^*_{O_f(x)}$.  We have the
following Lemma due to I. Assani \cite{assani}

(see
also \cite{weber} -
we refer to Jamison, Orey and Pruitt
\cite{jamisonoreypruitt}
for more on the random variable case,
and \cite{raoren} for
Birnbaum-Orlicz spaces and the ``$L \log_+ L$" notation).

\begin{lemma}
\label{lemma8}

(i) For any non-negative $f \in L^1( \mu )$ the
function $N_f(x)$ is weak-$(p,p)$
for all $p \in (1, \infty)$.  Further
\begin{equation}
\label{equation4.3i}
\lim _{m\to \infty} {\# \{ n: {f(\tau ^n(x))
\over n}>{1\over m} \}\over m}
=
\int _X f d\mu ,
\end{equation}
$\mu$ almost everywhere and also in
$L^1(\mu )$-norm.

(ii)  For $f \in L \log_+ L$

\begin{equation}
\label{equation4.3ii}
\left \| \left \| \left \{ {f(\tau ^n (x)) \over n}   \right \} \right  \| _{1, \infty} \right  \|_1 < \infty .
\end{equation}
\end{lemma}

Note that, for two positive constants $C_1 , C_2$,
$$
C_1 N_f (x) \leq
\left \| \left \{ {f(\tau ^n (x)) \over n} , n\geq 1\right \} \right  \|_{1, \infty}
\leq
C_2 N_f (x).
$$

We now turn to the proof of
Theorem \ref{theorem4}.

By Lemma \ref{lemma6},
$$
\sum _{k=1}^{\infty}
\bigl\|  \sup _{N_k \leq N < N_{k+1}}
|Y_N(\sigma ) - Y_{N_k}(\sigma )| \  \bigr\|_2^2 \leq
{C\over \pi} \int _{\mathbb{R}}
{|\zeta (\sigma + i \tau )|^2 \over 1+ \tau ^2} d \tau ,
$$
where as before $C$ is universal.
We note that, for
$\sigma \in ( {1\over 2} , 1 )$,
$$
 \int _{\mathbb{ R}}
{|\zeta (\sigma + i t )|^2 \over 1+ t^2} dt =
\int _0^{\infty} {\{x\} \over  x^{2 \sigma +1}}
dx
=
-{1\over \sigma} \left (
{ \zeta (2 \sigma ) \over 2} +
{ \zeta (2 \sigma -1) \over 2 \sigma -1} \right ).
$$
Here  {\color{black} again} $\{ x \} = x- [x]$
denotes the fractional
part of $x$.
The quantity in the bracket on the
right hand side is negative.
Further,
$$
{1\over 2 \pi} \int _{\mathbb{ R}}{ |\zeta ({1\over 2} + it) |^2 \over {1\over 4} +t^2} dt
= \log ( 2 \pi ) - \gamma .
$$
where $\gamma$ is Euler's constant.
See Prop. 7,  Cor. 8 and (1.26) in Coffey
\cite{coffey}.
Thus
\begin{equation}
\label{CsurPi}
 {C\over \pi} \int _{\mathbb{ R}}
{|\zeta (\sigma + i \tau )|^2 \over 1+ \tau ^2} d \tau \leq   { C \over \pi \sigma}
\left | {\zeta (2\sigma ) \over 2}+ {\zeta ( {2\sigma -1}) \over 2\sigma -1} \right |.
\end{equation}
completing the proof of Theorem \ref{theorem4}.
$\square$

We now turn to the proof of
Theorem \ref{theorem5}.
The first assertion is a consequence
of Lemma \ref{lemma8}
(i).  The limit is
$$
{1\over \pi} \int _{\mathbb{ R}}
{|\zeta (\sigma + i t )|^2 \over 1+ \tau ^2} d t .
$$
This integral is a special case of integrals
calculated in \cite{steuding}.

As a consequence of
Theorem 1.1 in
\cite{steuding2},

$$
{1\over \pi} \int _{\mathbb{ R}}
{|\zeta (\sigma + i t )|^2 \over 1+ \tau ^2} d t  = \zeta (\sigma +1) - {2 \over \sigma ( 2 - \sigma )}.
$$

For the second assertion, it follows from
Lemma \ref{lemma8}
(ii).
The function
$\zeta (s) \in L \log_{+} L$

is integrable, with respect to the
Cauchy  measure, because of the estimate
$$
\zeta (\sigma + it ) \ll _{\epsilon} t^{{1-\sigma \over 2}+ \epsilon}.
$$
See  Titchmarsh, \cite{titchmarsh}
section 5.1, for the details of this.
Therefore  Theorem \ref{theorem5}
is proved as required.
$\square$

To prove Theorem \ref{theorem1.4}
we need the following  two lemmas.

Suppose $(X,\beta ,\mu )$ is a measure space and
that $T : X \to X$ is a measure preserving map.
Let $(a_k)_{k= 0}^{\infty}$ be a
sequence of natural numbers and for any
measurable $f$ on $X$ set
\begin{equation}
\label{CNfak_definition}
\mbox{} \qquad \qquad C_Nf(x) \
:=
\ {1\over N}\sum _{k=0}^{N-1}
f(T^{a_k}x),
\qquad \qquad (N=1,2,\cdots)
\end{equation}
that is the ergodic averages
corresponding to the sequence and let
$$
Mf(x) \
:=
\ \sup _{N\geq 1}|C_Nf(x)|.
$$
Further from the data $(X,\beta , \mu , T)$
and $f$ we construct the
ergodic $q$-variation function
$$
V_qf(x) \ = \ (\sum _{N\geq 1}
|C_{N+1}f(x)-C_Nf(x)|^q)^{1\over q}.
\eqno (q \ \geq \ 1)
$$

Our first lemma is Theorem 1
from
\cite{nairweber1}.

\begin{lemma}
\label{lemma4.4}

Suppose for a sequence of natural numbers
$(a_k)_{k=0}^{\infty}$ that for some
$p \ > \ 1$ and
$\widetilde{C_p} \ > \ 0$ {\color{black}depending
only} on $p$ and $(X,\beta , \mu , T)$
we have
\begin{equation}
\label{leqno4.1}
\bigl\|Mf \bigr\|_p \ \leq \
\widetilde{C_p} \|f\|_p.
\end{equation}
Then there exists another constant $D_p \ > \ 0$ {\color{black}depending only} on $p$
and $(X,\beta , \mu , T)$ such that if $q \ > \ 1$ then
\begin{equation}
\label{leqno4.2}
\bigl\|V_qf \bigr\|_p \ \leq \
D_p \|f\|_p.
\end{equation}
Further suppose there is a constant
$\widehat{C} \ > \ 0$ {\color{black}depending only}
on $(X,\beta, \mu , T)$ such that we have
\begin{equation}
\label{leqno4.3}
\mu (\{ x : Mf (x) \ > \ \lambda \}) \ \leq \
{\widehat{C} \over \lambda }
\int _X|f|\, d\mu.
\end{equation}
Then  there is a constant $D \ > \ 0$ {\color{black}depending only} on
$(X,\beta , \mu , T)$ such that if $q \ > \ 1$ then
\begin{equation}
\label{leqno4.4}
\mu (\{ x : V_qf (x) \ > \ \lambda \}) \ \leq \
{D\over \lambda }\int _X|f|\, d\mu.
\end{equation}

\end{lemma}

\begin{proof}
Suppose $(X, {\cal L}, \eta)$ denote a
finite measure space i.e.
$\eta (X) < \infty$.
Now, let $(T_n)_{n\geq 1}$ denote a sequence of
linear transformations of
$L^p(X, {\cal L}, \eta )$ into measurable
functions on $X$, such that each $T_n$ is
continuous in measure, that is, such
that if $||f-f_m||_p $ tends to $0$
as $m$ tends to $\infty$, then
$||T_n f-T_nf_m||_p $  also tends
to $0$ as
$m$ tends to $\infty$.
Set $T^*f(x)= \sup _{n\geq 1}|T_nf(x)|$
for $f \in L^p(X, {\cal L}, \eta )$.
We will say the family
$(T_n)_{n\geq 1}$ commutes with
$w:X \to X$ if  $T^*f(w(x)) \leq  T^*g (x)$
where $g(x) = f(w(x))$ on $X$.  Now suppose
$\cal F$ is a family of ergodic $\eta$
preserving transformations on $X$, closed under
composition.
We will call $(T_n)_{n\geq 1}$ distributive
if it commutes with all the elements of
some family $\cal F$ on $X$.   We have the
following result of S. Sawyer
\cite{sawyer}.

\bigskip
\begin{lemma}
\label{lemma4.5}

Let $(T_n)_{n\geq 1}$ be a distributive
sequence of linear operators on
$L^p(X, {\cal L}, \eta )$, where
each $T_n$ is
continuous in measure and
maps  $L^p(X, {\cal L}, \eta )$ to measurable
functions on $X$.
If $p \in [1,2]$ and if
$T^*f(x) < \infty$ $\eta$ almost everywhere,
then there exists a uniform
constant $C>0$ such that
\begin{equation}
\label{equation4.5}
\eta
\bigl( \{ x : T^*f(x) \geq \lambda  \}
\bigr)
\leq {C\over \lambda ^p} \int _X|f(x)|^pd\eta ,
\end{equation}
for all $\lambda >0$ and
$f \in L^p(X, {\cal L}, \eta )$.
\end{lemma}

\bigskip
The conclusion of Lemma \ref{lemma4.5}
is that if $T^*f(x) < \infty$ $\eta$ almost everywhere, then the operator $T^*$ satisfies a weak$^* (p,p)$.
If there is a $C>0$ such that
$||T^*f||_p \leq C||f||_p$ we say
$T^*$ satisfies a
{\it strong $(p,p)$ inequality}.
It is easy to check that strong $(p,p)$
inequalities imply the corresponding
weak $(p,p)$ inequalities for
the operator $T^*$.
On the other hand it follows
from the Marcinkiewicz interpolation
theorem
\cite{steinweiss}
that  a weak $(p,p)$ inequality
implies a strong $(q,q)$ inequality
if $q>p$.
We now specialize \eqref{A_N_definition}
to the situation

where $f$ is defined on $X=\mathbb{R}$,
${\cal L}$ is the Lebesgue algebra
on $X=\mathbb{ R}$
and $\eta = \mu _{\alpha , \beta }$:
$$
T_N f(x)
:={1\over N}
\sum _{n=1}^N f(T_{\alpha , \beta}^{k_n}(x))
\eqno (n=1,2, \dots )
$$
with $f(x) = \zeta (\sigma + ix)$.
One checks readily that
$(T_N)_{N\geq 1}$

commutes with $(T^n_{\alpha , \beta })_{n\geq 1}$
and is therefore distributive.
In light of
Lemma \ref{lemma4.5},
we see that Theorem
\ref{theorem1.4}
follows by
recalling our estimate
$$
\int _{{\mathbb{ R}}} |\zeta (\sigma + i x) |^2
d\mu _{\alpha , \beta }
~\leq~
{C\over \pi}
\int _{{\mathbb{R}}}
{|\zeta (\sigma + i \tau )|^2 \over 1+ \tau ^2}
d \tau
~\leq~   { C \over \pi \sigma}
\left | {\zeta (2\sigma ) \over 2}+
{\zeta ( {2\sigma -1}) \over 2\sigma -1} \right |
$$
arising in \eqref{CsurPi}
in the Proof of
Theorem \ref{theorem5},
Theorem \ref{theorem1.4}
is proved.
\end{proof}

It is possible to say more about
specific sequences and families of sequences.
Suppose ${\cal N} =(N_k)_{k\geq 1}$
and
$(a_n)_{n\geq 1}$ are
sequences of natural numbers.
Consider the definition of $C_N$
in \eqref{CNfak_definition}.
Let
$$
Sf(x)
= S( {\cal N} ,f) (x)
:=
\ \Bigl( \sum _{k\geq 1}
\bigl|C_{N_{k+1}}f(x)-C_{N_k}f(x)\bigr|^2
\Bigr)^{1\over 2}.
$$
In the first instance, we are interested in
conditions under which there are constants $C>0$
such that
\begin{equation}
\label{leqno4.5}
\bigl\|Sf \bigr\|_2 \leq C \|f\|_2.
\end{equation}

We have the following lemma taken from
\cite{nair2}.

\begin{lemma}
\label{lemma4.6}

Suppose
$1 \ < \ a \ \leq \ {N_{k+1}\over N_k}
\ \leq \ b \ < \ \infty$ and
$a_k \ = \ \phi (n)$,
where $\phi$ is a non-constant
polynomial mapping the natural numbers to
themselves.
Then there is a constant
$C \ > \ 0$ such that
\eqref{leqno4.5}
holds.
\end{lemma}

The following lemmas are taken from
\cite{nairweber1}.

\begin{lemma}
\label{lemma4.7}

Suppose $1 \ < \ a \ \leq \ {N_{k+1}\over N_k}
\ \leq \ b \  < \ \infty$ and $a_k \ =
\ \phi (p_n)$, where $\phi$ is as in
Lemma \ref{lemma4.6}
and $p_n$ is the $n^{th}$ rational prime.
Then there is a constant
$C \ > \ 0$  such that
\eqref{leqno4.5}
holds.
\end{lemma}

Let
$\theta \ = \ (\theta _k )_{k\geq 1}$
be a ${\color{black}{\mathbb Z}}$ valued
sequence of independent identically distributed
random variables with basic probability space
$(\Omega , {\cal B}, P)$.
We assume the $\sigma$-algebra
${\cal B}$ is $P$ complete and that
there exists $\gamma \ > \ 0$ such that
${\color{black}{\mathbb E }}\bigl((\theta _1^+)^{\gamma }
\bigr) \ < \ \infty$.
Here we have used
$\theta _1^+$ to denote $\max (\theta _1,0)$.
Consider a strictly
increasing sequence $(q_k)_{k\geq 1}$ of natural
numbers for which
$S$ is a bounded map from $L^2$ to itself
when $a_k \ = \ q_k$ and
$N_k \ = \ [\rho ^k ]$ for some  $\rho \ > \ 0$,
that is  such that
\begin{equation}
\label{leqno4.6}
\Bigl\|
\bigl(\sum _{k\geq 1}
\bigl|
C_{N_{k+1}}(f) \ -
\ C_{N_k}(f)
\bigr|^2(x)
\bigr)^{1\over 2}
\Bigr\|_2
\ \leq \
C \bigl\|f \bigr\|_2.
\end{equation}
Also assume there exists $\delta \ \in \ (0,1)$
such that
\begin{equation}
\label{leqno4.7}
q_k \ = \ o(2^{k^{\delta }})
\end{equation}
with
\begin{equation}
\label{leqno4.8}
\limsup _{k\to \infty}
{\log \ q _{2k} \over \log \ q_k} \
< \ \infty

\end{equation}
and
$$
P(q_1 \ + \ \theta _1 \ \geq \ 0) \ = \ 1.
$$
Let
$$
C_N^{\theta}(f) \
:= \ {1\over N}
\sum _{k=1}^Nf(T^{q_k+\theta _k}x).
$$

\begin{lemma}
\label{lemma4.8}

If $(q_k)_{k\geq 1}$
satisfies \eqref{leqno4.6}, \eqref{leqno4.7}
and \eqref{leqno4.8},
then there is a set $\Omega _0$ contained
in $\Omega $ of full $P$ measure
such that if $\omega \ \in \ \Omega _0$ there exists
a constant
$C>0$ such that
\begin{equation}
\label{equation4.8}
\Bigl\|
\bigl(\sum _{k\geq 1}
\bigl|C_{N_{k+1}}^{\theta }(f) \ - \
C_{N_k}^{\theta }(f)
\bigr|^2(x)
\bigr)^{1\over 2}
\Bigr\|_2
\ \leq \  C\|f\|_2.
\end{equation}
\end{lemma}

Consider two strictly increasing sequences of
natural numbers $(q_k)_{k\geq 1}$
and ${\cal N} \ = \ (N_k)_{k\geq 1}$ such that
if $a_k \ = \ q_k$ then
\eqref{leqno4.6}
holds.
Let $I_q \ = \ [2^{c_q},2^{c_{q+1}})$
denote the $q^{th}$ interval
of the form $[2^a,2^{a+1})$ containing an element
of ${\cal N}$.  Let
$\Phi (N) \ = \
(\log [q_n \ + \ q \ + 2] )^{1\over 2}$
if $N \ \in \ I_q$
and assume
\begin{equation}
\label{leqno4.9}
\sum _{N \in \ {\cal N}} {\Phi ^2(N) \over N} \
< \ \infty .
\end{equation}
Let $\phi \ = \ (\phi _k )_{k \geq 1}$ be a
sequence of independent,
identically distributed random variables defined on
a basic probability space $(\Omega ,
\beta , P)$ with $\phi _1 \ \in
\ L^1(\Omega , \beta , P)$ such that
\begin{equation}
\label{leqno4.10}
{\color{black}{\mathbb E}}\Bigl\{ \sup _{N\in \
\textcolor{black}{\mathbb{N} \setminus\{0\}}
}
\bigl({\sum _{k=1}^N(\phi _k \ -
\ {\color{black}{\mathbb E}}(\phi _k))^2 \over N}
\bigr)^{1\over 2}
\Bigr\} \
< \ \infty .

\end{equation}
Let
$$
w_N^{\phi}(f) \
:= \ {1\over N} \sum _{k=1}^N
\phi _kf(T^{q_k}x).
\eqno (N \ = \ 1,2,\cdots )
$$

\begin{lemma}
\label{lemma4.9}

Suppose
$(q_k)_{k\geq 1}$, ${\cal N } \
= \
(N_k)_{k\geq 1}$ and $\phi \
= \ (\phi _ k)_{k \geq 1}$ satisfy
\eqref{leqno4.8}, \eqref{leqno4.9}
and
\eqref{leqno4.10}.

Then for
almost all $\omega$ in $\Omega$
\begin{equation}
\label{equation4.9}
\bigl\|(\sum _{k\geq 1} |w_{N_{k+1}}^{\phi _k}(f) \ - \
w_{N_k}^{\phi _k}(f)|^2)^{1\over 2}
\bigr\|_2
\ \leq \  C\|f\|_2.
\end{equation}
\end{lemma}
Some remarks about the nature of condition
\eqref{leqno4.10}
are in order.  If $N_k  \ =
\ [k^{\epsilon}]$ $(k \ = \ 1,2, \ldots )$,
with $\epsilon \ > \ 1$ then
\eqref{leqno4.10}
reduces to
$$
\sum _{k\geq 1}
{\log (q_{[k^{\epsilon}]}+ \log k) \over k^{\epsilon}}
\ < \ \infty,
$$
which is realised if $q_k \ = \ O(2^{k^{\delta}})$
for $\delta \ > \ 0$.
Also if $N_k \ = \ 2^k$
$(k \ = \ 1,2,\ldots )$ then
\eqref{leqno4.10}

reduces to
$$
\sum _{k\geq 1}{\log (q_{2^k}+ \log k) \over 2^k}
\ < \ \infty,
$$
which is realized if $q_k \ = \ O(2^{k^{\gamma}})$.
Given the earlier Lemma of this section these two
conditions are
not difficult to {\color{black}satisfy.

Set} $k_n \ = \
[g(n)]$ $(n \ = \ 1,2,\ldots )$ where $g$ is a
differentiable
function from $[0,\infty )$ to itself whose
derivative increases with
its argument.  Let $Z_M$ denote the
cardinality of
the set
$\{ n \ : \ k_n \ \leq \ M \}$ and suppose
for some function
$a \ : \ [1,\infty ) \ \to \ [1,\infty )$
increasing to infinity
as its argument does, that we set
$$
b(M) \
:= \ \sup _{{\color{black}\{\alpha \}} \in [{1\over a(M)},
{1\over 2})}
{\color{black}\left |\sum _{n:k_n \leq M} e^{2\pi ik_n \alpha } \right |.}
$$
Suppose also for some decreasing function
$c \ : \ (1,\infty ) \ \to
\ (0,\infty )$ that
$$
{b(M) \ + \ Z_{[a(M)]} \ + \
{M\over a(M)} \over Z_M}
\ \leq \ C \,c(M).
$$
Then if we have
$$
\sum _{s=1}^{\infty} c(\rho ^s) \ < \infty ,
$$
for every $\rho \ > \ 1$ we say $(k_n)_{n\geq 1}$
satisfies
{\em condition H}
(see examples in \textcolor{black}{S}ection \ref{S7}).
Let $b_k \ =
\ g^{-1}(k) \ - \ g^{-1}(k-1)$ where
$g^{-1}$ here denotes the inverse function
of $g$ on its set of
definition.  Also suppose that
there is a constant $C$ such that
\begin{equation}
\label{leqno4.11}
\bigl(g^{-1}({\color{black} \left [{1\over \{\alpha \}} \right ]}
)\bigr)^2
\Bigl(\sum _{{\color{black}\left  [{1\over \{\alpha \}} \right ]}N_k\geq 1}
\Bigl({1\over g^{-1}(N_k)}
\Bigr)^2
\Bigr)
\ \leq \ C.
\end{equation}
{

\begin{lemma}
\label{lemma4.11}

Suppose $(k_n)_{n\geq 1}$
satisfies condition H,
and \eqref{leqno4.11}
and that $1 \ < \ a \
\leq {N_{k+1}\over N_k} \ < \ b$.
Then
\eqref{leqno4.8}
holds for the corresponding square function.
\end{lemma}

In \cite{nair}
it is shown that examples of sequences of
integers which
satisfy conditions H include those given by
$g(n) \ = \ n^{\omega}$, for
non-integer $\omega \ > 1$, $g(n) \ =
\ e^{(\log  n)^{\gamma}}$ for
$\gamma \ \in (1, {3\over 2})$ and $g(n) \
= \ \alpha _k n^k \ + \ \cdots
\ + \ \alpha _1 n \ + \ \alpha _0$, where the real
numbers $\alpha _ 1, \cdots , \alpha _k$ are not
all multiples of the
same real number.
The reader will readily verify
that these examples
also satisfy conditions H and
\eqref{leqno4.10}.
An important point of note
is that, as shown in \cite{nair},
the ergodic averages,
for $a_n \ = \
k_n$ and $f$ in $L^p$ with $p \ > \ 1$ converge
to a $T$ invariant limit.

We want to show that if $(k_n)_{n\geq 1}$
satisfies conditions H,
\eqref{leqno4.10}
then \eqref{leqno4.5}
is satisfied with
$1 \ < \ \ a \ \leq {N_{k+1}\over N_k} \ \leq \ b$.

By identifying it with its characteristic
function  we may view a strictly
increasing sequence of natural numbers as a point
in the power set of the
natural numbers \textcolor{black}{(i.e. $2^{\mathbb{N}}$)},
or as a point
in the Cartesian product
$\Pi _{n=1}^{\infty}X_n$ where for each natural
number $n$, we have
set $X_n \ = \ \{ 0,1 \}$.  As
a consequence we may put a probability measure on
the space of strictly
increasing sequences of integers, as a product
measure $\pi$ by setting
$\pi _n (\{1 \}) \ = \ \sigma _n$ for
$\sigma _n \ \in \ [0,1]$ and
$\pi _n (\{ 0 \}) \ = \ 1 \ - \ \sigma _n$
and defining $\pi$ to be the
Cartesian product measure
$\Pi _{n=1}^{\infty}\pi _n$.  For a strictly
increasing sequence of integers $(N_k)_{k\geq 1}$
suppose also that
\begin{equation}
\label{leqno4.12}
\sum _{k\geq 1}
\Bigl({\log _2 \ N_k \over
\sum _{n\leq N_k} \sigma _n}
\Bigr)
 \
 < \ \infty ,
\end{equation}
for arbitrary real $\alpha $ that we have some
constant $C$ dependent
only on $\pi$ such that
\begin{equation}
\label{leqno4.13}
{1\over \left(\sum _{n\leq {\color{black} \left [{1\over \{\alpha \} } \right ]}}
\sigma _n
\right)^2}
\,
\displaystyle
\sum _{N_k{\color{black} \left [{1\over \{\alpha \}} \right ]}\geq 1}
\Bigl({1\over
\sum _{n\leq N_k}\sigma _n}
\Bigr)^2 \
< \ C.
\end{equation}
Then with regard to the probability measure
just defined, we have
the following lemmas.

\begin{lemma}
\label{lemma4.12}

For ergodic averages with regard to almost all
strictly increasing sequences $(a_k)_{k\geq 1}$
with respect to $\pi$,
if $(N_k)_{k\geq 1}$ and $(\sigma _k )_{k\geq 1}$
satisfy
\eqref{leqno4.12} and \eqref{leqno4.13}

above then \eqref{leqno4.8}
holds.
\end{lemma}

\begin{lemma}
\label{lemma4.13}

Fix a natural number
$d \ >  \ 2$ and let $(k_n)_
{n\geq 1}$ denote the set
\begin{equation}
\label{equation4.13}
\Lambda  \ = \ \{ n \ \in \ {\mathbb{ N}}: n \
= \ \sum _{j\geq 1}
q_j d^j\ with \ q_j \ \in \{ 0,1 \} \},
\end{equation}
ordered by size.
Then if
$N_k \ = \ d^k \ (k\ = \ 1,2,\cdots )$
then \eqref{leqno4.5}
holds.
\end{lemma}

For $s=\sigma + i t$ set
$$
C_N(s , \zeta, x )
:= {1\over N} \sum _{n=1}^N
\zeta (\sigma +iT^{k_n}(x) )

\qquad \qquad (N=1,2, \ldots )
$$
and set
$$
S(s, \zeta, x) :=
\ \Bigl( \sum _{k\geq 1}
\bigl|C_{N_{k+1}}(s , \zeta, x )-
C_{N_k}(s, \zeta, x )
\bigr|^2
\Bigr)^{1\over 2}.
$$

We have the following theorem.

\begin{theorem}
\label{theorem4.14}

If $\sigma \in ({1\over 2}, 1)$
we have $C>0$
such that
\begin{equation}
\label{equation4.14}
\bigl\|S(s, \zeta, .)
\bigr\|_2^2
~\leq~
{C\over \pi \sigma }
\left|{\zeta(2\sigma ) \over 2 \sigma } +
{\zeta(2\sigma -1 ) \over 2\sigma - 1}
\right|,
\end{equation}
for all the families and sequences
$(k_n)_{n\geq 1}$
and ${\cal N}$ listed in
Lemmas \ref{lemma4.6} --
\ref{lemma4.13}.
\end{theorem}

The situation is different if we replace
$L^2$-norms by $L^1$-norms.

\begin{lemma}
\label{lemma4.15}

Suppose $(X,\beta , \mu, T )$ is an ergodic
and measure preserving transformation with $\mu$
non-ergodic.  Suppose $(k_n)_{n\geq 1}$ is Hartman
uniform distributed and $L^p$- good universal for
fixed $p \ \geq \ 1$.
Then for any non-constant function $f$ on
$(X,\beta ,\mu )$ we set
$$
C_{N}( f) = \frac{1}{N}
\sum_{n = 1}^{N} f(T^{k_n}(x))
\qquad \qquad (N=1,2, \ldots ) .
$$
Then
we have
\begin{equation}
\label{equation4.15}
\sum _{N \geq 1}
\bigl|C_{N+1}(f)-C_{N}( f)\bigr| \
= \ + \infty ,
\qquad \mu ~\mbox{almost everywhere}.
\end{equation}
\end{lemma}

Applying Lemma \ref{lemma4.15}
to
$f(x) = \zeta ( \sigma + i x)$ for
$\sigma \in ({1\over 2} , 1)$
and $ x \in \mathbb{ R}$ we get
Theorem \ref{theorem1.5}.

\section{Comparing dynamical and  probabilistic models}
\label{S6}

Let $Y=\Pi _{n=1}^{\infty}\Omega$,
that is the space of sequences
$(X_1, X_2, \cdots )$ in $\Omega$.
Let $p_1$ denote
the projection $p_1: Y\to \Omega$ defined
by $p_1((X_1,X_2, \cdots )) = X_1$.
Also let $S:Y \to Y$ denote the shift map
defined on $Y$ by $S((X_1,X2, \cdots , ))
= (X_2, X_3, \cdots )$.
It is routine to check  that $S$ preserves the infinite
product measure
$\mu ^{\infty} = \Pi _{n=1}^{\infty}\mu$ on $Y$.
That the shift map $T$ is also ergodic
with respect to this infinite product
measure is consequence of the
{\color{black}Kolmogorov} zero one law.
Now define $f$ on $Y$ by
 $f(\omega _1 , \omega _2 , \cdots )
 = X_1(\omega _1 )$.
 This means that  $X_n(\omega ) = f(S^{n-1}y) $ where
$y = (\omega _1 , \omega _2, \cdots )$ and $S$
denotes the above shift map.
Also a simple computation shows that
$\int _Yf(y)d\mu ^{\infty} = {\cal E}(X_1)$.
This means that the strong law of
large numbers follows from an
application of Birkhoff's pointwise ergodic theorem and
the weak law of large numbers from an application of
Von Neumann's norm ergodic theorem respectively.

The upshot of this is that, under quite
weak hypotheses, the comparison
between the random model described
in \cite{weber}
and
the dynamical model in \cite{srichan}
is actually a comparison
between two different
dynamical systems.
Now suppose that
$(k_n)_{n\geq 1}$ is Hartman uniformly
distributed and $L^p$ good universal
for fixed $p \in [1, \infty)$
and $\mu$ is the Cauchy distribution
$\mu _{\alpha , \beta }$ then
\begin{equation}
\label{(6.1)}
\lim _{N \to \infty} {1\over N} \sum _{n=0}^{N-1} f(s+iX_{k_n}( \omega ))
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{f(s+ i \tau ) \over \alpha ^2 +
(\tau - \beta )^2}  d\tau ,
\end{equation}
for almost all $\omega$ in $\mathbb{ R}$.
We can specialise this to
\begin{equation}
\label{(6.1)bis}
\lim _{N\to \infty} {1\over N} \sum _{n=1}^N \log  {\color{black} \left | \zeta \left  ({1\over 2}
+ {1\over 2}i X_{k_n}(\omega ) \right ) \right | }
=
\sum _{\rho: \Re (\rho ) {\color{black}>} {1\over 2}}
\log \Bigl| {\rho \over 1- \rho } \Bigr| .
\end{equation}

Again, the Riemann {\color{black} Hypothesis} follows if
either side is zero.
As above similar observations can be
obtained other \textcolor{black}{zeta} functions and $L${\color{black} -functions}.

The condition of good universality
is an assumption about all
dynamical systems.
We don't need to assume so much
and can deduce our conclusions
from the properties of one transformation.

The following Theorem offers
that link between  the two models. {\color{black}See
also
\cite{jonesolsen}

where similar ideas appear.}

\begin{theorem}
\label{theorem_1equal2}
Consider two ergodic dynamical systems
$(X_1, \beta _1 , \mu _1, T_1)$ and
$(X_2, \beta _2 , \mu _2, T_2)$ both on separable measure spaces.
Suppose that $\mu_1$ and $\mu _2$
are non-atomic.
Then if for a particular sequence of
integers $(k_n)_{n\geq 1}$ for each
$f_1 \in L^p (X_1, \beta , \mu _1)$
for all $p>1$ we have
\begin{equation}
\label{1equal2}
\lim _{N\to \infty}{1\over N}
\sum _{n=1}^Nf_1(T_1^{k_n}x_1)
=
\int _{X_1} f_1(x_1) d\mu _1 ,
\end{equation}
$\mu_1$ almost everywhere,
then the same is true with $1$ replaced by $2$.
\end{theorem}

The condition of non-atomicity of
$\mu_1$ and $\mu _2$ is not strictly
necessary for the proof but it simplifies
the proof somewhat and
our intended applications are  to non-atomic dynamical sytems.

\begin{proof}
Let  $(X, \beta , \mu , T)$ denote a dynamical
system  and for sequence of natural numbers
$(k_n)_{n\geq 1}$ let
$$
m(f) = \sup _{N\geq 1}
\Bigl| {1\over N} \sum _{n=1}^N f(T^{k_n}x)
\Bigr|.
$$
A special case of a theorem of S. Sawyer
\cite{sawyer},
tells us, after the {\color{black} hypothesis} of
Theorem \ref{theorem_1equal2},
with $m = m_1$,
that
there exists $C'_p> 0$ such that
$$
\mu \bigl( \{ x_1 \in X_1 :m_1(f_1)(x_1)  \geq \lambda \}
\bigr) \leq {C_p \| f_1\|_p \over \lambda } .
$$
Another way to say this is that
the operator $m_1$ satisfies a
weak-$(p,p)$ bound.
A stronger assertion is that
there exists $C_p*>0$
such that
\begin{equation}
\label{(6.2)}
\bigl\|m_1(f_1) \bigr\|_p \leq C_p^*\| f_1\|_p.
\end{equation}
Here we say $m_1$ satisfies a strong-$(p,p)$
inequality.
For any fixed $p$, a strong-$(p,p)$ inequality
implies  the corresponding weak-$(p,p)$
inequality.  On the other hand
as a consequence of the
Marcinkiewicz interpolation theorem
\cite{steinweiss}
weak-$(p_a ,p_a)$ implies strong-$(p_b, p_b)$
if $p_a < p_b$.
This means that the fact that
the weak-$(p,p)$ of $m$ for all $p>1$
is equivalent to it being
strong-$(p,p)$ for all $p>1$.

We now show that inequality
\eqref{(6.1)} implies
\begin{equation}
\label{(6.3)}
\bigl\|m_2(f_2) \bigr\|_p
\, \leq
\, C_p^*\| f_2\|_p
\end{equation}

Suppose that the dynamical systems
$(X_1, \beta _1 , \mu _1, T_1)$
and $(X_2, \beta _2 , \mu _2, T_2)$
are both ergodic.
Now the  argument used to deduce
Theorem \ref{theorem1}
from \eqref{(3.1)} together with
the ergodicy  $(X_2, \beta _2 , \mu_2 ,T_2)$
readily implies the conclusion of
Theorem \ref{theorem9}.

We now prove \eqref{(6.3)}.
As both the dynamical systems
$(X_1, \beta _1 , \mu _1, T_1)$
and $(X_2, \beta _2 , \mu _2, T_2)$
are ergodic,
the Rokhlin-Halmos lemma says that
given any integer $N\geq 1$ and $\epsilon > 0$,
there exist sets $E_i \subset X_i$ $(i=1,2)$
with $E_i, T_i^{-1}E_i, \ldots , T_i^{-(N-1)}E_i$
disjoint and
$$\mu _i (E_i \cup T_i^{-1}E_i \ldots
\cup T_i^{-(N-1)}E_i) < 1- \epsilon .$$

Let $\gamma = {\mu _1(E_2) \over  \mu _2 (E_2)}$.
Also for a set $D$ and a function $f$ let $f_D$
denote the restriction of $f$ to $D$.

Let us observe that there always exists
a bijection from $E_1$ to $E_2$

that preserves the measure from $\mu_1$
to $\gamma \mu _2$.
This is always possible by the measure isomorphism
theorem between separable spaces.
Let us call
$\delta$ such a 1-1 measure map
from $E_1$ to $E_2$. It is not canonical.

Now set
$$
F_i^N= E_i \cup T_i^{-1}E_i \ldots  \cup T_i^{-(N-1)}E_i,  \eqno (i=1,2)
$$
and extend the definition of
$ \Delta |_{E_1} : =
\delta$ to $F_1$ by setting
$\Delta (x)
=
(T_2^k \delta T_1^{-k})(x)$ for
$x \in T^kE_1$.
Now set
$$
Wf(y) := f(\Delta ^{-1}y)\gamma ^{1\over p}.
$$
Notice ${\rm supp} \  Wf(y) \subset T_2^kE$
if ${\rm supp} \ f(y) \subset T_1^kE$.
Direct computation now gives for
${\rm supp} \ f(y) \subset T_1^kE$
$$
\int _{T^kE_1} |f|^pd\mu_1 =
\int _{T^kE_2} \bigl|Wf \bigr|^pd\mu_2,
$$
and so
\begin{equation}
\label{F1f_F2Sf}
\int _{F_1} |f|^pd\mu_1 =
\int _{F_2} \bigl|Wf \bigr|^pd\mu_2.
\end{equation}

Let
$$
C_{i,l} (f)(x) =
{1\over l}\sum _{n=1}^lf(T_i^{k_n}x), \eqno (l=1, 2, \ldots )
$$
and set
$$
m_{i,N} (f)(x) = \sup _{1\leq l   \leq N}
| C_{i,l} (f)(x) |,  \eqno (N=1, 2, \ldots )
$$
and evidently
$$
m_i(f)(x) = \lim _{N\to \infty}m_{i,N}(f)(x).
$$
In proving \eqref{(6.3)}
by splitting $f$ into its real
and imaginary parts and each of those into their
positive and non-negative parts
we may assume $f\geq 0$.
Suppose $x \in T_1^kE_1$ and for
the transformation $T_1$ we have
$m_N(f)(x)\geq 0$.
Then there exists $l \in [1, N]$ such that
$$
m_{i, l}(f)(x) = C_{i,l} (f)(x).
$$
Let $y=Wx$.
Then a direct computation shows that for the
transformation $T_2$ we have
$$
C_{2,l}(Wf)(y) =W(m_{1,N}(f))(x).
$$
From this we deduce that
$$
m_{2,N}(Wf)(y) \geq m_{1, l}(f)(x).
$$
Thus
$$
\bigl\| m_{2,N}(f) \bigr\|_p^p =
\int _{X_2} \bigl| m_{2,N}(f)(x) \bigr|^p d\mu_2
$$
which is
$$
\leq \int _{F_2} \bigl| m_{2,N}(f)(x) \bigr|^pd\mu_2
+ \epsilon .
$$
and we know $W$ fixes measure on
$F_2$ and hence $L^p$ norms so we have
$$
\int _{F_2} \bigl| W(m_{2,N}(f))(y) \bigr|^pd\mu_2
+ \epsilon
= \bigl\| (m_{2,N}(W(f)))(y) \bigr\|_p^pd\mu_2
+ \epsilon
$$
$$
\leq \, c \bigl\| W(f) \bigr\|_p^p+ \epsilon ~
\leq  \,c \| f \|_p^pd + \epsilon .
$$
Now let $N \to \infty$ and
let $\epsilon \to 0$ and
the proof of the theorem is complete.

\end{proof}

The ergodicity of the random dynamical system
is implied by the {\color{black} Kolmogorov} $0-1$ law.
If we choose $ \mu = \mu _{\alpha , \beta}$,
we see that Theorem \ref{theorem1}
is equivalent to \eqref{(6.1)}.
From this we deduce all the applications of
Theorems \ref{theorem_1equal2}

with $T^{k_n}$ replaced by $X_{k_n}$.

\section{Hartman uniformly distributed and good universal sequences}
\label{S7}

In this section we give  some examples of
$L^p$-good universal sequences
for some $p\geq~\!\!\!1$.
The examples 1, {\color{black}3--6} are Hartman uniformly
distributed.  Example 2 is not Hartman
uniformly distributed in general.

\begin{enumerate}
\item[1.-] {\sl The natural numbers:}

The sequence $(n)_{n=1}^\infty$ is
$L^1$-good universal. This is
Birkhoff's pointwise ergodic theorem.

Let $\phi$ be any non-constant polynomial mapping the natural numbers to themselves.
Note that if
$n\in \mathbb{ N}$,
then $n^2 \not\equiv 3\bmod 4$,
so in general the sequences
$(\phi(n))_{n=1}^\infty$ and
$(\phi(p_n))_{n=1}^\infty$ are
not Hartman uniformly distributed.
We do, however, know that if
$\gamma\in \mathbb{ R}\setminus \mathbb{ Q},$ then
$(\phi(n)\gamma
)_{n=1}^\infty$ and
$(\phi(p_n)\gamma)_{n=1}^\infty$ are
uniformly distributed modulo $1$.

\item[2.] {\sl Condition ${\rm H}$:}

Sequences $(k_n)_{n=1}^\infty$ that are
both $L^p$-good universal and Hartman uniformly
distributed can be constructed as follows.
Set $k_n=[\tau(n)]$ $(n=1,2,\dots c)$,
where $\tau:[1,\infty)\to[1,\infty)$
is a differentiable function whose derivative
increases with its argument.
Let $\Omega_m$ denote the cardinality
of the set $\{n\colon a_n\le m\}$,
and suppose, for some function
$\varphi:[1,\infty)\to[1,\infty)$
increasing to infinity as its argument
does, that we set
$$
\varrho(m) = \sup_{\{z\}\in\bigl[
{1 \over \varphi(m)},{1 \over 2}\bigr)}
\Bigl|\sum_{n\colon k_n\le m} e(zk_n)\Bigr|.
$$
Suppose also, for some decreasing
function $\rho:[1,\infty)\to[1,\infty)$
and some positive constant $\omega>0$,
that
$$
{\varrho(m)+\Omega_{[\varphi(m)]}+
{m \over \varphi(m)}}{\Omega_m}
\le
\omega\rho(m).
$$
Then if we have
$$
\sum_{n=1}^\infty \rho(\theta^n)
<\infty
$$
for all $\theta>0$, we say that
$(a_n)_{n=1}^\infty$ satisfies
condition ${\rm H},$ see \cite{nair}.

Sequences satisfying condition
${\rm H}$ are known to be both
Hartman uniformly distributed and
$L^p$-good universal.
Specific sequences of integers
that satisfy condition ${\rm H}$ include
$a_n=[\tau(n)]$ $(n=1,2,\dots c)$ where:

        [I.] $\tau(n)=n^\gamma$ if $\gamma>1$ and $\gamma\notin \mathbb{ N}.$

        [II.] $\tau(n)=e^{((\log n)^{\gamma})}$
        for $\gamma\in(1,{3 \over 2}).$
        $\quad$

        [III.] $\tau(n)=b_kn^k+\dots b+b_1n+b_0$ for $b_k,\dots c, b_1$ not all rational multiplies of the same real number.

        [IV.] {\sl Hardy fields:} By a Hardy field, we mean a closed subfield (under differentiation) of the ring of germs at $+\infty$ of continuous real-valued functions with addition and multiplication taken to be pointwise. Let $\cal{H}$ denote the union of all Hardy fields. Conditions for $(a_n)_{n=1}^\infty=([\psi(n)])_{n=1}^\infty,$ where $\psi\in\cal{H}$
to satisfy condition H are given by the hypotheses of Theorems 3.4, 3.5 and 3.8. in
\cite{boshernitzanetal}.
Note the term ergodic is used
in this paper in place of the older term Hartman uniformly distributed.

\item[3.] {\sl A random example:}

Suppose that
$S=(k_n)_{n=1}^\infty$ is a strictly
increasing sequence of natural numbers.
By identifying $S$ with its characteristic
function $\chi_S,$ we may view it as a
point in $\Lambda=\{0,1\}^{\mathbb{ N}}$,
the set of maps from $\mathbb{ N}$ to $\{0,1\}$.
We may endow $\Lambda$ with a
probability measure by viewing it as
a Cartesian product
$\Lambda=\prod_{n=1}^\infty X_n$,
where, for each natural number $n$,
we have $X_n=\{ 0,1\}$ and
specify the probability $\nu_n$ on
$X_n$ by $\nu_n(\{1\})
=
\omega_n$ with
$0\le \omega_n\le 1$ and
$\nu_n(\{0\})=1-\omega_n$ such that
$\lim_{n\to\infty}\omega_n n=\infty$.
The desired probability measure on
$\Lambda$ is the corresponding
product measure
$\nu=\prod_{n=1}^\infty\nu_n$.
The underlying $\sigma$-algebra
$\cal{A}$ is that generated
by the cylinders
$$
\bigl\{(\Delta_n)_{n=1}^\infty\in\Lambda\colon \Delta_{n_1}
=
\alpha_{n_1},\dots c,
\Delta_{n_k}=\alpha_{n_k}
\bigr\}
 $$
for all possible choices of
$n_1,\dots c, n_k$ and
$\alpha_{n_1},\dots c,\alpha_{n_k}$.
Then almost every point
$(a_n)_{n=1}^\infty$ in $\Lambda$,
with respect to the measure $\nu$,
is Hartman uniformly distributed
(see Proposition 8.2 (i) in Bourgain
\cite{bourgain}).

Hartman uniformly distributed
sequences are called ergodic sequences
in \cite{bourgain}.

\item[4.] {\sl Block sequences:}

Suppose that $(a_n)_{n=1}^\infty=
\bigcup_{n=1}^\infty[d_n,e_n]$ is
ordered by absolute value for disjoint
$([d_n,e_n])_{n=1}^\infty$ with
$d_{n-1}=O(e_n)$ as $n$ tends to infinity.
Note that this allows the possibility that
$(a_n)_{n=1}^\infty$ is zero density.
This example is an immediate consequence
of Tempelman's semigroup ergodic theorem.
See page 218 of  \cite{bellowlosert}.
Being a group average ergodic theorem
this pointwise limit must be invariant,
which ensures
that the block sequence must be
Hartman uniformly distributed.

\item[5.]{\sl Random perturbation of good sequences:}

Suppose that $(a_n)_{n=1}^\infty$ is an $L^p$-good
universal sequence which is also Hartman uniformly
distributed.
Let $\theta=(\theta_n)_{n=1}^\infty$
be a sequence of $\mathbb{ N}$-valued
independent, identically distributed
random variables with basic probability
space $(Y,\cal{A},\cal{P}),$ and
a $\cal{P}$-complete $\sigma$-field
$\cal{A}$.
Let $ \mathbb{ E}$ denote expectation
with respect to the basic probability
space $(Y,\cal{A},\cal{P})$.
Assume that there exist
$0<\alpha<1$ and $\beta>1/\alpha$
such that
$$
a_n=O(e^{n^\alpha}) \mathbb{ E}\log_+^\beta|\theta_1| <\infty.
$$
Then $(k_n+\theta_n(\omega))_{n=1}^\infty$ is
both $L^p$-good universal and
Hartman uniformly distributed
\cite{nairweber2}.

\end{enumerate}

\section{Moving averages}
\label{S8}

In this paragraph we show that the assumption
of a (univariate)
sequence
which has the property to be $L^p$ good universal and Hartman uniformly distributed
can be removed from the Theorems
\ref{theorem9},
\ref{theorem11},
\ref{theorem12},
\ref{theorem14},
\ref{theorem15},
\ref{theorem16},
and replaced
by the assumption of
a (bivariate) Stoltz sequence.

\subsection{Stoltz sequences and {\color{black}Theorem 7.2}}
\label{S8.1}

Let $Z$
be a collection of points in
$\mathbb{ Z}\times \mathbb{
N}$ {\rm and let}
$$Z^h \ :=
\ \{ (n,k) \ : \ (n,k) \
\in \ Z \ {\rm and} \ k \ \geq  \ h \},$$
$$
Z^h_{\alpha} \ := \ \{ (z,s) \ \in \ \mathbb{ Z}^2 \
: \
|z \ - \ y| \ < \ \alpha (s \ - \ r) \ {\rm for}
\ {\rm some} \ (y,r)
\ \in Z^h \}
$$
and
$$
\mbox{}\qquad
\qquad \qquad
Z^h_{\alpha }(\lambda ) \
:= \ \{ n : \
(n, \lambda ) \ \in \
Z^h_{\alpha } \}
\qquad \qquad \qquad( \lambda \in \mathbb{ N} ).
$$
Geometrically we can think of $Z^1 _{ \alpha }$ as
the lattice points contained
in the union of all solid cones with aperture
$\alpha$ and vertex contained in
$Z^1= Z$.

We say a sequence of
pairs of natural numbers
$(n_l ,k_l )_{l=1}^{\infty}$ is \it Stoltz \rm  if
there exists a collection of points $Z$ in
$\mathbb{ Z}\times \mathbb{N}$, and a
function $h \ = \ h(t)$ tending to infinity with
$t$ such
that $(n_l,k_l )_{l=t}^{\infty} \ \in \ Z^{h(t)}$
and
there exist $h_0$, $\alpha _0$ and $A \ > \ 0$ such
that for all integers $\lambda \ > \ 0$ we have
$|Z^{h_0}_{\alpha _0}(\lambda)| \ \leq \ A\lambda$.
This technical
condition is interesting because of the following
theorem \cite{bellowjonesrosenblatt}, which will be used in the proof of Theorem
\ref{theorem3}.  \textcolor{black}{The Stoltz condition is related to the
``cone condition" of Nagel and Stein
\cite{nagelstein} and
of Sueiro
\cite{sueiro}}.

\begin{theorem}
\label{theorem2}

Let  $(X,\beta,\mu, T )$ denote a
dynamical system, with set $X$, a $\sigma$-algebra of its
subsets $\beta$, a measure $\mu$ defined on the measurable
space $(X,\beta)$ such that $\mu (X) \ = \ 1$ and a
measurable, measure preserving map $T: X \to X$.  Suppose $g$ is in $L^1(X,\beta , \mu )$ and that
the sequence of pairs on natural numbers $(n_l ,
k_l)_{l=1}^{\infty}$ is Stoltz.
Then

\begin{equation}
\label{limit_Stolz_measurable}
\widetilde{m_g}(x)
:= \lim _{l\rightarrow \infty }{1\over
k_l}\sum _{j=1}^{k_l}g(T^{n_l+j}x)
\end{equation}
exists almost everywhere with respect to $\mu$.
\end{theorem}

\begin{theorem}
\label{theorem3}

Let $f$ be a meromorphic  function on
$\mathbb{ H}_c$ satisfying conditions (1),
(2) and (3)
of Theorem \ref{theorem1}.
Then if $(n_l , k_l)_{l\geq 1}$ is Stoltz,
for any $s \in \mathbb{ H}_{c}
\backslash \mathbb{ L}_{\sigma_0}$ ,
we have
\begin{equation}
\label{limit_Stolz_meromorphic}
\lim _{l\rightarrow \infty }{1\over
k_l}\sum _{j=1}^{k_l}
f(s+iT^{n_l +
j}_{\alpha , \beta } (x))
= {\alpha \over \pi } \int _{\mathbb{ R}}
{f(s+ i \tau ) \over \alpha ^2 +
(\tau - \beta )^2} \, d\tau
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

\begin{proof}

Let
\begin{equation}
\label{mlf_definition}
\mbox{} \qquad
\widetilde{m_{l,f}}(x)
:= {1\over
k_l}\sum _{j=1}^{k_l}  f(s+iT^{n_l + j}_{\alpha , \beta } (x))
\qquad (l=1,2, \ldots )
\end{equation}
and let
\begin{equation}
\label{mf_definition}
\widetilde{m_f}(x)
:= \lim _{l\to \infty}{1\over
k_l}\sum _{j=1}^{k_l}  f(s+iT^{n_l + j}_{\alpha , \beta } (x)).
\end{equation}
Notice that
$$
\widetilde{m_{l,f}}(s+iT_{\alpha ,\beta} (x))
- \widetilde{m_{l,f}}(x)
=
{1\over k_l}( f(s+iT_{\alpha , \beta}^{n_l+k_l+1}(x)) -
f(s+iT_{\alpha , \beta}^{n_l+1}(x))).
$$
This means that
$\widetilde{m_{f}}(s+iT_{\alpha , \beta }(x))
=
\widetilde{m_{f}}(s+i (x))$ $\mu _{\alpha , \beta}$
almost everywhere.
As an ergodic dynamical system constant along
almost all orbits must be constant, we must have $\widetilde{m_f}(x) =
\int _{\mathbb{ R}} f(s+ix)d
\mu _{\alpha , \beta }(x)$.
We have therefore shown that
$$
\lim _{l\rightarrow \infty }{1\over
k_l}\sum _{j=1}^{k_l}  f(s+iT^{n_l
+j}_{\alpha , \beta } (x))
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{f(s+ i \tau ) \over \alpha ^2 +
(\tau - \beta )^2}  d\tau ,
$$
for almost all $x$ in $\mathbb{ R}$, as required.
\end{proof}

\subsection{Applications and moving average ergodic Theorems}
\label{S8.2}

We will forgo the proof of the
Theorems below as they follow from
those of Theorems
\ref{theorem11},
\ref{theorem12},
\ref{theorem14},
\ref{theorem15},
\ref{theorem16} and
\ref{theorem_1equal2}
respectively via minor modification, in a
similar way, using Theorem
\ref{theorem2} and Theorem \ref{theorem3}.

\begin{theorem}
\label{theorem11stoltz}

Suppose
$(n_q , k_q)_{q\geq 1}$ is Stoltz.
Suppose $k$ is any non-negative integer.
Then the statement, for any natural number
$l$,
\begin{equation}
\label{limitzetak_Talphabeta_iterates}
\lim _{q \to \infty} {1\over k_q}
\sum _{j=1}^{k_q}
\bigl|
\zeta ^{(k)} (s+iT^{n_q + j}_{\alpha , \beta }( x))
\bigr|^l
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{|\zeta ^{(k)} (s+ i \tau )|^l \over \alpha ^2 +(\tau - \beta )^2} \,  d\tau
\end{equation}
for $\mu _{\alpha , \beta }$ -almost all $x$
in $\mathbb{ R}$, is equivalent
to the {\color{black} Lindel\"of}  {\color{black} Hypothesis}.
\end{theorem}

\begin{theorem}
\label{theorem12stoltz}

Suppose $(n_q , k_q)_{q \geq 1}$ is Stoltz.
Then
for almost all $x$ in $\mathbb{ R}$
with respect to Lebesgue measure we have
\begin{equation}
\label{zeta_Titerates_zeroes}
\lim _{q \to \infty} {1\over k_q}
\sum _{j=1}^{k_q} \log
\bigl| \zeta ({1\over 2} +
{1\over 2}i T^{n_q + j}x)
\bigr|
=
 \sum _{\rho: \Re (\rho ) {\color{black}>} {1\over 2}}
\log
\Bigl| {\rho \over 1- \rho } \Bigr| .
\end{equation}
\end{theorem}
Again, if either side is zero, this
is equivalent to the Riemann {\color{black} Hypothesis}.

We now consider Dirichlet $L$-{\color{black}functions}.

\begin{theorem}
\label{theorem14stoltz}

Let $L(s, \chi )$ denote the $L$-series
associated to the character $\chi$.
Suppose $(n_q , k_q)_{q \geq 1}$ is Stoltz,
and let $k$ be a nonnegative integer.
Then,

(i) if $\chi$ is non-principal,
for $s \in \mathbb{ H}_{-{ 1\over 2}}\backslash
\mathbb{ L}_{1}$ we have
$$
\lim _{q \to \infty} {1\over k_q}
\sum _{q=1}^{k_q}
L ^{(k)} (s+iT^{n_q + j}_{\alpha , \beta }( x), \chi ) =
{\alpha \over \pi } \int _{\mathbb{ R}}
{L ^{(k)} (s+ i \tau, \chi  ) \over \alpha ^2 +(\tau - \beta )^2} \, d\tau
\hspace{3cm}\mbox{}
$$
\begin{equation}
\label{lseries_chinonprincipalstoltz}
\mbox{} \hspace{1cm}
=
  ~{L ^{(k)} (s +\alpha +i\beta  ,  \chi  )}
\qquad \mbox{for almost all $x$ in $\mathbb{ R}$},
\end{equation}
(ii) if $\chi$ is principal, for
$s \in \mathbb{ H}_{-{1\over 2}}\backslash
\mathbb{ L}_{1}$ we have
\begin{equation}
\label{lseries_chiprincipalstoltz}
\lim _{q \to \infty} {1\over k_q}
\sum _{j=1}^{k_q}
L ^{(k)} (s+iT^{n_q + j}_{\alpha , \beta }( x), \chi ) =
{\alpha \over \pi } \int _{\mathbb{ R}}
{L ^{(k)} (s+ i \tau, \chi  ) \over \alpha ^2 +
(\tau - \beta )^2} \, d\tau ,
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

We now  consider the Hurwitz zeta function.

\begin{theorem}
\label{theorem16_stoltz}

Suppose $(n_q , k_q)_{q \geq 1}$ is
Stoltz.
For any $s$ such that  $ \Re (s) > -{1\over 2},
s \not= 1$, $0 \leq a < 1$
and $k$ a non-negative integer,  we have
\begin{equation}
\label{limHurwitzk_iteratesTalphabetastoltz}
\lim _{q \to \infty} {1\over k_q}
\sum _{j=1}^{k_q}
\zeta ^{(k)} (s+iT^{n_q + j}_{\alpha , \beta}( x), a)
=
{\alpha \over \pi } \int _{\mathbb{ R}}
{\zeta ^{(k)} (s+ i \tau , a ) \over \alpha ^2 +(\tau - \beta )^2} \, d\tau  ,
\end{equation}
for almost all $x$ in $\mathbb{ R}$.
\end{theorem}

In \eqref{limHurwitzk_iteratesTalphabetastoltz}
the right hand side is given by
\eqref{lk_alphabeta_hurwitz},
\eqref{gammak_alphabeta_hurwitz}
and
\eqref{lzero_alphabeta_hurwitz}.

The analogue of Theorem
\ref{theorem_1equal2}
is now the following.

\begin{theorem}
\label{theorem_1equal2stoltz}
Consider two ergodic dynamical systems
$(X_1, \beta _1 , \mu _1, T_1)$ and
$(X_2, \beta _2 , \mu _2, T_2)$,
both on separable measure spaces.
Suppose that $\mu_1$ and $\mu _2$
are non-atomic.
Then if for a Stoltz sequence of integers
$(n_q , k_q)_{q \geq 1}$
and each
$f_1 \in L^{p}(X_1 , \beta_1 , \mu_1)$
 we have
\begin{equation}
\label{1equal2stoltz}
\lim _{q \to \infty}{1\over k_q}
\sum _{j=1}^{k_q}
f_1(T_1^{n_q + j}x_1)
=
\int _{X_1} f_1(x_1) d\mu _1 ,
\end{equation}
$\mu_1$ almost everywhere,
then the same is true with $1$ replaced by $2$.
\end{theorem}

\section{Sublinearity and the Riemann zeta function}
\label{S9}

In the same vein as Theorem \ref{theorem4},
Theorem \ref{theorem5} and Theorem \ref{theorem1.4},
now with peculiar sublinear sequences,
the following Theorem can be deduced.

The following Lemma is taken from
\cite{lifshitsweber2}.

Let $I_A$ denote the indicator function of the set $A$.

\begin{lemma}
\label{Lemma9.1}

Suppose $k_q= k(q),  q = 1, 2, \dots $,
with $k$ sub-linear such that $\sup _u k'(u)$
is absolutely bounded.
Also assume
$$
K(\theta ) = {1\over |\theta |}
\int _{\min ({1\over |\theta |},1)}
{k'(u) \over |u|^2}du +
k\left ({1\over | \theta |} \right )|\theta |
\,I_{\{ |\theta | \leq 1\}},
$$
is uniformly bounded on $\mathbb{R}$.
Suppose $(X, \beta, \mu , T)$ is  a dynamical system
and $f \in L^2(X, \beta , \mu )$ and set
$$
g^k_n(f(x))={1\over n}
\sum _{j=k_n}^{k_n+n-1}f(T^{j}(x)).
$$
Then, for any strictly increasing
sequence of natural numbers
$(n_q)_{q\geq 1}$, there exists $C> 0$ such that
$$
\Bigl\|
\Bigl( \sum _{q \geq 1}
| g^k_{n_{q+1}}(f) - g^k_{n_{q}}(f)|^2
\Bigr)^{1\over 2} \Bigr\|_2
\leq C \| f \|_2.
$$\end{lemma}

We now specialize to the case, $f(x)= \zeta ( \sigma +ix)$ with $\sigma \in [{1\over 2} , 1)$.

\begin{theorem}
\label{theorem9.2}
Suppose $k$ and  $K $ are as in Lemma \ref{Lemma9.1}.

Now

set, with $\sigma \in [{1\over 2} , 1)$,
$$
G^{k}_{n, \sigma} (x) = G^k_n(\zeta (\sigma +ix))={1\over n} \sum _{j=k_n}^{k_n+n-1}\zeta (\sigma + iT^{j}(x)).
$$
Then, for any strictly increasing sequence of
natural numbers $(n_q)_{q \geq 1}$,

there exists a constant $C> 0$ such that
$$
\Bigl\|
\Bigl( \sum _{q \geq 1}
\bigl| G^k_{n_{q+1}, \sigma }(\zeta ) - G^k_{n_{q}, \sigma }(\zeta )
\bigr|^2
\Bigr)^{1\over 2}
\Bigr\|_2
\leq { C \over \pi \sigma}
\left | {\zeta (2\sigma ) \over 2}+ {\zeta ( {2\sigma -1}) \over 2\sigma -1} \right |.
$$
\end{theorem}

\section*{Acknowledgements}
We thank the referee for very detailed comments, substantially improving the presentation of this paper, and for bringing to our attention important additional references of material significance to this topic.

config: arxiv


TITLE: Disproving a function exists
QUESTION [1 upvotes]: Prove/Disprove: There exists a function $f: \Bbb R   \to\Bbb R$ such that
$$
 \arctan(f(x)) =  2x/(\cos^2(x) + 3)
$$
for every x ∈ $\Bbb R$
I know that it's not true because arctan is limited between $\frac{-0.5}{\pi} \leq x \leq \frac{0.5}{\pi}$, but how can you fully prove it without just giving an example? seeing that it is an "exists" proof

REPLY [1 votes]: Exactly what you did.
Suppose it exists. Then for $x = 100$ (some big number), $\arctan f(x) = \frac{200}{\cos^2 100 + 3} > \frac{200}{4} = 50 > \max \arctan$, contradiction.

REPLY [0 votes]: I don't understand your request. Take $x>\pi$. Then$$\frac{2x}{\cos^2(x)+3}\geqslant\frac{2x}4=\frac x2>\frac\pi2.$$Therefore, for such a $x$ you cannot possible have$$\arctan\bigl(f(x)\bigr)=\frac{2x}{\cos^2(x)+3},$$since$$(\forall y\in\mathbb{R}):\arctan(y)<\frac\pi2.$$What can possibly be wrong with this proof?


TITLE: Changing variables between two different metric ansatzes in the calculation of the Klein-Gordon equation
QUESTION [1 upvotes]: My question concerns changing variables in the calculation of the Klein-Gordon equation for a scalar field given two different "guesses" for the metric.
I consider the following Einstein tensor, which describes an action with a scalar field $\phi(r)$ with potential $V(\phi)$:
$$
R_{\mu \nu}-\kappa \left(\partial_\mu \phi\partial_\nu \phi+g_{\mu \nu}V(\phi)\right)=0
$$
Now, we'll propose a metric of the form
$$
ds^2=-f(r)dt^2+f^{-1}(r)dr^2+a^2(r)d\sigma^2
$$
This gives us a Klein-Gordon equation of the form (Eqn. 1)
$$
\square \phi=g^{11}\phi''-(g^{00}\Gamma_{00}^1+g^{11}\Gamma_{11}^1+g^{22}\Gamma_{22}^1+g^{33}\Gamma_{33}^1)\phi'\notag\\
=f(r)\phi''(r)+\left(f'(r)+2\frac{a'(r)}{a(r)}f(r)\right)\phi'(r)=\frac{dV}{d\phi}
$$
Now, we can also define the metric in the form
$$
ds^2=p(r')\left\{
-b(r')dt^2+\frac{1}{b(r')}dr'^2+r'^2d\sigma^2
\right\}
$$
The Klein-Gordon equation now yields (Eqn. 2)
$$\frac{b(r')}{p(r')}\phi''(r')+\phi'(r')\left\{\frac{b(r')p'(r
)}{p^2(r')}+\frac{2b(r')}{r'p(r')}+\frac{b'(r')}{p(r')}\right\}=\frac{dV}{d\phi} $$
where now the derivative is taken with respect to r'.
My goal is to go from Eqn 1. from Eqn 2. with the following change of variables:
$$f(r')=p(r')b(r'),\quad  a(r')=r'\sqrt{p(r')},\quad \frac{dr'}{dr}=\frac{1}{p(r')}$$
For example, for the second derivative term,
$$f\phi''=pb \frac{d^2\phi}{dr^2}\notag\\
=pb\frac{d}{dr}\frac{dr'}{dr}\frac{d\phi}{dr'}\notag\\
=pb \frac{d}{dr'}\left(\frac{1}{p}\frac{d\phi}{dr'}\right)\frac{1}{p}\notag\\
=-\frac{p'b}{p^2}\phi'+\frac{b}{p}\phi'' $$
where the variables are implied and primed-derivative notation is reserved for the $r'$ coordinate. The term proportional to $\phi''$ matches the form in Eqn. 2, but the term proportional to $\phi'$ seems a bit off. Doing the exact same thing for the other terms in $\phi'$ and adding them altogether, I find that
$$f'+2\frac{a'}{a}f=\frac{2b}{r'}+\frac{2bp'}{p}+b' $$
Putting this altogether, I find
$$\frac{b(r')}{p(r')}\phi''(r')+\phi'(r')\left\{
-\frac{b(r')p'(r')}{p^2(r')}+\frac{2b(r')}{r'}+\frac{2b(r') p'(r')}{p(r')}+b'(r')
\right\} $$
Although I am close, I am off by a few signs and factors. However, I have checked it several times and all seems to be fine. Is my mapping incorrect? When I change variables in the derivative, am I doing something incorrect, or am I missing something more subtle?

REPLY [1 votes]: I have discovered the issue: the derivative of $\phi'$ was not transformed properly. The correct calculation is given below:
$$\left(f'+2\frac{a'}{a}f\right)\phi'\notag\\
=\left(b\frac{p'}{p}+b'+\frac{2b}{r'}+\frac{bp'}{p}\right)\frac{1}{p}\phi'\notag\\
=\frac{bp'}{p^2}+\frac{b'}{p}+\frac{2b}{r'p}+\frac{bp'}{p^2} $$
The first term cancels the $\frac{d\phi}{dr'}$ term that pops out of $\frac{d^2\phi}{dr^2}$, yielding the correct Eqn. 2 given in the question.


TITLE: CCC and point separating weight
QUESTION [1 upvotes]: CCC means countable chain condition;
A cover $\cal A$ of  a set $E$ is separating if for each $p\in E$, $\bigcap \{A: A \in \mathcal{A}, p\in A\}=\{p\}.$ The point separating weight of $X$, denoted $psw(X)$, is the smallest infinite cardinal $\kappa$ such that $X$ has a separating open cover $\cal V$ with ord$(p, \cal V)\le \kappa$ for each $p\in X$.
Let $X$ be a CCC space and has countable point separating weight. Is the cardinality of $X$ always at most $2^\omega$?
Thanks for any help:)

REPLY [2 votes]: $\newcommand{\supp}{\operatorname{supp}}$The cardinality of the space can by arbitrarily large.
Let $\kappa$ be any infinite cardinal. Let $\varphi:[\kappa]^{<\omega}\to\kappa$ be any injection with the property that $\varphi(F)\notin F$ for each $F\in[\kappa]^{<\omega}$. For each $\xi<\kappa$ let $X_\xi$ be a copy of $[0,1]$, and let $X=\prod_{\xi<\kappa}X_\xi$, with the product topology. For $F\in[\kappa]^{<\omega}$ let
$$X_F=\left\{x\in X:x_{\varphi(F)}=1\text{ and }\forall\xi\in F\Big(x_\xi\in(0,1)\Big)\text{ and }\forall\xi\in\kappa\setminus\big(F\cup\{\varphi(F)\}\big)(x_\xi=0)\right\}\;.$$
Finally, let $$Y=\bigcup_{F\in[\kappa]^{<\omega}}X_F$$
with the subspace topology inherited from $X$, and for each $x\in Y$ let $\supp(x)$ be the unique $F\in[\kappa]^{<\omega}$ such that $x\in X_F$.
Let $\mathscr{I}=\{(p,q):p<q\text{ and }p,q\in\Bbb Q\cap(0,1)\}$, and let $\mathscr{J}=\{(p,1]:p\in\Bbb Q\cap(0,1)\}.$ For each $F\in[\kappa]^{<\omega}$, function $I:F\to\mathscr{I}$, and $J\in\mathscr{J}$ let
$$B(F,I,J)=\left\{x\in Y:\forall\xi\in F\Big(x_\xi\in I(\xi)\text{ and }x_{\varphi(F)}\in J\Big)\right\}\;,$$
and let $$\mathscr{B}=\left\{B(F,I,J):F\in[\kappa]^{<\omega}\text{ and }I:F\to\mathscr{I}\text{ and }J\in\mathscr{J}\right\}\;;$$ clearly $\mathscr{B}$ is an open cover of $Y$. Suppose that $x\in B(F,I,J)\in\mathscr{B}$. Then $x_\xi\ne 0$ for each $\xi\in F\cup\{\varphi(F)\}$, so $F\cup\{\varphi(F)\}\subseteq\supp(x)$, and it follows immediately that $\operatorname{ord}(x,\mathscr{B})=\omega$. It’s also clear that for each $x\in Y$ we have $\bigcap\{B\in\mathscr{B}:x\in B\}=\{x\}$, so $\operatorname{psw}(Y)=\omega$.
Finally, $X$ is a product of separable spaces, so $c(X)=\omega$, and $Y$ is dense in $X$, so $c(Y)=\omega$ as well.


text:
{\bf Problem.} Triangle $ABC$ is a right triangle with legs $AB$ and $AC$. Points $X$ and $Y$ lie on legs $AB$ and $AC$, respectively, so that $AX:XB = AY:YC = 1:2$. If $BY = 16$ units, and $CX = 28$ units, what is the length of hypotenuse $BC$? Express your answer in simplest radical form.
{\bf Level.} Level 5
{\bf Type.} Geometry
{\bf Solution.} Let $AB = x$ and $AC = y$. Then we can write two Pythagorean equations from the information given: $(x/3)^2 + y^2 = 28^2$ and $x^2 + (y/3)^2 = 16^2$. These equations become $x^2/9 + y^2 = 784$ and $x^2 + y^2/9 = 256$. Multiplying them both by 9, we get $x^2 + 9y^2= 7056$ and $9x^2 + y^2
= 2304$. Now we add the two equations to get $10x^2 + 10y^2 = 9360$, which can be reduced to $x^2
+ y^2 = 936$. We do not need to solve for $x$ and $y$ since 936 is the square of the hypotenuse $BC$. The length is thus $\sqrt{936} = \sqrt{(36 \times 26)} = \sqrt{36} \times \sqrt{26} = \boxed{6\sqrt{26}}$ units.

set_name: MATH


TITLE: Domain of a two variables function
QUESTION [1 upvotes]: I want to draw the domain of a two variables function. I did it, but not sure my answer is correct. I can't load images due to being new here, so I will describe what I got, and hopefully you can verify. The function is:
$$f(x,y)=\frac{\sqrt{x}\sqrt{9-x^{2}-y^{2}}}{\ln(\frac{x^{2}}{4}+\frac{y^{2}}{16}-1)}$$
My domain contains 2 ellipses, one circle, for positive values of x or zero only. Am I correct ? My main question is, are there 2 ellipses in this domain or only 1 ? I think 2, because you have the inside of the ln which needs to be > 0 and you have the ln itself that should not be 0, meaning the inside is not 1

REPLY [0 votes]: I've chalked up your description (in Paint, no less) below.
I agree that you have to stay either on or inside the brown circle. Also, you need to stay on the half-plane corresponding to the non-negative part of the $x$-axis.
Regarding the inner ellipse, you have to stay strictly outside this ellipse in order for $\frac{x^2}{4}+\frac{y^2}{16}>1$ to be true, and as you yourself point out, you should also avoid landing on the ellipse $\frac{x^2}{4}+\frac{y^2}{16}=2$.


TITLE: Set of unitary operators such that $U|0\rangle = |\psi\rangle$
QUESTION [2 upvotes]: In quantum computing, one of the central areas of study is to determine efficient quantum circuits - described by unitaries - to prepare a state $|\psi\rangle$ from the initial computational basis state $|0\cdots 0\rangle$.
I am currently interested the mathematical abstraction of this problem; to be specific, if we have a $d$ dimensional Hilbert space, an initial state $|0\rangle$ and a final state $|\psi\rangle$, can we say anything about the set of unitaries $U$ such that $U|0\rangle = |\psi\rangle$? In particular, I am wondering about the following two things:

Does this set of unitaries form a submanifold of $U(d)$? (It clearly does not form a Lie subgroup.)
If we know a single unitary $U_0$ such that $U_0|0\rangle = |\psi\rangle$, can we obtain other unitaries with this property from $U_0$? Can we obtain all of the unitaries with this property from $U_0$?

My guess to the first question is that this set of unitaries forms a submanifold of $U(d)$ of dimension $d^2 - d$ isomorphic to $U(d) \ / \ \mathbb{C}^d$, but I am completely unsure about the second. Are there known answers to these questions, or in general is there anything that can be said about this class of unitaries?

REPLY [2 votes]: Say your state space is $D = 2^d$ dimensions. Without loss of generality, consider an orthonormal basis of states we will label by $| i \rangle$, where $i = 1 \ldots D$, where $|1\rangle = | \psi \rangle$, and $|i \rangle$ for $i > 1$ are other random states.
In this basis, a few things become clear. If we have any block diagonal matrix of the form
$$
U' = \begin{pmatrix} 1 &  \mathbf{0} \\ \mathbf{0} & U_{D-1}\end{pmatrix}
$$
where $U_{D-1}$ is any $(D-1) \times (D-1)$ matrix, then we clearly have
$$
U' |\psi\rangle = U' |1\rangle = |1 \rangle = |\psi \rangle.
$$
Therefore, we have a whole $U(D-1)$ worth of transformations that preserve $|\psi \rangle$. If we have your matrix $U_0$, composing it with one of these matrices $U'$ will preserve the property that $(U' U_0) |0 \ldots 0 \rangle = |\psi \rangle$. In fact, once we have $U_0$, we can get all the other such matrices by simply multiplying it in this way $U' U_0$.