text:
\begin{document}
\maketitle

\begin{abstract}
The total irregularity of a simple undirected graph $G$ is denoted by $irr_t(G)$ and is defined as $irr_t(G) =\frac{1}{2}\sum \limits_{u,v \in V(G)}|d(u) - d(v)|$. In this paper, the concept called edge-transformation in relation to total irregularity of simple undirected graphs with at least one cut edge is introduced. We also introduce the concept of an edge-joint between two simple undirected graphs. We also introduce the concept of total irregularity in respect of in-degree and out-degree  in simple directed graphs. These invariants are called total in-irregularity and total out-irregularity  respectively.  In this paper, we initiate a study on these parameters of given simple undirected graphs and simple digraphs.
\end{abstract}

\ni \textbf{Key Words:} Total irregularity, branch-transformation, edge-transformation, edge-joint, total in-irregularity, total out-irregularity.

\vspace{0.2cm}

\ni \textbf{Mathematics Subject Classification: 05C07, 05C20, 05C38, 05C70, 05C75.}

\section{Introduction}

For general notations and concepts in graph theory, we refer to \cite{BM1}, \cite{FH} and \cite{DBW} and for digraph theory, we further refer to \cite{CL1} and \cite{JG1}. All graphs mentioned in this paper are simple, connected and finite graphs, unless mentioned otherwise. Also, except for Section \ref{Sec-4}, all the graphs mentioned here are undirected graphs.

A graph $G$ is said to be \textit{regular} if the degree of all vertices are equal. A graph that is not regular is called an \textit{irregular graph}. The total irregularity of a given simple connected graph is defined in \cite{MOA} as follows.

\begin{definition}{\rm
\cite{MOA} The \textit{imbalance} of an edge $e=uv$ in a given graph $G$ is defined as $|d(u)-d(v)|$. The \textit{total irregularity} of a graph $G$,  denoted by $irr_t(G)$, is defined as $irr_t(G) =\frac{1}{2}\sum \limits_{u,v \in V(G)}|d(u)-d(v)|$. }
\end{definition}

If the vertices of a graph $G$ on $n$ vertices are labelled as $v_i, i = 1, 2, 3, \ldots, n$, then the definition may be $irr_t(G) = \frac{1}{2}\sum \limits_{i=1}^{n} \sum \limits_{j=1}^{n}|d(v_i) - d(v_j)| = \sum \limits_{i=1}^{n} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|$ or $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d(v_i) - d(v_j)|$. For a graph on a singular vertex (\emph{1-null} graph or $K_1$), we define $irr_t(G) = 0$.
 Clearly, $irr_t(G) = 0$ if and only if $G$ is regular.

The notion of \textit{branch-transformation} of a graph has been introduced in \cite{ZYY} as follows.

\begin{definition}{\rm
\cite{ZYY} Let $G$ be a graph with at least two pendent vertices. Without loss of generality, let $u$ be a vertex of $G$ with $d_G(u)\ge 3$, $T$ be a hanging tree of $G$ connecting to $u$ with $|V(T)|\ge 1$ and $v$ be a pendant vertex of $G$ with $v\notin T$. Let $G'$ be the graph obtained from $G$ by deleting $T$ from vertex $u$ and attaching it to vertex $v$. We call the transformation from $G$ to $G'$ a \textit{branch-transformation} on $G$ from vertex $u$ to vertex $v$.}
\end{definition}

Certain studies on irregularities and total irregularities of given graphs and the properties graphs related to these irregularities have been studied in \cite{ABD,ACD,AD,MOA,DS1,HR} and \cite{ZYY}. Motivated from these studies, in this paper, we initiate a study on certain types of total irregularities of given graphs and directed graphs. We introduce the notion of edge-transformation in relation to total irregularity of simple graphs with at least one cut edge as well as an edge-joint between two graphs. We also introduce the notion of total irregularity with respect to in-degree and out-degree in directed graphs.

\section{Total Irregularity Resulting from Edge-Joints}

Consider a graph $G$ on $n$ vertices with two connected components $G_1$ and $G_2$. Therefore, $G = G_1 \cup G_2$.  Hence, the total irregularity of $G$ is given by $irr_t(G) = irr_t(G_1) + irr_t(G_2) + \sum\limits_{i=1}^{r}\sum\limits_{j=1}^{s}|d(u_i)- d(v_j)|$, where $u_i \in V(G_1)$, $v_j \in V(G_2)$ and $r=|V(G_1)|$ and $s=|V(G_2)|$.

The concept of an edge-joint between two simple undirected graphs $G$ and $H$ is defined below.

\begin{definition}{\em
The \textit{edge-joint} of two graphs $G$ and $H$ is the graph obtained by adding one edge, say $uv$, where $u \in V(G), v \in V(H)$, and is denoted by $G\rightsquigarrow_{uv}H$.}
\end{definition}

\begin{remark}
It is to be noted that $G\rightsquigarrow_{uv}H = G \cup H + uv$ and $G\rightsquigarrow_{uv}H \simeq H\rightsquigarrow_{vu}G$.
\end{remark}

\ni Now, we make the following standard definitions and notation.

\begin{definition}\label{D-VSS1}{\rm
Let $G$ be a graph on $n$ vertices with two connected components $G_1$ and $G_2$ whose vertex sets are $V(G_1) = \{u_i: 1 \leq i \leq r\}$ and $V(G_2) = \{v_j: 1 \leq j \leq s\}$. We fix the vertices $u_1$ from $G_1$ and $v_1$ from $G_2$. Now, we define the vertex subsets $V_1 = \{u_x: d_{G_1}(u_x) \le d_{G_1}(u_1), x\ne 1\}$; $V_2 = \{u_y: d_{G_1}(u_y) > d_{G_1}(u_1)\}$ and let $|V_1|=a$ and  $|V_2|=b$. Then, choose $V_3 = \{v_x:d_{G_2}(v_x) \le d_{G_1}(u_1)\}$ and $V_4 = \{v_y: d_{G_2}(v_y) > d_{G_1}(u_1)\}$, where $|V_3| = a^{\ast}$ and $|V_4| = b^{\ast}$. Similarly, let $V_5 = \{v_z: d_{G_2}(v_z) \le d_{G_2}(v_1),~z \ne 1\}$ and $V_6 = \{v_w: d_{G_2}(v_w) > d_{G_2}(v_1)\}$ where $|V_5| = c$ and $|V_6| = d$ and choose $V_7= \{u_z: d_{G_1}(u_z) \le d_{G_2}(v_1)\}$ and $V_8 = \{u_w: d_{G_1}(u_w) > d_{G_2}(v_1)\}$ where $|V_7| = c^{\ast}$ and $|V_8| = d^{\ast}$.}
\end{definition}

In view of the above definition, we have the relation between the cardinalities of the the above mentioned vertex subsets of $V(G_1)$ and $V(G_2)$ as follows.

\begin{remark}\label{Rem-2.3}{\rm
With regard to definition \ref{D-VSS1}, define the variables $b=r-a$, $d=s-c=n-r-c$, $b^{\ast} = r-a^{\ast}$ and $d^{\ast} = s-c^{\ast} = n-r-c^{\ast}$.}
\end{remark}

\begin{theorem}\label{Thm-2.1}
Let $G$ be a graph on $n$ vertices with two connected components $G_1$ and $G_2$, where $V(G_1) =\{u_i: 1\le i \le r\}$ and $V(G_2) =\{v_j: 1\le j\le s\}$ . Also, let $G' = G_1\rightsquigarrow_{u_1v_1}G_2$. Then, we have
$irr_t(G') = irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)| + 2n-2(b + b^{\ast}+d+d^{\ast})-2$  or
$irr_t(G') = irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)| + 2(a + a^{\ast} + c + c^{\ast}) -2n + 2$.
\end{theorem}
\begin{proof}
Clearly, for the graph $G = G_1 \cup G_2$, we have $irr_t(G) = irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)|$ with $|V(G_1)| = r$ and $|V(G_2)| = s$.

By increasing $d_{G_1}(u_1)$ by 1 we increase the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_1}}^{a}|d_{G_1}(u_1) - d_{G_1}(w_j)|$ by exactly $(a - 1)$. It also reduces the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_2}}^{b}|d_{G_1}(u_1) - d_{G_1}(w_j)|$ by exactly $b$. It also increases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_3}}^{a^{\ast}}|d_{G_1}(u_1) - d_{G_2}(w_j)|$ by exactly $a^{\ast}$ and decreases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_4}}^{b^{\ast}}|d_{G_1}(u_1) - d_{G_2}(w_j)|$ by exactly $b^{\ast}$. Furthermore, by increasing $d_{G_2}(v_1)$ by 1, we increase the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_5}}^{c}|d_{G_2}(v_1) - d_{G_2}(w_j)|$ by exactly $(c - 1)$. It also reduces the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_6}}^{d}|d_{G_1}(u_1) - d_{G_2}(w_j)|$ by exactly $d$. It also increases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_7}}^{c^{\ast}}|d_{G_1}(u_1) - d_{G_1}(w_j)|$ by exactly $c^{\ast}$ and decreases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_8}}^{d^{\ast}}|d_{G_1}(u_1) - d_{G_1}(w_j)|$ by exactly $d^{\ast}$.

Hence, we have an interim result as follows.

$irr_t(G') = irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_k{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)| + (a - 1) - b + a^{\ast} - b^{\ast} + (c - 1)-d + c^{\ast} - d^{\ast}\\  =  irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)| + (a-b) + (a^{\ast}- b^{\ast}) + (c - d) + (c^{\ast} - d^{\ast}) - 2$.

By substituting the variables $b, d, b^{\ast}$, and $d^{\ast}$ as defined in Definition \ref{D-VSS1} the final result is as follows.

$irr_t(G') = irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)| + 2n - 2(b + b^{\ast} + d + d^{\ast}) - 2$, or;
$irr_t(G') = irr_t(G_1) + irr_t(G_2) + \sum \limits_{i=1}^{r} \sum\limits_{j=1}^{s}|d_{G_1}(u_i)- d_{G_2}(v_j)| + 2(a + a^{\ast} + c + c^{\ast}) -2n + 2$, follows.
\end{proof}

\ni Clearly $irr_t(G')$ is edge dependent in general but we have the following Corollary.

\begin{corollary}\label{Cor-2.6}
Let the degree sequence of graphs $G_1$ and $G_2$ be $(d_{G_1}(u_1) \leq d_{G_1}(u_2) \leq d_{G_1}(u_3) \leq \ldots\leq d_{G_1}(u_n))$ and $(d_{G_2}(v_1) \leq d_{G_2}(v_2) \leq d_{G_2}(v_3) \leq \ldots\ldots \leq d_{G_2}(v_m))$ respectively. If $d_{G_1}(u_i) = d_{G_2}(v_j)$ for some $i,j$ and $d_{G_1}(u_k) = d_{G_2}(v_l)$ for some $k,l$ and $G' = G_1\rightsquigarrow_{u_iv_l}G_2$ and $G'' = G_1\rightsquigarrow_{u_kv_j}G_2$ then, $irr_t(G') = irr_t(G'')$.
\end{corollary}
\begin{proof}
Begin the proof by choosing any vertex degree value $t_1$ in the degree sequence of $G_1$ and identify largest vertex index say, $i$ for which $d_{G_1}(u_i) = t_1$. Similarly, choose any vertex degree value $t_2$ in the degree sequence of $G_2$ and identify largest vertex index say, $l$ for which $d_{G_2}(v_l) = t_2$. Here, we have to consider the following cases.

\ni{\em Case 1:} With respect to $G' = G_1\rightsquigarrow_{u_iv_l}G_2$, using definition \ref{D-VSS1}, set the values as follows.

\begin{enumerate}\itemsep0mm
\item[(i)] $|V_1| = a = i-1$,
\item[(ii)] $|V_2| = b = n-i$,
\item[(iii)] $|V_3| = a^{\ast} = j$,
\item[(iv)] $|V_4| = b^{\ast} = m-j$
\item[(v)] $|V_5| = c = l - 1$,
\item[(vi)] $|V_6| = d = m- l$,
\item[(vii)] $|v_7| = c^{\ast} = k$,
\item[(viii)] $|V_8| = d^{\ast} = n-k$.
\end{enumerate}

Therefore, we have $2(n+m) -2((n-i) + (m-j) +(m- l) + (n-k)) -2 = 2(i+j+k+l -(n+m)) - 2$.

\ni{\em Case 2:} In respect of $G'' = G_1\rightsquigarrow_{u_kv_j}G_2$ and using definition \ref{D-VSS1}, set the values as follows.

\begin{enumerate}\itemsep0mm
\item[(i)] $|V_1| = a = k-1$,
\item[(ii)] $|V_2| = b = n- k$,
\item[(iii)] $|V_3| = a^{\ast} = l$,
\item[(iv)] $|V_4| = b^{\ast} = m-l$
\item[(v)] $|V_5| = c = j - 1$,
\item[(vi)] $|V_6| = d = m- j$,
\item[(vii)] $|v_7| = c^{\ast} = i$,
\item[(viii)] $|V_8| = d^{\ast} = n- i$.
\end{enumerate}

Therefore, here we have $2(n+m)-2((n-k)+(m-l)+(m-j)+(n-i))-2=2(k+l+j+i-(n+m))-2$.

Since Case 1 and Case 2 yields the same result, the result $irr_t(G') = irr_t(G'')$ follows from \ref{Thm-2.1}.
\end{proof}

An immediate consequence of Corollary \ref{Cor-2.6} is that for regular graphs $G_1$ and $G_2$ we have $irr_t(G_1\rightsquigarrow_{vu}G_2)_{\substack{u \in V(G_1)\\ v \in V(G_2)}}$ is a constant. This result is proved in the following proposition.

\begin{proposition}
For the regular graphs $G_1, G_2$ on $n, m$ vertices respectively with $d_{G_1}(u) \geq d_{G_2}(v)$ we have
\begin{equation*}
irr_t(G_1\rightsquigarrow_{uv}G_2)=
\begin{cases}
2(n+m) - 2, & \text {if $d_{G_1}(u) = d_{G_2}(v)$,}\\
n\cdot m|d_{G_1}(u)- d_{G_2}(v)| + 2(n-1), & \text {if $d_{G_1}(u) > d_{G_2}(v)$.}
\end{cases}
\end{equation*}
\end{proposition}
\begin{proof}
The proof follows immediately from definitions \ref{D-VSS1}, Remark \ref{Rem-2.3} and Corollary \ref{Cor-2.6}.
\end{proof}

We note that if $G_1$ and $G_2$ are of equal $k$-regularity, then $irr_t(G_1\rightsquigarrow_{uv}G_2)$ is independent of the \textit{$k$- degree} of the vertices.

\section{Total Irregularity Due to Edge-Transformation}

Consider a graph $G$ on $n = l_1 + l_2$ vertices and a cut edge $u_1v_1$. Let $G = (G_1 \cup G_2) + u_1v_1$, $u_1 \in V(G_1) = \{u_i: 1 \leq i \leq l_1\}$ and $v_1 \in V(G_2) = \{v_i: 1 \leq i \leq l_2\}$. Edge-transformation with respect to $u_1$ will be the graph $G^{u_iv_1}$ obtained by deleting the edge $u_1v_1$ and adding the edge $u_iv_1$ for any $i \neq 1$. We call $G_1$ the {\em master graph} and $G_2$ the {\em slave graph}.

\vspace{0.2cm}

Let us now introduce the notion of edge-transformation partitioning of a vertex set of a given graph as follows.

\begin{definition}\label{D-VSS2a}{\rm
The \textit{edge-transformation partitioning} of the vertex set $V(G)$ of a graph $G$ on $n$ vertices with at least one cut edge say $u_1v_1$, is defined to be $V_h = \{u_i, v_k:d_{G_1}(u_i) = d_{G_1}(u_1) -1\}$ and $d_{G_2}(v_k) = d_{G_1}(u_1) -1\}\cup \{u_1\}, h= |V_h|$, and $V_s = \{ u_i, v_k: d_{G_1}(u_i) > d_{G_1}(u_1)-1\}$ and $d_{G_2}(v_k) > d_{G_1}(u_1)-1\}, s=|V_s|$ and $V_t = \{ u_i, v_k: d_{G_1}(u_i) < d_{G_1}(u_1)-1\}$ and $d_{G_2}(v_k) < d_{G_1}(u_1)-1\}, t=|V_t|$.}
\end{definition}

\ni Invoking Definition \ref{D-VSS2a}, we now define certain vertex sets in $G$ as given below.

\begin{definition}{\rm
We define the certain important sets as follows. Let $V_{s_1} = \{u_j, v_k: d_{G_1}(u_j) \leq d_{G_1}(u_i)$ and $d_{G_2}(v_k) \leq d_{G_1}(u_i)$ and $u_j, v_k \in V_s \}, m = |V_{s_1}|$, and $V_{s_2} = \{u_j, v_k: d_{G_1}(u_j) > d_{G_1}(u_i)$ and $d_{G_2}(v_k) > d_{G_1}(u_i)$ and $u_j, v_k \in V_s \}, l = |V_{s_2}|$, and $V_{t_1} = \{u_j, v_k: d_{G_1}(u_j) \leq d(u_i)$ and $d_{G_2}(v_k) \leq d_{G_1}(u_i)$ and $u_j, v_k \in V_t \}, m_1 = |V_{t_1}|$, and $V_{t_2} = \{u_j, v_k: d_{G_1}(u_j) > d_{G_1}(u_i)$ and $d_{G_2}(v_k) > d_{G_1}(u_i)$ and $u_j, v_k \in V_t \}, l_1 = |V_{t_2}|$.}
\end{definition}

\ni In view of the above definitions, we propose the following theorem

\begin{theorem}\label{Thm-3.3}
For a graph $G$ with a cut edge $u_1v_1$, let $G-u_1v_1 = G_1 \cup G_2$. After edge-transformation in respect of $v_1$ we have
\begin{equation*}
irr_t(G^{u_iv_1})=
\begin{cases}
irr_t(G), & \text{if} \quad d_{G_1}(u_i) = d_{G_1}(u_1) -1,\\
irr_t(G) + 2m, & \text{if}\quad d_{G_1}(u_i) > d_{G_1}(u_1)-1,\\
irr_t(G) - 2(h + l_1), & \text{if}\quad d_{G_1}(u_i) < d_{G_1}(u_1)-1.
\end{cases}
\end{equation*}
\end{theorem}
\begin{proof}
If $d_{G_1}(u_i) = d_{G_1}(u_1) -1$,  then reducing $d_{G_1}(u_1)$ by 1, reduces the partial sum
$\sum \limits_{\substack{j=1\\w_j \in V_h}}^{h}|d_{G_1}(u_1)-d(w_j)|$ by exactly $(h-1)$. It also increases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_s}}^{s}|d_{G_1}(u_1)-d(w_j)|$ by exactly $s$ and finally it reduces the the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_t}}^{t}|d_{G_1}(u_1)-d(w_j)|$ by exactly $t$.

\ni {\em Case 1:} By increasing $d_{G_1}(u_i), u_i \in V_h$ by $1$, the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_h}}^{h}|d_{G_1}(u_i)-d(w_j)|$ increases by exactly $(h-1)$. It also decreases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_s}}^{s}|d_{G_1}(u_i)-d(w_j)|$ by exactly $s$ and finally it increases the the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_t}}^{t}|d_{G_1}(u_i)-d(w_j)|$ by exactly $t$. Hence, the result, $irr_t(G^{u_iv_1})=irr_t(G)-(h -1)+s-t +((h-1)-s+t)=irr_t(G)$ follows.

\ni {\em Case 2:} By increasing $d_{G_1}(u_i), u_i \in V_s$ by 1, the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_h}}^{h}|d_{G_1}(u_i)-d(w_j)|$ increases by exactly $h$. It also changes the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_s}}^{s}|d_{G_1}(u_i) - d(w_j)|$ by exactly $(m-1)-l$ and finally it increases the the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_t}}^{t}|d_{G_1}(u_i)-d(w_j)|$ by exactly $t$. Hence, the result, $irr_t(G^{u_iv_1}) = irr_t(G) - (h -1) + s - t + h + (m-1) - l + t =  irr_t(G) + 2m$ follows.

\ni {\em Case 3:} By increasing $d_{G_1}(u_i), v_i \in V_t$ by 1, the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_t}}^{h}|d_{G_1}(u_i)-d(w_j)|$ decreases by exactly $h$. It also decreases the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_t}}^{s}|d_{G_1}(u_i)-d(w_j)|$ by exactly $s$ and finally it changes the the partial sum $\sum \limits_{\substack{j=1\\w_j \in V_t}}^{t}|d_{G_1}(u_i)-d(w_j)|$ by exactly $(m_1-1)-l_1$.

Hence, the result $irr_t(G^{u_iv_1})=irr_t(G)-(h-1)+s-t-(h-1)-2-s+(m_1-1)-l_1=irr_t(G)-2(h+l_1)$ follows.
\end{proof}

It is to be noted Theorem \ref{Thm-3.3} provides an alternate proof for the following that lemma provided in \cite{ZYY}.

\begin{lemma}
{\rm \cite{ZYY}} Let $G'$ be the graph obtained from $G$ by branch-transformation from $u$ to $v$. Then $irr_t(G) > irr_t(G')$.
\end{lemma}

\ni Theorem \ref{Thm-3.3} can be extended to multi graphs also as explained in the following result.

\begin {corollary}
If multiple edges or loops are allowed in the graph or if edge-transformation is performed in a simple graph without a cut edge to give $G^{w_iv_1}_{w_i \in G}$ then, we have
\begin{equation*}
irr_t(G^{u_iv_1})=
\begin{cases}
irr_t(G), & \text{if} \quad d_{G_1}(u_i) = d_{G_1}(u_1) -1,\\
irr_t(G) + 2m, & \text{if}\quad d_{G_1}(u_i) > d_{G_1}(u_1)-1,\\
irr_t(G) - 2(h + l_1), & \text{if}\quad d_{G_1}(u_i) < d_{G_1}(u_1)-1.
\end{cases}
\end{equation*}
\end{corollary}
\begin{proof}
The proof of this theorem follows immediately as a consequence of Theorem \ref{Thm-3.3}.
\end{proof}

\section{Total Irregularities of Directed Graphs}\label{Sec-4}

In this section, we extend the concept of total irregularities of graphs mentioned in above sections to directed graphs. Since the edges of a digraph $D$ are directed edges and the vertices of $D$ has two types of degrees, in-degrees and out-degrees, we need to define two types of total irregularities for a digraph, which are called total in-degree irregularities and total out-degree irregularities.

\ni Let the vertices of a simple directed graph $D^\rightarrow$ on $n$ vertices be labelled as $v_i; i= 1,2,3,\ldots,n$ and let $d^+_{D^\rightarrow}(v_i)=d^+(v_i)$ and $d^-_{D^\rightarrow}(v_i)=d^-(v_i)$. Then, the notion of total in-irregularity of a given directed graph is introduced as follows.

\begin{definition}{\rm
The {\em total in-irregularity} of a directed graph $D$ with respect to the in-degree of all vertices of $D$, denoted by $irr_t^-(D^\rightarrow)$, is defined as $irr_t^-(D^\rightarrow)=\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}|d^-(v_i)-d^-(v_j)| =\sum\limits_{i=1}^{n} \sum \limits_{j=i+1}^{n}|d^-(v_i) - d^-(v_j)|$ or $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d^-(v_i) - d^-(v_j)|$.}
\end{definition}

\ni Similarly, the total out-irregularity of a digraph can also be defined as follows.

\begin{definition}{\rm
The {\em total out-irregularity} of a directed graph $D$ with respect to the out-degree of all vertices of $D$, denoted by $irr_t^+(D^\rightarrow)$, is defined as $irr_t^+(G^\rightarrow)=\frac{1}{2}\sum\limits_{i=1}^{n} \sum \limits_{j=1}^{n}|d^+(v_i) - d^+(v_j)| = \sum \limits_{i=1}^{n} \sum \limits_{j=i+1}^{n}|d^+(v_i)-d^+(v_j)|$ or $\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d^+(v_i)-d^+(v_j)|$. }
\end{definition}

Re-orientation of an arc or arc-transformation of an arc will find application in most classical applications of directed graphs like tournaments, transportation problems, flow analysis or alike.

\subsection{Total Irregularities of Directed Paths and Cycles}

The total in-irregularity and the total out-irregularity of a directed path are determined in the following proposition.

\begin{proposition}
For a directed path $P_n^\rightarrow$ which is consecutively directed from left to right for which vertices $v_1, v_n$ are called the \textit{start-vertex} and the \textit{end-vertex} respectively, we have
\begin{enumerate}\itemsep0mm
\item[(i)] $irr_t^-(P_n^\rightarrow) = irr_t^+(P_n^\rightarrow)=n-1$,
\item[(ii)]
\begin{equation*}
irr^-_t(P_n^{\rightarrow}) =
\begin{cases}
n-1, & \text{if the orientation of}~ (v_1, v_2) \text{is reversed}, \\
3n-5, & \text{if the orientation of}~ (v_i, v_{i+1}), 2 \leq i \leq (n-1) \\ & \text{is reversed}
\end{cases}
\end{equation*}
\item[(iii)] \begin{equation*}
irr^+_t(P_n^{\rightarrow}) =
\begin{cases}
n-1 , & \text {if the orientation of}~ (v_{n-1}, v_n) \text{is reversed},\\
3n-5, & \text {if the orientation of}~ (v_i,v_{i+1}), \leq i\leq n-2 \\ &   \text{is reversed}.
\end{cases}
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{proof}
The proof is obvious from the definition of total in-irregularity and total out-irregularity of a given digraph.
\end{proof}

The total in-irregularity and the total out-irregularity of a directed cycle are determined in the following proposition.

\begin{proposition}
For a directed cycle $C_n^\rightarrow$ which is consecutively directed clockwise we have
\begin{enumerate}\itemsep0mm
\item[(i)] $irr_t^-(C_n^\rightarrow)=irr_t^+(C_n^\rightarrow)=0$,
\item[(ii)] $irr_t^-(C_n^\rightarrow)=irr_t^+(C_n^\rightarrow)=2(n-1)$, if we reverse the orientation of any arc.
\end{enumerate}
\end{proposition}
\begin{proof}
The proof is obvious from the definition of total in-irregularity and total out-irregularity of a given digraph.
\end{proof}

\ni Through a simple change of Definition \ref{D-VSS2a} the in-arc-transformation partitioning in respect of $v_1$ and the out-arc-transformation partitioning in respect of $v_1$ can be defined.

\begin{definition}\label{Def-4.3}{\rm
The \textit{in-arc-transformation partitioning} with respect to a vertex $v_i$ of the vertex set $V(G)$ of a simple connected directed graph $G^\rightarrow$ on $n$ vertices is defined to be $V_h = \{v_i:d^-(v_i) = (d^-(v_1) -1)\} \cup \{v_1\}, h= |V_h|$, and $V_s = \{ v_i: d^-(v_i) > (d^-(v_1)-1)\}, s=|V_s|$ and $V_t = \{ v_i: d^-(v_i) < (d^-(v_1)-1)\}, t=|V_t|$.}
\end{definition}

\ni In view of Definition \ref{Def-4.3}, we define the following sets

\begin{definition}\label{Def-4.4}{\rm
Invoking the above definition, some vertex sets of a given digraph are defined as follows. $V_{s_1} = \{v_j:| d^-(v_j) \leq d^-(v_i), v_j \in V_s\}, m = |V_{s_1}|$ and $V_{s_2} = \{v_j: d^-(v_j) > d^-(v_i), v_j \in V_s\}, l = |V_{s_2}|$ and $V_{t_1} = \{v_j: d^-(v_j) \leq d^-(v_i), v_j \in V_t \}, m_1 = |V_{t_1}|$ and $V_{t_2} = \{v_j: d^-(v_j) > d^-(v_i), v_j \in V_t\}, l_1 = |V_{t_2}|$.}
\end{definition}

\begin{definition}\label{Def-4.5} {\rm
The \textit{out-arc-transformation partitioning} with respect to a vertex $v_i$ of the vertex set $V(G^\rightarrow)$ of a simple connected directed graph $G^\rightarrow$ on n vertices is defined to be $V_{h^{\ast}} = \{v_i:d^+(v_i) = (d^+(v_1) -1)\} \cup \{v_1\}, h^{\ast}= |V_{h^{\ast}}|$, and $V_{s^{\ast}} = \{ v_i: d^+(v_i) > (d^+(v_1)-1)\}, s^{\ast}=|V_{s^{\ast}}|$ and $V_{t^{\ast}} = \{ v_i: d^+(v_i) < (d^+(v_1)-1)\}, t^{\ast}=|V_{t^{\ast}}|$.}
\end{definition}

\ni In view of Definition \ref{Def-4.5}, we define the following sets

\begin{definition}\label{Def-4.6}{\rm
Invoking the above definition, some vertex sets of a given digraph are defined as follows. $V_{s^{\ast}_1} = \{v_j: d^+(v_j) \leq d^+(v_i), v_j \in V_s\}, m^{\ast} = |V_{s^{\ast}_1}|$ and $V_{s^{\ast}_2} = \{v_j: d^+(v_j) > d^+(v_i), v_j \in V_s\}, l^{\ast} = |V_{s^{\ast}_2}|$ and $V_{t^{\ast}_1} = \{v_j: d^+(v_j) \leq d^+(v_i), v_j \in V_t \}, m^{\ast}_1 = |V_{t^{\ast}_1}|$ and $V_{t^{\ast}_2} = \{v_j: d^+(v_j) > d^+(v_i), v_j \in V_t\}, l^{\ast}_1 = |V_{t^{\ast}_2}|$. }
\end{definition}

\ni Analogous to Theorem \ref{Thm-3.3}, we propose the following result.

\begin{proposition}
Consider a simple connected directed graph $G$. After in-arc-transformation in respect of $v_1$ we have
\begin{enumerate}
\item[(i)]
\begin{equation*}
irr_t^-(G^{v_iu_1})=
\begin{cases}
irr_t^-(G) , & \text{if}\quad d^-(v_i) = d^-(v_1) -1\\
irr_t^-(G) + 2m, & \text{if}\quad d^-(v_i) > d^-(v_1)-1, \\
irr_t^-(G) - 2(h + l_1), & \text{if}\quad d^-(v_i) < d^-(v_1)-1
\end{cases}
\end{equation*}
and
\item[(ii)]
\begin{equation*}
irr_t^+(G^{v_iu_1})=
\begin{cases}
irr_t^+(G) , & \text {if}\quad d^+(v_i)=d^+(v_1)-1,\\
irr_t^+(G) + 2m^{\ast}, & \text {if}\quad d^+(v_i) > d^+(v_1)-1,\\
irr_t^+(G) - 2(h^{\ast} + l^{\ast}_1), & \text {if}\quad d^+(v_i) < d^+(v_1)-1.
\end{cases}
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{proof}
The proof is similar to Theorem \ref{Thm-3.3}.
\end{proof}

\subsection{Total Irregularities of Directed Complete Graphs}

In this section, we initiate a study on the two types of irregularities of directed complete graphs. Consider a complete undirected graph $K_n$ and label the vertices $v_1, v_2, v_3, \ldots, v_n$. Assign direction the edges of $K_n$ to get a directed graph, with $K_n$ as its underlying graph, in such a way that the edge $v_iv_j$ becomes the  arc $(v_i,v_j)$ of this directed graph if $i<j$. We denote this directed graph by $K^\rightarrow_n$. The following lemma discusses the two types of irregularities of $K^\rightarrow_n$.

\begin{lemma}\label{Lem-4.8}
 For the directed complete graph $K^{\rightarrow}_n$, the total irregularities are given by $irr_t^+ (K^{\rightarrow}_n) = irr_t^- (K^\rightarrow_n)= \sum \limits_{i=1}^{n-1}\sum \limits_{j=1}^{i}j =\frac{1}{6}n(n^2-1)$.
\end{lemma}
\begin{proof}
The orientation results in an in-degree sequence $(0,1,2,\ldots,(n-1))$ and an out-degree sequence $(n-1,n-2,n-3,\ldots,0)$. Choose the $k$-th entry of the in-degree sequence. We know that the $k$-th term is given by $\sum\limits_{j=k+1}^{n} |d^-(v_k)-d^-(v_j)|=\sum \limits_{i=1}^{n-(k+1)}i$. Also, we have $irr_t^-=\sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d^-(v_i) - d^-(v_j)|$ and hence $irr_t^-(K^\rightarrow_n)=\sum\limits_{i=1}^{n-1}i+\sum \limits_{i=1}^{n-2}i+\ldots +\sum\limits_{i=1}^{n-(n-1)}i=\sum\limits_{i=1}^{n-1}\sum\limits_{j=1}^{i}j=\frac{1}{6}n(n^2-1)$. Furthermore, since the out-degree sequence is a mirror image of the in-degree sequence and $irr_t^+ = \sum \limits_{i=1}^{n-1} \sum \limits_{j=i+1}^{n}|d^+(v_i) - d^+(v_j)|$, the result follows similarly.
\end{proof}

\ni A general application this study can be the following.

Consider any connected undirected graph $G$ on $n$ vertices and label its vertices randomly by $v_1, v_2, v_3, \ldots, v_n$. Assign direction to the edges of the graph $G$ to be arcs according to the condition mentioned above and refer to the directed graph as the {\em root directed graph}, $G^\rightarrow_{root-graph}$. Then, calculate both $irr_t^+(G^\rightarrow_{root-graph})$ and $irr_t^-(G^\rightarrow_{root-graph})$. In a {\em derivative graph} $G^\rightarrow_{derivative}$ identify all arcs which were re-oriented or subjected to arc-transformation and apply the applicable results to recursively determine the total in-irregularity and total out-irregularity.

Consider the complete bipartite graph $K_{(m, n)}$ and call the $m$ vertices in the first bipartition by {\em left-side vertices} and the $n$ vertices in the second bipartition by {\em right-side vertices}. Assign directions to the edges of $K_{m,n}$ strictly from {\em left-side vertices} to \textit{right-side vertices} to obtain $K^{l \rightarrow r}_{m, n}$.

\begin{proposition}\label{Prop-4.9}
For the directed graph $K^{l \rightarrow r}_{m,n}$, we have $irr^-_t(K^{l \rightarrow r}_{m,n}) = m^2n$ and $irr^+_t(K^{l \rightarrow r}_{m,n}) = mn^2$.
\end{proposition}
\begin{proof}
The orientation of the directed complete bipartite graph $K^{l \rightarrow r}_{m,n}$ results in the in-degree sequence $(\underbrace{0, 0, \ldots, 0,}_{m-entries} \underbrace{m, m, \ldots, m}_{n-enties})$ and the out-degree sequence $( \underbrace{n,n,\ldots,n,}_{m-entries}\\ \underbrace{0, 0, \ldots, 0}_{n-enties})$. Here, we have the following cases.

\ni \textit{Case 1:} For the above mentioned in-degree sequence of $K^{l \rightarrow r}_{m,n}$, we have the sum $\sum\limits_{i=1}^{(m+n)-1}\sum\limits_{j=(i+1)}^{(m+n)}|d^-(v_i)-d^-(v_j)|$ results in the value $m$, ($mn$ times) and $0$, ($(m+n)-2$ times). Hence, $irr_t^-(K^{l \rightarrow r}_{(m,n)}) = m^2n$.

\ni \textit{Case 2:} For the above mentioned out-degree sequence of $K^{l \rightarrow r}_{m,n}$, we have the sum $\sum\limits_ {i=1}^{(m+n)-1}\sum\limits_{j= (i+1)}^{(m+n)}|d^+(v_i)-d^+(v_j)|$ results in the value $n$, ($mn$ times) and $0$, ($(m+n)-2$) times).  Hence, $irr_t^+(K^{l \rightarrow r}_{(1, n)}) = mn^2$. This completes the proof.
\end{proof}

Invoking from Proposition \ref{Prop-4.9}, we note that for the directed bipartite graph $K^{l \rightarrow r}_{1,n}$, we have $irr^-_t(K^{l \rightarrow r}_{1,n}) = n$ and $irr^+_t(K^{l \rightarrow r}_{(1, n)}) = n^2$ and $irr^-_t(K^{l \rightarrow r}_{m,1}) = m^2$ and $irr^+_t(K^{l\rightarrow r}_{m,1})=m$.

\ni The following is a challenging and interesting problem in this context.

\begin{problem}{\rm
Describe an efficient algorithm to determine $irr_t^- (G^\rightarrow_{derivative})$ and $irr_t^+(G^\rightarrow_{derivative})$  from $irr_t^-(G^\rightarrow_{root-graph})$ and $irr_t^+(G^\rightarrow_{root-graph})$.}
\end{problem}

\section{Conclusion}

In this paper, we have studied certain types of total irregularities of certain graphs and digraphs. More problems in this area still remain unsettled. More studies on different types of irregularities for different graph classes, graph operations, graph products and on certain associated graphs such as line graphs and total graphs of given graphs and digraphs remain open. All these facts indicates that there is a wide scope for further investigations in this area.

config: arxiv


text:
\begin{document}
\title{On the moments of the moments of $\zeta(1/2+it)$}

\begin{abstract}
Taking $t$ at random, uniformly from $[0,T]$, we consider the $k$th moment, with respect to $t$, of the random variable corresponding to the $2\beta$th moment of
$\zeta(1/2+ix)$ over the interval $x\in(t, t+1]$, where $\zeta(s)$ is the Riemann zeta function.   We call these the `moments of moments' of the Riemann zeta function, and present a conjecture for their asymptotics, when $T\to\infty$, for integer $k,\beta$.  This is motivated by comparisons with results for the moments of moments of the characteristic polynomials of random unitary matrices and is shown to follow from a conjecture for the shifted moments of $\zeta(s)$ due to Conrey, Farmer, Keating, Rubinstein, and Snaith~\cite{cfkrs2}.  Specifically, we prove that a function which, the shifted-moment conjecture of~\cite{cfkrs2} implies, is a close approximation to the moments of moments of the zeta function does satisfy the asymptotic formula that we conjecture.  We motivate as well similar conjectures for the moments of moments for other families of primitive $L$-functions.
\end{abstract}

\author{E. C. Bailey}
\address{School of Mathematics, University of Bristol, Bristol, BS8 1UG, United Kingdom}
\email{e.c.bailey@bristol.ac.uk}

\author{J. P. Keating}
\address{Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom}
\email{jon.keating@maths.ox.ac.uk}

\maketitle

\section{Introduction}\label{sec:intro}

\subsection{Moments of moments}\label{sec:mom}

Moments of $L$-functions play a central role in analytic number theory.  It is a long-standing conjecture for the Riemann zeta function, $\zeta(s)$, that for the $2\beta$th moment on the critical line $\RE(s)=1/2$, as $T\to\infty$

\begin{equation}\label{zeta_moment}
  M_\beta(T)\coloneqq\frac{1}{T}\int_0^T|\zeta(\tfrac{1}{2}+it)|^{2\beta}dt\sim a_\beta c_\beta \left(\log \frac{T }{2\pi}\right)^{\beta^2},
\end{equation}
where $a_\beta$ is an arithmetic factor given in terms of an Euler product and $c_\beta$ is discussed below (see also, for example,~\cite{confar00}).  Hardy and Littlewood~\cite{harlit18} proved this asymptotic for $M_1(T)$, and Ingham~\cite{ing26} proved it for $M_2(T)$.  Their formulae, together with heuristic calculations by Conrey and Ghosh~\cite{congho92} and Conrey and Gonek~\cite{congon01} for $M_3(T)$ and $M_4(T)$ respectively, suggest that $c_\beta \cdot (\beta^2)!\in\mathbb{N}$ for $\beta\in\mathbb{N}$.  Beyond $\beta=2$ there are no rigorous asymptotic results for $M_\beta(T)$.  Ramachandra~\cite{ram80} and Heath-Brown~\cite{heabro81} have established the lower bound $M_\beta(T)\gg (\log\frac{T}{2\pi})^{\beta^2}$ for positive, rational $\beta$, and Soundararajan and Radziwi{\l}{\l}~\cite{sourad13} extended the result to all $\beta\geq 1$.  Upper bounds of the correct size are known, conditional on the Riemann hypothesis, due to arguments of Soundararajan~\cite{sou09} and Harper~\cite{har13}.

There is now a relatively well developed understanding of the conjectural connections between the Riemann zeta function and the characteristic polynomials of random unitary matrices (cf.~\cite{keasna00a, cfkrs2}).  The notation we use for such a characteristic polynomial is
\begin{equation}
  P_N(A,\theta)=\det(I-Ae^{-i\theta}),
\end{equation}
for $A\in U(N)$, the group of $N\times N$ unitary matrices.  By comparing densities of eigenvalues and zeros, one identifies the matrix size $N$ with a height $\log\frac{T}{2\pi}$ up the critical line.  Under this analogy, the random matrix average corresponding to~\eqref{zeta_moment} is
\begin{equation}
  \mathcal{M}_\beta(N)\coloneqq\int_{U(N)}|P_N(A,\theta)|^{2\beta}dA,
\end{equation}
where the measure $dA$ is the Haar measure on $U(N)$. $\mathcal{M}_\beta(N)$ can be computed when $\RE(\beta)>-1/2$, giving the following result.
\begin{thm}[Keating-Snaith~\cite{keasna00a}]
  For $\RE(\beta)>-1/2$,
  \[\mathcal{M}_\beta(N)=\prod_{j=1}^N\frac{\Gamma(j)\Gamma(j+2\beta)}{(\Gamma(j+\beta))^2}.\]
\end{thm}
It follows that asymptotically, as $N\to\infty$,
\begin{equation}\label{rmt_moment}
  \mathcal{M}_\beta(N)\sim \frac{\mathcal{G}^2(1+\beta)}{\mathcal{G}(1+2\beta)}N^{\beta^2}\,
\end{equation}
where $\mathcal{G}(s)$ is the Barnes $\mathcal{G}$-function.  One sees that by identifying $N$ with $\log \frac{T}{2\pi}$, $\mathcal{M}_\beta(N)$ grows asymptotically like $M_\beta(T)$, leading to the conjecture~\cite{keasna00a} that
\begin{equation}\label{coefficient_conj}
  c_\beta=\frac{\mathcal{G}^2(1+\beta)}{\mathcal{G}(1+2\beta)}.
\end{equation}
This matches the known values when $\beta=1$~\cite{harlit18} and $\beta=2$~\cite{ing26}, as well as the values previously conjectured for $\beta=3$~\cite{congho92} and $\beta=4$~\cite{congon01} using heuristic number-theoretic methods.  Recently, these heuristic number-theoretic methods have been extended to calculate all integer moments of the zeta function on its critical line, giving results that agree precisely with~\eqref{coefficient_conj} and with its extension to include lower order terms~\cite{cfkrs2} -- see~\cite{ck1, ck2, ck3, ck4, ck5}.  (The correctness of the formula in the function field setting follows, in the appropriate limit, from equidistribution; cf.~\cite{katsar99}.)

In \eqref{zeta_moment}, the moments are defined with respect to an average over an asymptotically long stretch of the critical line.  Recently there has been interest in the fluctuations in the values of moments defined with respect to averages over random short intervals, e.g.~intervals of constant length.  The moments of these fluctuations are therefore the {\it moments of moments}.   These moments of moments have played a central role in our understanding of the extreme values taken by the zeta function over short intervals~\cite{fyodorov12, fyodorov14, harper1, harper2}, and a precise conjecture for the local maximum is due to Fyodorov and Keating~\cite{fyodorov14}. Recent work of Najnudel~\cite{Najnudel} and Arguin et al.~\cite{abbrs} has led to a verification to leading order of this conjecture, with an almost sharp upper bound, including the predicted subleading order, due to Harper~\cite{harper2}.  At time of writing, the strongest result is due to Arguin et al.~\cite{argbourad20}, whose work settles the upper bound part of the conjecture including the predicted tail. Extreme values and moments over mesoscopic and macroscopic intervals, as well as the transition between the two, are also of interest and have been studied by Arguin et al.~\cite{argouirad}.

Precisely, we define the moments of moments here as follows.

\begin{definition}
  For $T>0$ and $\RE(\beta)>-1/2$
  \begin{align}\label{def:mom_zeta}
    \mom_{\zeta_T}(k,\beta)\coloneqq \frac{1}{T}\int_0^T\left(\int_{t}^{t+1}|\zeta(\tfrac{1}{2}+ih)|^{2\beta}dh\right)^kdt.
  \end{align}
\end{definition}
Choosing to average over intervals of length $1$ is simply for notational convenience\footnote{See also~\cite{fyodorov12, fyodorov14} where the moments of moments were introduced; there the intervals are instead of length $2\pi$.}; the results stated in section~\ref{sec:results} hold for any interval that is $O(1)$ as $T\rightarrow\infty$.

The corresponding `moments of moments' for the characteristic polynomials of random unitary matrices may similarly be defined by
\begin{equation}\label{unitary_mom}
  \mom_{U(N)}(k,\beta)\coloneqq\int_{U(N)}\left(\frac{1}{2\pi}\int_0^{2\pi}|P_N(A,\theta)|^{2\beta}d\theta\right)^kdA.
\end{equation}
These have recently been computed for integer values of $k,\beta$ in~\cite{baikea19, ak19}.  This suggests a conjecture for $\mom_{\zeta_T}(k,\beta)$, which we state in section~\ref{sec:results}.  We show that this conjecture follows from another conjecture due to Conrey et al.~\cite{cfkrs2} concerning the shifted moments of the zeta function.  Our proof of this follows similar lines to the corresponding proof in~\cite{baikea19}, which is based on an expression derived in~\cite{cfkrs1} for the shifted moments of characteristic polynomials of random matrices.  Moreover, by appealing to results of Assiotis, Bailey, and Keating~\cite{abk19} for the other classical compact matrix groups, in section~\ref{sec:l-function_conj} we give conjectures of moments of moments for other families of $L$-functions.

\subsection{Statement of Result}\label{sec:results}

In order to develop a conjecture for the asymptotics of $\mom_{\zeta_T}(k,\beta)$ for large $T$, we use the large $N$ behaviour of $\mom_{U(N)}(k,\beta)$. Fyodorov, Hiary and Keating~\cite{fyodorov12}, and Fyodorov and Keating~\cite{fyodorov14} conjectured the asymptotic form\footnote{Here and throughout, we write $A(t)\sim B(t)$ as $t\rightarrow\infty$ for $A(t)/B(t)\rightarrow 1$ as $t\rightarrow\infty$.} of $\mom_{U(N)}(k,\beta)$ as matrix size $N\to\infty$,
\begin{equation}\label{conj:FK}
  \mom_{U(N)}(k,\beta)\sim
  \begin{cases}
    \left(\frac{\mathcal{G}^2(1+\beta)}{\mathcal{G}(1+2\beta)\Gamma(1-\beta^2)}\right)^k
    \Gamma(1-k\beta^2)N^{k\beta^2},
    &\text{if }k<1/\beta^2,
    \\
    c_{k, \beta}N^{k^2\beta^2-k+1},
    &\text{if }k>1/\beta^2,
  \end{cases}
\end{equation}
where again $\mathcal{G}(s)$ is the Barnes $\mathcal{G}$-function and $c_{k,\beta}$ is an unspecified function of $k$ and $\beta$.

For integer $k,\beta$, Bailey and Keating~\cite{baikea19} determined the asymptotic growth of $\mom_{U(N)}(k,\beta)$:
\begin{thm}[Bailey-Keating~\cite{baikea19}]
  For $k,\beta\in\mathbb{N}$,
  \begin{equation}\label{eq:baikea}
    \mom_{U(N)}(k,\beta)=c_{k,\beta}N^{k^2\beta^2-k+1}(1+O\left(\tfrac{1}{N}\right)),
  \end{equation}
  where the leading order coefficient $c_{k,\beta}$ is explicitly given in terms of an integral over a certain simplex.  Furthermore, $\mom_{U(N)}(k,\beta)$ is a polynomial in $N$.
\end{thm}

Via an alternative representation (involving Toeplitz determinants and a Riemann-Hilbert analysis) Claeys and Krasovsky~\cite{CK} calculated $\mom_{U(N)}(2,\beta)$ for $\beta>-1/4$, and determined an alternative expression for the coefficient $c_{2,\beta}$.  This calculation was recently extended by Fahs~\cite{fahs} to all $k\in\mathbb{N}$ and non-negative, real $\beta$, though without an explicit expression for $c_{k,\beta}$.  For integer values of $k$ and $\beta$, Assiotis and Keating showed how one of the approaches developed in~\cite{baikea19} leads to a connection with lattice point counts and gave yet another expression for $c_{k,\beta}$, as a volume of a certain region involving continuous Gelfand-Tsetlin patterns with constraints.  In a recent paper~\cite{baikea21}, the present authors computed the analogue of \eqref{unitary_mom} for a different logarithmically correlated process - branching random walks - with $k\in\mathbb{N}$ and $\beta\in\mathbb{R}$.  The resulting expression agrees asymptotically with \eqref{eq:baikea}.

By making the natural exchanges in \eqref{eq:baikea}, we therefore expect the following to hold.
\begin{conjecture}
  Let $k,\beta\in\mathbb{N}$.  Then,
  \begin{equation}\label{conj:main}
    \mom_{\zeta_T}(k,\beta)=\alpha_{k,\beta}c_{k,\beta}\left(\log\tfrac{T}{2\pi}\right)^{k^2\beta^2-k+1}\left(1+O_{k,\beta}\left(\log^{-1} T\right)\right),
  \end{equation}
  where $c_{k,\beta}$ is the same coefficient appearing in \eqref{eq:baikea}, and $\alpha_{k,\beta}$ contains the arithmetic information and takes the form of an Euler product over primes (cf. \eqref{zeta_moment}).
\end{conjecture}
A similar conjecture, as well as further motivation for studying moments of moments, can be found in Fyodorov and Keating~\cite{fyodorov14}.

Furthermore, we expect that the moments of moments, when suitably smoothed, will have the general form $\operatorname{Poly}_{k^2\beta^2-k+1}\left(\log\tfrac{T}{2\pi}\right)+O_{k,\beta}\left(T^{-\delta}\right)$, for some $\delta>0$, where $\operatorname{Poly}_n(x)$ is a polynomial of degree $n$ in the variable $x$.

For integer $k$ we can rewrite $\mom_{\zeta_T}(k,\beta)$ by expanding and switching the order of integration,
\begin{equation}\label{eq:mom_expanded}
  \mom_{\zeta_T}(k,\beta)=\frac{1}{T}\int_{0}^{1}\cdots\int_{0}^{1}\int_0^T\prod_{j=1}^k|\zeta(\tfrac{1}{2}+i(t+h_j))|^{2\beta}dt dh_1\cdots dh_k.
\end{equation}

We now apply a conjecture of Conrey et al.~\cite{cfkrs2} concerning shifted moments of the zeta function\footnote{The conjecture in~\cite{cfkrs2} is more general in that it allows for different shifts, as well as integrating against different weight functions.} such as those appearing in the inner integral in \eqref{eq:mom_expanded}.  Specialising this conjecture to our situation (and to our notation) we have the following.

\begin{conjecture}[Conrey et al.~\cite{cfkrs2}]
  Take $k, \beta\in\mathbb{N}$, and $\underline{h}=(h_1,\dots,h_k)$ with $h_j\in\mathbb{R}$.  Then
  \begin{equation}\label{conj:cfkrs}
    \frac{1}{T}\int_{0}^T\prod_{j=1}^k|\zeta(\tfrac{1}{2}+i(t+h_j))|^{2\beta}dt=\frac{1}{T}\int_0^TP_{k,\beta}(\log\tfrac{t}{2\pi},\underline{h})dt+O(T^{-\delta}),
  \end{equation}
  for some $\delta>0$, where
  \begin{equation}\label{def:p}
    P_{k,\beta}(x;\underline{h})\coloneqq\frac{(-1)^{k\beta}}{(k\beta)!^2}\frac{1}{(2\pi i)^{2k\beta}}\oint\cdots\oint\frac{G(z_1,\dots,z_{2k\beta})\Delta(z_1,\dots,z_{2k\beta})^2}{e^{\frac{x}{2}\sum_{j=1}^{k\beta}z_{k\beta+j}-z_j}\prod_{j=1}^{2k\beta}\prod_{l=1}^{k}(z_j-ih_l)^{2\beta}}dz_1\cdots dz_{2k\beta}
  \end{equation}
  in which $\Delta(x_1,\dots,x_n)=\prod_{i<j}(x_j-x_i)$ is the Vandermonde determinant, and the contours are small circles surrounding the poles.  Additionally,
  \begin{equation}\label{def:g_func}
    G(z_1,\dots,z_{2k\beta})\coloneqq A_{k\beta}(z_1,\dots,z_{2k\beta})\prod_{1\leq i\leq k\beta < j\leq 2k\beta}\zeta(1+z_i-z_j)
  \end{equation}
  which includes the Euler product\footnote{Throughout we write $e(\theta)=\exp(2\pi i\theta)$.}
  \begin{equation}\label{def_A}
    A_{k\beta}(\underline{z})\coloneqq\prod_p \prod_{1\leq l\leq k\beta < m\leq 2k\beta} (1-p^{z_m-z_l-1})\int_0^1\prod_{j=1}^{k\beta}\left(1-\frac{e(\theta)}{p^{\frac{1}{2}+z_j}}\right)^{-1}\left(1-\frac{e(-\theta)}{p^{\frac{1}{2}-z_{k\beta+j}}}\right)^{-1}d\theta.
  \end{equation}
\end{conjecture}

In light of this, we define
\begin{equation}\label{mom_p_def}
  \mom_{P_{k,\beta}}(T)\coloneqq  \frac{1}{T}\int_{0}^{1}\cdots\int_{0}^{1}\int_0^TP_{k,\beta}(\log\tfrac{t}{2\pi},\underline{h})dtdh_1\cdots dh_k.
\end{equation}
Under the conjecture of Conrey et al., $\mom_{P_{k,\beta}}(T)$ should approximate $\mom_{\zeta_T}(k,\beta)$ up to a power saving in $T$.  In \eqref{mom_p_def}, the length of the integral over $h_1,\dots, h_k$ has been chosen to be $1$ in order to be explicit.  It is readily apparent from the proof that the techniques extend to any $O(1)$ interval (with respect to $T\rightarrow\infty$).

Our main result is the following.
\begin{theorem}\label{thm:main}
  For $k,\beta\in\mathbb{N}$, one has that as $T\rightarrow\infty$
  \begin{equation}
    \mom_{P_{k,\beta}}(T)=\alpha_{k,\beta}\gamma_{k,\beta}\left(\log\tfrac{T}{2\pi}\right)^{k^2\beta^2-k+1}(1+O\left(\log^{-1}T\right)),
  \end{equation}
  where the term $\alpha_{k,\beta}=A_{k\beta}(0,\dots,0)$ contains the arithmetic information and $A_{k\beta}$ is as defined in \eqref{def_A}, and $\gamma_{k,\beta}$ denotes the remaining, non-arithmetic, contribution (further details can be found within the proof, see in particular \eqref{eq:gamma_coeff} and \eqref{pre_incomplete_gamma}).
\end{theorem}

Therefore $ \mom_{P_{k,\beta}}(T)$, which, according to the conjecture for the shifted moments of~\cite{cfkrs2} is a close approximation to $\mom_{\zeta_T}(k,\beta)$, does satisfy the same asymptotic formula as we conjecture holds for the moments of moments of the zeta function.

\subsection{Conjectures for other families of $L$-functions}\label{sec:l-function_conj}

Following Katz and Sarnak~\cite{katsar99} (see also~\cite{confar00, keasna00b, cfkrs2}), we can also consider other families of $L$-functions.  These families are split by `symmetry type',  indicating which random matrix ensemble (unitary, symplectic, or orthogonal) should be used as comparison.

Here we give an example from each family of the relevant `moments of moments', and the corresponding conjecture for their growth.  For each symmetry type, the moments of moments comprise an average through the family and an average over small shifts near a symmetry point. The conjectures for the symplectic and orthogonal families are derived from the relevant random matrix calculations of Assiotis et al.~\cite{abk19}, which we briefly recap here.

Let $\mathscr{G}(N)\in\{ U(N), \operatorname{Sp}(2N), \operatorname{SO}(2N)\}$, where $\operatorname{Sp}(2N)$ is the group of $2N\times 2N$ unitary symplectic matrices, and $\operatorname{SO}(2N)$ the group of $2N\times 2N$ orthogonal matrices with unit determinant.  A generalisation of the definition given by \eqref{unitary_mom} is the following.
\begin{equation}
  \mom_{\mathscr{G}(N)}(k,\beta)\coloneqq\int_{\mathscr{G}(N)}\left(\frac{1}{2\pi}\int_0^{2\pi}|P_N(A,\theta)|^{2\beta}d\theta\right)^kdA,
\end{equation}
where the external average is over the relevant Haar measure for the chosen matrix group.

Assiotis et al.~\cite{abk19} computed integer moments of moments for $\mathscr{G}(N)=\operatorname{Sp}(2N)$ and $\mathscr{G}(N)=\operatorname{SO}(2N)$, and their results are summarised here.

\begin{thm}[Assiotis et al.~\cite{abk19}]\label{MainTheoremSymplectic}
  Let $\mathscr{G}(N)=\operatorname{Sp}(2N)$. Let $k,\beta \in \mathbb{N}$. Then,
  \begin{align}\label{eq:mom_sympl}
    \mom_{\operatorname{Sp}(2N)}\left(k,\beta\right)=\mathfrak{c}_{k,\beta}N^{k\beta(2k\beta+1)-k}\left(1+O_{k,\beta}\left(\tfrac{1}{N}\right)\right),
  \end{align}
  \sloppy where the leading order term coefficient $\mathfrak{c}_{k,\beta}$ is the volume of a certain convex region\footnote{Differing from the coefficient appearing in~\eqref{eq:baikea}.}. Moreover, $\mom_{\operatorname{Sp}(2N)}\left(k,\beta\right)$ is a polynomial function in $N$.
\end{thm}

\begin{thm}[Assiotis et al.~\cite{abk19}]\label{MainTheoremOrthogonal}
  Let $\mathscr{G}(N)=\operatorname{SO}(2N)$.  Let $k, \beta\in\mathbb{N}$.  Then,
  \begin{align}
    \mom_{\operatorname{SO}(2N)}(1,1)&=2(N+1)\\
    \intertext{otherwise,}
    \mom_{\operatorname{SO}(2N)}(k,\beta)&=\tilde{\mathfrak{c}}_{k,\beta}N^{k\beta(2k\beta-1)-k}\left(1+O_{k,\beta}\left(\tfrac{1}{N}\right)\right),\label{eq:mom_orthog}
  \end{align}
  where the leading order term coefficient $\tilde{\mathfrak{c}}_{k,\beta}$ is given as a sum of volumes of certain convex regions\footnote{Again, differing from $c_{k,\beta}$ in \eqref{eq:baikea}, as well as $\mathfrak{c}_{k,\beta}$ given in \eqref{eq:mom_sympl}.}. Moreover, $\mom_{\operatorname{SO}(2N)}(k,\beta)$ is a polynomial function in $N$.
\end{thm}
Note that these results differ from the unitary case, where the degree of the polynomial is $k^2\beta^2-k+1$,  see~\eqref{eq:baikea}.

Returning to the families of $L$-functions, the archetypal example of a unitary $L$-function is the Riemann zeta function, as discussed in section~\ref{sec:mom}. We rewrite the family here for context,
\begin{equation}\label{ex:unitary_family}
  \{\zeta(\tfrac{1}{2}+it)\; |\; t\geq 0\}.
\end{equation}
To each family there is an associated `height' by which the elements are ordered.  For the family given by \eqref{ex:unitary_family}, one orders by the height up the critical line, $t$. The moments of moments are defined by \eqref{def:mom_zeta}, and their conjectured asymptotic is given by \eqref{conj:main}.

Quadratic Dirichlet $L$-functions provide an example of a symplectic family:
\begin{equation}\label{ex:symplectic_family}
  \{L(s, \chi_d)\; |\;\ d\text{ a fundamental discriminant, } \chi_d(n)=\left(\tfrac{d}{n}\right)\},
\end{equation}
where the family is ordered by $|d|$. For $\RE(s)>1$
\begin{equation}
  L(s, \chi_d)\coloneqq\sum_{n=1}^\infty \frac{\chi_d(n)}{n^s},
\end{equation}
and otherwise $L(s,\chi_d)$ is defined by its analytic continuation.  The moments of moments are defined as
\begin{equation}\label{mom_symplectic}
  \mom_{L_{\chi_d}}(k,\beta)\coloneqq\frac{1}{D^*}\sideset{}{^*}\sum_{|d|\leq D}\left(\int_{0}^{1}L(\tfrac{1}{2}+it,\chi_d)^{2\beta} dt\right)^k,
\end{equation}
where the sum is only over fundamental discriminants $d$, and $D^*$ denotes the number of terms in the sum. We hence conjecture that for $k, \beta\in \mathbb{N}$, as $D\rightarrow\infty$,
\begin{equation}
  \mom_{L_{\chi_d}}(k,\beta)=\eta_{k,\beta}\mathfrak{c}_{k,\beta}\left(\log D\right)^{k\beta(2k\beta+1)-k}\left(1+O_{k,\beta}\left(\log^{-1}D\right)\right),
\end{equation}
where $\mathfrak{c}_{k,\beta}$ corresponds to the leading order coefficient in \eqref{eq:mom_sympl}, and $\eta_{k,\beta}$ contains the arithmetic information (see for example~\cite{cfkrs2}).

For an orthogonal example we take the family of twisted elliptic curve $L$-functions,
\begin{equation}\label{ex:orthogonal_family}
  \{L_E(s, \chi_d)\; |\ d\text{ a fundamental discriminant, } \chi_d(n)=\left(\tfrac{d}{n}\right)\},
\end{equation}
which is ordered by $|d|$ and where the $L$-function is defined by
\begin{equation}
  L_E(s,\chi_d)\coloneqq\sum_{n=1}^\infty \frac{a_n \chi_d(n)}{n^s}
\end{equation}
for $\RE(s)>1$ and by analytic continuation otherwise. The coefficients $a_n$ are the weightings from the $L$-function associated to $E$, normalised so that the critical line is shifted to $\RE(s)=1/2$. The moments of moments are then defined as
\begin{equation}\label{mom_orthogonal}
  \mom_{L_{E}}(k,\beta)\coloneqq\frac{1}{D^*}\sideset{}{^*}\sum_{|d|\leq D}\left(\int_{0}^{1}L_E(\tfrac{1}{2}+it,\chi_d)^{2\beta} dt\right)^k,
\end{equation}
where again the sum is only over fundamental discriminants $d$, and $D^*$ denotes the number of terms in the sum. We hence conjecture that for $k, \beta\in \mathbb{N}$ and $k\neq 1\neq \beta$, as $D\rightarrow\infty$,
\begin{equation}
  \mom_{L_{E}}(k,\beta)=\xi_{k,\beta}\tilde{\mathfrak{c}}_{k,\beta}\left(\log D\right)^{k\beta(2k\beta-1)-k}\left(1+O_{k,\beta}\left(\log^{-1}D\right)\right),
\end{equation}
where $\tilde{\mathfrak{c}}_{k,\beta}$ corresponds to the leading order coefficient in \eqref{eq:mom_orthog}, and $\xi_{k,\beta}$ contains the arithmetic information (again, see for example~\cite{cfkrs2}).

We note that for both the symplectic and the orthogonal families defined above one could also prove theorems akin to theorem~\ref{thm:main} using the techniques described in sections~\ref{sec:structure} and~\ref{sec:proofs}.  This is due to the fact that \cite{cfkrs2} also provides conjectures for the shifted moments of the $L$-functions in question.  However, such calculations follow near verbatim those for the Riemann zeta function, so we do not report the details.

\section{Structure of proof of theorem~\ref{thm:main}}\label{sec:structure}

Within this section, we outline the proof of theorem~\ref{thm:main}.  We require a series of lemmas whose proofs are presented in section~\ref{sec:proofs}.

To contextualise the proof, we note that the general expression for shifted moments of the zeta function conjectured by Conrey et al.~\cite{cfkrs2} (cf. \eqref{conj:cfkrs}) has an exact correspondence to characteristic polynomials in the random matrix case~\cite{cfkrs1} (quoted as theorem~$1.5.2.$ in~\cite{cfkrs2}). A generalised version of the random matrix theorem was used by the present authors when proving \eqref{eq:baikea}.  Hence, the approach outlined in this section follows similar lines to the proof of theorem~$1.2$ in~\cite{baikea19}.  However, in several places we will need different steps to account for the arithmetic factors involved.

We first recall the definition of $P_{k,\beta}(x;\underline{h})$ from \eqref{def:p},
\begin{equation}
    P_{k,\beta}(x;\underline{h})=\frac{(-1)^{k\beta}}{(k\beta)!^2}\frac{1}{(2\pi i)^{2k\beta}}\oint\cdots\oint\frac{G(z_1,\dots,z_{2k\beta})\Delta(z_1,\dots,z_{2k\beta})^2}{e^{\frac{x}{2}\sum_{j=1}^{k\beta}z_{k\beta+j}-z_j}\prod_{j=1}^{2k\beta}\prod_{l=1}^{k}(z_j-ih_l)^{2\beta}}dz_1\cdots dz_{2k\beta}
\end{equation}
where one integrates over small circles surrounding the poles and
\begin{equation}
  G(z_1,\dots,z_{2k\beta})= A_{k\beta}(z_1,\dots,z_{2k\beta})\prod_{1\leq i\leq k\beta < j\leq 2k\beta}\zeta(1+z_i-z_j)
\end{equation}
which includes the Euler product
\begin{equation}
  A_{k\beta}(\underline{z})=\prod_p \prod_{1\leq l\leq k\beta < m\leq 2k\beta} (1-p^{z_m-z_l-1})\int_0^1\prod_{j=1}^{k\beta}\left(1-\frac{e(\theta)}{p^{\frac{1}{2}+z_j}}\right)^{-1}\left(1-\frac{e(-\theta)}{p^{\frac{1}{2}-z_{k\beta+j}}}\right)^{-1}d\theta.
\end{equation}

Since it will be useful, we also here record an alternative form for $A_{k\beta}(\cdot)$ also due to Conrey et al.~\cite{cfkrs2} (a special case of corollary 2.6.2 in their paper, found there at (2.6.15)),
\begin{equation}\label{alt_a}
  A_{k\beta}(z_1,\dots,z_{2k\beta})=\prod_p\sum_{m=1}^{k\beta}\prod_{n\neq m}\frac{\prod_{j=1}^{k\beta}\left(1-\frac{1}{p^{1+z_j-z_{k\beta+n}}}\right)}{1-p^{z_{k\beta+n}-z_{k\beta+m}}},
\end{equation}
where additionally they show that each local factor is actually a polynomial in $p^{-1}, p^{-z_j}$, and $p^{z_{k\beta+j}}$ for $j=1,\dots,k\beta$.

We begin by decomposing the multiple contour integral $P_{k,\beta}(x,\underline{h})$ in to a sum of multiple contour integrals, and apply a variable change so that each contour shifts to small circles surrounding the origin. Eventually (see \eqref{conj:cfkrs}--\eqref{def:p}), we will want to set $x=\log(t/2\pi)$.

\begin{lemma}\label{integral_decomposition}
  Asymptotically for large $x$,
  \begin{align}
    P_{k,\beta}(x;\underline{h})&\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2(2\pi i)^{2k\beta}}\left(\frac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}\nonumber\\
    &\qquad\quad\times\int_{\Gamma_0}\cdots\int_{\Gamma_0}\prod_{\substack{ m\leq k\beta<n\\\mu_n\neq \mu_m}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\right)f(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m,
  \end{align}
  where $c_{\underline{l}}(k,\beta)$ is a product of binomial coefficients determined within the proof and
  \begin{equation}\label{def:f_func}
    f(\underline{v};\underline{l})\coloneqq\frac{e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(v_n-v_m\right)^2}{\prod_{\substack{m\leq k\beta<n\\\mu_n= \mu_m}}(v_m-v_n)\prod_{m=1}^{2k\beta}v_m^{2\beta}}.
  \end{equation}
  The contours denoted by $\Gamma_0$ are small circles around the origin, and the $\mu_i$ variables are related to the $h_j$ variables - for the exact definition see \eqref{mu_vector}. Further, the following sets are useful to define, and the former appears in the statement of the lemma,
  \begin{align}
    \cS_{k,\beta;\underline{l}}&\coloneqq\{(m,n):1\leq m\leq k\beta<n\leq 2k\beta, \mu_m=\mu_n\}\label{S_set},\\
    \cT_{k,\beta;\underline{l}}&\coloneqq\{(m,n):1\leq m\leq k\beta<n\leq 2k\beta, \mu_m\neq\mu_n\}.\label{T_set}
  \end{align}
\end{lemma}

The next lemma relies on lemma~\ref{integral_decomposition} and determines the remaining $x\equiv \log(t/2\pi)$ dependence in the integrand of $P_{k,\beta}(x;\underline{h})$ after calculating the contribution from the integration over the $h_1,\dots,h_k$.

\begin{lemma}\label{correct_size}
  As $T\rightarrow\infty$,
  \begin{equation}
    \mom_{P_{k,\beta}}(T)\sim \alpha_{k,\beta}\gamma_{k,\beta} \left(\log\tfrac{T}{2\pi}\right)^{k^2\beta^2-k+1},
  \end{equation}
  where $\alpha_{k,\beta}= A_{k\beta}(0,\dots,0)$,
  \begin{align}
    \gamma_{k,\beta}&=\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2(2\pi i)^{2k\beta}}\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\Psi_{k,\beta}(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m, \label{eq:gamma_coeff}
  \end{align}
  $c_{\underline{l}}(k,\beta)$ and $f(\underline{v};\underline{l})$ are given by \eqref{binom_prod} and \eqref{def:f_func}, and $\Psi_{k,\beta}(\underline{v};\underline{l})$ is a multiple integral defined within the proof, see~\eqref{pre_incomplete_gamma}.
\end{lemma}

The final lemma establishes non-vanishing of the leading order coefficient.

\begin{lemma}\label{positivity_coefficient}
  For $k, \beta\in\mathbb{N}$, the coefficient $\alpha_{k,\beta}\gamma_{k,\beta}$ is non-zero.
\end{lemma}

By combining the statements of lemma~\ref{correct_size} and lemma~\ref{positivity_coefficient}, we arrive at the statement of theorem~\ref{thm:main}

\begin{equation}
  \mom_{P_{k,\beta}}(T)=\alpha_{k,\beta}\gamma_{k,\beta}\left(\log\tfrac{T}{2\pi}\right)^{k^2\beta^2-k+1}\left(1+O\left(\log^{-1}\tfrac{T}{2\pi}\right)\right),
\end{equation}
for $k,\beta\in\mathbb{N}$.

\section{Proofs of lemmas~\ref{integral_decomposition}--\ref{positivity_coefficient}}\label{sec:proofs}

In this section we give the proofs of lemma~\ref{integral_decomposition}, lemma~\ref{correct_size}, and lemma~\ref{positivity_coefficient} from section~\ref{sec:structure}.

\begin{proof}[Proof of lemma~\ref{integral_decomposition}]

We recall the statement of lemma~\ref{integral_decomposition}.

\begin{lemmum}
   Asymptotically for large $x$,
  \begin{align}
    P_{k,\beta}(x;\underline{h})&\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2(2\pi i)^{2k\beta}}\left(\frac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}\nonumber\\
    &\qquad\quad\times\int_{\Gamma_0}\cdots\int_{\Gamma_0}\prod_{\substack{ m\leq k\beta<n\\\mu_n\neq \mu_m}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\right)f(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m,
  \end{align}
  where $c_{\underline{l}}(k,\beta)$ is a product of binomial coefficients determined in the proof and
  \begin{equation}
    f(\underline{v};\underline{l})=\frac{e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(v_n-v_m\right)^2}{\prod_{\substack{m\leq k\beta<n\\\mu_n= \mu_m}}(v_m-v_n)\prod_{m=1}^{2k\beta}v_m^{2\beta}}.
  \end{equation}
\end{lemmum}

Recall the definition of $P_{k,\beta}(x,\underline{h})$ from \eqref{def:p},
\begin{equation}\label{def:p_proof}
  P_{k,\beta}(x;\underline{h})=\frac{(-1)^{k\beta}}{(k\beta)!^2}\frac{1}{(2\pi i)^{2k\beta}}\oint\cdots\oint\frac{G(z_1,\dots,z_{2k\beta})\Delta(z_1,\dots,z_{2k\beta})^2}{e^{\frac{x}{2}\sum_{j=1}^{k\beta}z_{k\beta+j}-z_j}\prod_{j=1}^{2k\beta}\prod_{l=1}^{k}(z_j-ih_l)^{2\beta}}dz_1\cdots dz_{2k\beta}
\end{equation}
where the contours are small circles around the poles, and $G(\underline)$ is given by \eqref{def:g_func}.

We start by decomposing the $2k\beta$ contours in \eqref{def:p_proof} so that each contour consists of $k$ small circles encompassing the poles at $ih_1,\dots,ih_k$ with connecting straight lines whose contributions cancel out.  Denote a small circular contour encircling $ih_j$ by $\Gamma_{i h_j}$.  Then,
\begin{align}
  P_{k,\beta}(x,\underline{h})&=\frac{(-1)^{k\beta}}{(k\beta)!^2}\frac{1}{(2\pi i)^{2k\beta}}\sum_{\eps_j\in\{1,\dots,k\}}Q_{k,\beta}(x,\underline{h};\eps_1,\dots,\eps_{2k\beta}),\label{p_decomp}
\end{align}
where
\begin{equation}\label{eq:qMCI}
  Q_{k,\beta}(x,\underline{h};\eps_1,\dots,\eps_{2k\beta})=\int_{\Gamma_{ih_{\eps_1}}}\cdots\int_{\Gamma_{ih_{\eps_{2k\beta}}}}\frac{G(z_1,\dots,z_{2k\beta})\Delta(z_1,\dots,z_{2k\beta})^2}{e^{\frac{x}{2}\sum_{j=1}^{k\beta}z_{k\beta+j}-z_j}\prod_{j=1}^{2k\beta}\prod_{l=1}^{k}(z_j-ih_l)^{2\beta}}dz_1\cdots dz_{2k\beta}
\end{equation}
is the multiple contour integral $P_{k,\beta}(x,\underline{h})$ with the $2k\beta$ contours each specialised around one of the $k$ poles determined by the vector $\underline{\eps}=(\eps_1,\dots,\eps_{2k\beta})$.

Due to the symmetric nature of the integrand, many of the summands evaluate to zero. Such a fact was established in lemma~$3.2$ in~\cite{baikea19}, and is paraphrased here.
\begin{lemmum}
  Write $\underline{\eps}=(\eps_1,\dots,\eps_{2k\beta})$ for a choice of contours appearing as a summand in \eqref{p_decomp}.  Then the only choices of $\underline{\eps}$ leading to a non-zero summand are those where each pole is equally represented in $\underline{\eps}$.  That is: if $n_i$ counts the number of occurrences of $i$ in $\underline{\eps}$, for $i\in\{1,\dots,k\}$, then the summand corresponding to the choice $\underline{\eps}$ is identically zero unless $n_1=\cdots=n_{k}=2\beta$.
\end{lemmum}

\begin{remark}
  The above lemma was formulated in~\cite{baikea19} for a slightly different integral, namely
  \begin{equation}\label{eq:oldMCI}
    \int_{\Gamma_{ih_{\eps_1}}}\cdots\int_{\Gamma_{ih{\eps_{2k\beta}}}}\frac{e^{-N(z_{k\beta+1}+\cdots+z_{2k\beta})}\Delta(z_1,\dots,z_{2k\beta})^2dz_1\cdots dz_{2k\beta}}{\prod_{m\leq k\beta<n}\left(1-e^{z_n-z_m}\right)\prod_{m=1}^{2k\beta}\prod_{n=1}^{k}(z_m-ih_n)^{2\beta}}.
  \end{equation}
  However, the structure of \eqref{eq:oldMCI} only differs trivially to that of \eqref{eq:qMCI}.  The key difference is between the terms of the form $(1-\exp({z_n-z_m}))^{-1}$ in \eqref{eq:oldMCI} versus $\zeta(1+z_m-z_n)$ in \eqref{eq:qMCI}.  However, they share the same analytic structure since both have a simple pole at $z_n=z_m$. Therefore the proof presented in~\cite{baikea19} holds as well for~\eqref{eq:qMCI}.
\end{remark}

Hence \eqref{p_decomp} becomes
\begin{equation}
  P_{k,\beta}(x,\underline{h})=\frac{(-1)^{k\beta}}{(k\beta)!^2}\frac{1}{(2\pi i)^{2k\beta}}\sum_{l_1=0}^{2\beta}\cdots\sum_{l_{k-1}=0}^{2\beta}c_{\underline{l}}(k,\beta)Q_{k,\beta}(x,\underline{h};\underline{l}),\label{p_decomp_specialised}
\end{equation}
where $\underline{l}=(l_1,\dots,l_{k-1})$ and $Q_{k,\beta}(x,\underline{h};\underline{l})$ is the integral $Q_{k,\beta}(x,\underline{h};\underline{\eps})$ with contours given by
\[\underline{\eps}=(\overbrace{1,\dots,1}^{l_1},\overbrace{2,\dots,2}^{l_2},\dots,\overbrace{k-1,\dots,k-1}^{l_{k-1}},\overbrace{k,\dots,k}^{2\beta},\overbrace{k-1,\dots,k-1}^{2\beta-l_{k-1}},\dots,\overbrace{1,\dots,1}^{2\beta-l_1}),\]
and the coefficient $c_{\underline{l}}(k,\beta)$ is a product of binomial coefficients:
\begin{align}
  c_{\underline{l}}(k,\beta)&=\binom{k\beta}{l_1}\binom{k\beta-l_1}{l_2}\binom{k\beta-(l_1+l_2)}{l_3}\cdots\binom{k\beta-\sum_{m=1}^{k-2}l_m}{l_{k-1}}\nonumber\\
  &\quad\times\binom{k\beta}{2\beta-l_1}\binom{(k-2)\beta+l_1}{2\beta-l_2}\cdots\binom{k\beta-\sum_{m=1}^{k-2}(2\beta-l_m)}{2\beta-l_{k-1}}.\label{binom_prod}
\end{align}

We now apply the following change of variables so to shift all the contours to be small circles around the origin
\[z_n=\frac{2v_n}{x}+i\mu_n,\]
where
\begin{equation}\label{mu_vector}
  \mu_n=
  \begin{cases}
    h_1,&\text{if }n\in\{1,\dots,l_1\}\cup\{2(k-1)\beta+1+l_1,\dots,2k\beta\}\\
    h_2,&\text{if }n\in\{l_1+1,\dots,l_1+l_2\}\cup\{2(k-2)\beta+1+l_1+l_2,\dots,2(k-1)\beta+l_1\}\\
    \vdots&\quad\vdots\\
    h_{k-1},&\text{if }n\in\{\sum_{m=1}^{k-2}l_m+1,\dots,\sum_{m=1}^{k-1}l_m\}\cup\{2\beta+1+\sum_{m=1}^{k-1}l_m,\dots,4\beta+\sum_{m=1}^{k-2}l_m\}\\
    h_k,&\text{if }n\in\{\sum_{m=1}^{k-1}l_m+1,\dots,\sum_{m=1}^{k-1}l_m+2\beta\}.\end{cases}
\end{equation}
Note that $\mu_n$ is dependent on the choice of $\underline{l}$ (i.e. the exact ordering of the contours), but this is suppressed from the notation for simplicity.

Using the fact that the $\zeta$-function has a simple pole at $1$ with residue $1$, the resulting integrand of $Q_{k,\beta}(x,\underline{h};\underline{l})$ as $x\rightarrow\infty$ is
\begin{align}
  \big(1+O\big(\tfrac{1}{x}&\big)\big)A_{k\beta}\big(\tfrac{2v_1}{x}+i\mu_1,\dots,\tfrac{2v_{2k\beta}}{x}+i\mu_{2k\beta}\big)e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\nonumber\\
  &\qquad\times\prod_{\substack{m<n\\\mu_m\neq\mu_n}}(i\mu_n-i\mu_{n})^2\prod_{\substack{1\leq m\leq k\beta<n\leq 2k\beta\\\mu_n\neq \mu_m}}\zeta\big(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\big)\nonumber\\
  &\qquad\times\frac{\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(\tfrac{2(v_n-v_m)}{x}\right)^2}{\prod_{\substack{1\leq m\leq k\beta<n\leq 2k\beta\\\mu_n= \mu_m}}\left(\tfrac{2(v_m-v_n)}{x}\right)\prod_{m=1}^{2k\beta}\prod_{n=1}^{k}\left(\tfrac{2v_m}{x}-i(h_n-\mu_m)\right)^{2\beta}}\prod_{m=1}^{2k\beta}\tfrac{2dv_m}{x}\nonumber\\
  &=\left(1+O\left(\tfrac{1}{x}\right)\right)\left(\tfrac{x}{2}\right)^{-2k\beta}A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\nonumber\\
  &\qquad\times\prod_{\substack{m<n\\\mu_m\neq\mu_n}}(i\mu_m-i\mu_n)^2\prod_{\substack{m\leq k\beta<n\\\mu_n\neq \mu_m}}\zeta\big(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\big)\nonumber\\
  &\qquad\times\frac{\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(\tfrac{2(v_n-v_m)}{x}\right)^2\prod_{m=1}^{2k\beta}\left(\tfrac{2v_m}{x}\right)^{-2\beta}}{\prod_{\substack{m\leq k\beta<n\\\mu_n= \mu_m}}\left(\tfrac{2(v_m-v_n)}{x}\right)\prod_{\substack{m<n\\\mu_m\neq\mu_n}}(i\mu_m-i\mu_n)^2}\prod_{m=1}^{2k\beta}dv_m\\
  &=\left(1+O\left(\tfrac{1}{x}\right)\right)\left(\tfrac{x}{2}\right)^{4k\beta^2-2k\beta}A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\nonumber\\
  &\qquad\times\prod_{\substack{ m\leq k\beta<n\\\mu_n\neq \mu_m}}\zeta\big(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\big)\frac{\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(\tfrac{2(v_n-v_m)}{x}\right)^2}{\prod_{\substack{m\leq k\beta<n\\\mu_n= \mu_m}}\tfrac{2(v_m-v_n)}{x}}\prod_{m=1}^{2k\beta}\tfrac{dv_m}{v_m^{2\beta}}.\label{var_sub_final}
\end{align}

The power of $x$ coming from the terms that originated from the Vandermonde (i.e. the numerator of the fraction in \eqref{var_sub_final}) is calculated to be $2k\beta(2\beta-1)$. We now extract the $x$ dependence remaining in the denominator of the integrand~\eqref{var_sub_final}.  First, define the following sets,
\begin{align}
  \cS_{k,\beta;\underline{l}}&\coloneqq\{(m,n):1\leq m\leq k\beta<n\leq 2k\beta, \mu_m=\mu_n\}\\
  \cT_{k,\beta;\underline{l}}&\coloneqq\{(m,n):1\leq m\leq k\beta<n\leq 2k\beta, \mu_m\neq\mu_n\}.
\end{align}
Thus, $|\cS_{k,\beta;\underline{l}}|+|\cT_{k,\beta;\underline{l}}|=k^2\beta^2$, and the pairs $(m,n)\in \cS_{k,\beta;\underline{l}}$ are precisely those involved in the denominator of \eqref{var_sub_final}. As it will be useful, we also briefly record that
\[(-1)^{|\cS_{k,\beta;\underline{l}}|}=(-1)^{\sum_{j=1}^kl_j(2\beta-l_j)}=(-1)^{k\beta}.\]
Hence the integrand of $Q_{k,\beta}(x,\underline{h};\underline{l})$ is
\begin{align}
  &\left(1+O\left(\tfrac{1}{x}\right)\right)\left(\tfrac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}(-1)^{k\beta}A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\nonumber\\
  &\qquad\times\prod_{(m,n)\in \cT_{k,\beta;\underline{l}}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\right)\frac{\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(v_n-v_m\right)^2}{\prod_{\substack{m\leq k\beta<n\\\mu_n= \mu_m}}(v_m-v_n)}\prod_{m=1}^{2k\beta}\tfrac{dv_m}{v_m^{2\beta}}.
\end{align}
Thus, as $x\rightarrow\infty$,
\begin{align}
  P_{k,\beta}(x,\underline{h})&\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2(2\pi i)^{2k\beta}}\left(\tfrac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}\nonumber\\
  &\quad\times\int_{\Gamma_0}\cdots\int_{\Gamma_0}\prod_{(m,n)\in \cT_{k,\beta;\underline{l}}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\right)f(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m,
\end{align}
where the terms in the integrand with no $h$ dependence have been collectively denoted by $f(\underline{v};\underline{l})$, so
\begin{equation}
  f(\underline{v};\underline{l})\coloneqq\frac{e^{\sum_{j=1}^{k\beta}v_j-v_{k\beta+j}}\prod_{\substack{m<n\\\mu_m=\mu_n}}\left(v_n-v_m\right)^2}{\prod_{\substack{m\leq k\beta<n\\\mu_n= \mu_m}}(v_m-v_n)\prod_{m=1}^{2k\beta}v_m^{2\beta}},\label{none_h_terms}
\end{equation}
which concludes the proof.

\end{proof}

\begin{proof}[Proof of lemma~\ref{correct_size}]

  We recall the statement of lemma~\ref{correct_size}.

  \begin{lemmum}
    \[\mom_{P_{k,\beta}}(T)\sim \alpha_{k,\beta}\gamma_{k,\beta} \left(\log\tfrac{T}{2\pi}\right)^{k^2\beta^2-k+1},\]
    where $\alpha_{k,\beta}\coloneqq A_{k\beta}(0,\dots,0)$,
    \begin{align*}
      \gamma_{k,\beta}&\coloneqq\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2(2\pi i)^{2k\beta}}\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\Psi_{k,\beta}(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m,
    \end{align*}
    $c_{\underline{l}}(k,\beta)$ and $f(\underline{v};\underline{l})$ are given by \eqref{binom_prod} and \eqref{def:f_func} and $\Psi_{k,\beta}(\underline{v};\underline{l})$ is a multiple integral defined within the proof, see~\eqref{pre_incomplete_gamma}.
  \end{lemmum}

  Set $x\coloneqq\log(t/2\pi)$. By lemma~\ref{integral_decomposition}, we have that
    \begin{align}
    \mom_{P_{k,\beta}}(T)&= \frac{1}{T}\int_{0}^{1}\cdots\int_{0}^{1}\int_0^TP_{k,\beta}(\log\tfrac{t}{2\pi},\underline{h})dtdh_1\cdots dh_k\nonumber\\
    &\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{T(k\beta)!^2(2\pi i)^{2k\beta}}\nonumber\\
    &\qquad\times \int_{0}^{1}\cdots\int_{0}^{1}\int_0^T A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta}) e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}\left(\tfrac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}\nonumber\\
    &\qquad\times\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\prod_{(m,n)\in\cT_{k,\beta;\underline{l}}}\zeta\big(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\big)\prod_{m=1}^{2k\beta}dv_mdt \prod_{n=1}^kdh_n.\label{mom_lemma1}
    \end{align}

  We begin by firstly rewriting the product of zeta functions in the innermost integrand of \eqref{mom_lemma1} using the definition of $\mu_1,\dots, \mu_{2k\beta}$, see \eqref{mu_vector},
  \begin{align}
    \prod_{\substack{ (m,n)\in\cT_{k,\beta;\underline{l}}}}&\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(\mu_m-\mu_n)\right)\nonumber\\
    &=\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(h_\sigma-h_\tau)\right)\prod_{(m,n)\in\cV^-_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}-i(h_\sigma-h_\tau)\right).\label{zeta_expanded}
  \end{align}
  where the sets $\cV^+_{\sigma,\tau}$, $\cV^-_{\sigma,\tau}$ partition $\cT_{k,\beta;\underline{l}}$,
  \begin{align}
    \cV^+_{\sigma,\tau}&\coloneqq\{(m,n)\in \cT_{k,\beta;\underline{l}}:\mu_m-\mu_n=h_\sigma-h_\tau\},\label{V_subset1}\\
    \cV^-_{\sigma,\tau}&\coloneqq\{(m,n)\in \cT_{k,\beta;\underline{l}}:\mu_m-\mu_n=h_\tau-h_\sigma\}.\label{V_subset2}
  \end{align}

  Combining \eqref{mom_lemma1} and \eqref{zeta_expanded} we find
  \begin{align}
    \mom_{P_{k,\beta}}(T)&\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{T(k\beta)!^2(2\pi i)^{2k\beta}}\nonumber\\
    &\qquad\times \int_{0}^{1}\cdots\int_{0}^{1}\int_0^T A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta}) e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}\left(\tfrac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}\nonumber\\
    &\qquad\quad\times\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(h_\sigma-h_\tau)\right)\nonumber\\
    &\qquad\qquad\qquad\times \prod_{(m,n)\in\cV^-_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}-i(h_\sigma-h_\tau)\right)\prod_{m=1}^{2k\beta}dv_mdt\prod_{n=1}^kdh_n.
    \end{align}
  Next we switch the order of integration
  \begin{align}
    \mom_{P_{k,\beta}}(T)&\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{T(k\beta)!^2(2\pi i)^{2k\beta}}\int_0^T\left(\tfrac{x}{2}\right)^{|\cS_{k,\beta;\underline{l}}|}\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\nonumber\\
    &\qquad\times\int_{0}^{1}\cdots\int_{0}^{1} A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(h_\sigma-h_\tau)\right)\nonumber\\
    &\qquad\qquad\times \prod_{(m,n)\in\cV^-_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}-i(h_\sigma-h_\tau)\right)e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}\prod_{n=1}^kdh_n\prod_{m=1}^{2k\beta}dv_mdt,\label{switched_int}
  \end{align}
  and focus on the inner integrals over $h_1,\dots, h_k$.

  We use the structure of $(\mu_1,\dots,\mu_{2k\beta})$ in order to write the exponential term in the innermost integrand of \eqref{switched_int} explicitly in terms of $h_1,\dots,h_{2k\beta}$,

  \begin{equation}
    \exp\left(\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}\right)=\exp\left(ix\sum_{j=1}^{k-1}(l_j-\beta)(h_j-h_k)\right).
  \end{equation}

  Then, we use the Laurent expansion for the zeta function around its pole,
  \begin{equation}
    \zeta(1+s)=\frac{1}{s}+\sum_{j=0}^{\infty}c_js^j,
  \end{equation}
  where $c_j=(-1)^j\mathfrak{s}_j/j!$ and $\mathfrak{s}_j$ are the Stieltjes constants (so $c_0$ is the Euler-Mascheroni constant).

  In this calculation, we will be taking $s\sim 1/x$, so the main contribution to \eqref{switched_int} comes from just taking the first term in the Laurent series for each zeta function in the integrand. Hence the integral over the $h_1,\dots,h_k$ becomes

  \begin{align}
    \int_{0}^{1}&\cdots\int_{0}^{1} A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}+i(h_\sigma-h_\tau)\right)\nonumber\\
    &\times \prod_{(m,n)\in\cV^-_{\sigma,\tau}}\zeta\left(1+\tfrac{2(v_m-v_n)}{x}-i(h_\sigma-h_\tau)\right)e^{\frac{ix}{2}\sum_{j=1}^{k\beta}\mu_j-\mu_{k\beta+j}}dh_1\cdots dh_k\nonumber\\
    &\sim\int_{0}^{1}\cdots\int_{0}^{1} \frac{A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{ix\sum_{j=1}^{k-1}(l_j-\beta)(h_j-h_k)}dh_1\cdots dh_k}{\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\left(\tfrac{2(v_m-v_n)}{x}+i(h_\sigma-h_\tau)\right)\prod_{(m,n)\in\cV^-_{\sigma,\tau}}\left(\tfrac{2(v_m-v_n)}{x}-i(h_\sigma-h_\tau)\right)}.\label{h_int_zeta_exp}
  \end{align}
  Observe that each term in the integrand of \eqref{h_int_zeta_exp} is a function of the differences $\delta_j\coloneqq h_j-h_k$, $j=1,\dots,k$.  For the exponential term this is immediate.  For $A_{k\beta}(\cdot)$ we use \eqref{alt_a}, writing $A_{k\beta;p}(\cdot)$ for each local factor,
  \begin{align}
    A_{k\beta;p}(i\mu_1,\dots,i\mu_{2k\beta})=\sum_{m=1}^{k\beta}\prod_{n\neq m}\frac{\prod_{j=1}^{k\beta}\left(1-\frac{1}{p^{1+i(\mu_j-h_k)}p^{-i(\mu_{k\beta+n}-h_k)}}\right)}{1-p^{i(\mu_{k\beta+n}-h_k)}p^{-i(\mu_{k\beta+m}-h_k)}}.
  \end{align}
  Similarly for the terms in the denominator, we write $h_\sigma-h_\tau=(h_\sigma-h_k)-(h_\tau-h_k)$.  We now perform a change of variables $2\delta_j=x(h_j-h_k)$ and use the fact that $A_{k\beta}$ is analytic in a neighbourhood of $(0,\dots,0)$.  Hence
  \begin{align}
    \int_{0}^{1}&\cdots\int_{0}^{1}\frac{A_{k\beta}(i\mu_1,\dots,i\mu_{2k\beta})e^{ix\sum_{j=1}^{k-1}(l_j-\beta)(h_j-h_k)}dh_1\cdots dh_k}{\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\left(\tfrac{2(v_m-v_n)}{x}+i(h_\sigma-h_\tau)\right)\prod_{(m,n)\in\cV^-_{\sigma,\tau}}\left(\tfrac{2(v_m-v_n)}{x}-i(h_\sigma-h_\tau)\right)}\nonumber\\
    &\sim \int_{0}^{\frac{x}{2}}\cdots\int_{0}^{\frac{x}{2}} \frac{\left(\frac{x}{2}\right)^{|\cT_{k,\beta;\underline{l}}|-{k-1}}A_{k\beta}\left(0,\dots,0\right)e^{2i\sum_{j=1}^{k-1}(l_j-\beta)\delta_j}d\delta_1\cdots d\delta_{k-1}}{\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\left(v_m-v_n+i(\delta_\sigma-\delta_\tau)\right)\prod_{(m,n)\in\cV^-_{\sigma,\tau}}\left(v_m-v_n-i(\delta_\sigma-\delta_\tau)\right)}\\
    &\sim\left(\frac{x}{2}\right)^{|\cT_{k,\beta;\underline{l}}|-{k-1}}A_{k\beta}\left(0,\dots,0\right)\Psi_{k,\beta}(\underline{v};\underline{l}),
  \end{align}
  where
  \begin{align}
    \Psi_{k,\beta}(\underline{v};\underline{l})\coloneqq\int_{0}^{\infty}\cdots&\int_{0}^{\infty} \frac{e^{2i\sum_{j=1}^{k-1}(l_j-\beta)\delta_j}}{\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^+_{\sigma,\tau}}\left(v_m-v_n+i(\delta_\sigma-\delta_\tau)\right)}\nonumber\\
    &\qquad\qquad\times\frac{d\delta_1\cdots d\delta_{k-1}}{\prod_{1\leq\sigma<\tau\leq k}\prod_{(m,n)\in\cV^-_{\sigma,\tau}}\left(v_m-v_n-i(\delta_\sigma-\delta_\tau)\right)}. \label{pre_incomplete_gamma}
  \end{align}

  Then $\Psi_{k,\beta}(\underline{v};\underline{l})$ defined by \eqref{pre_incomplete_gamma} no longer depends on $x$.  We will further analyse its contribution to $\mom_{P_{k,\beta}}(T)$ in the proof of lemma~\ref{positivity_coefficient} below.

  Incorporating \eqref{pre_incomplete_gamma} in to \eqref{switched_int}, and recalling that $|\cS_{k,\beta;\underline{l}}|+|\cT_{k,\beta;\underline{l}}|=k^2\beta^2$ and $x=\log(t/2\pi)$, we find that
  \begin{align}
    \mom_{P_{k,\beta}}(T)&\sim\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)A_{k\beta}(0,\dots,0)}{T(k\beta)!^2(2\pi i)^{2k\beta}}\int_0^T\left(\frac{x}{2}\right)^{k^2\beta^2-k+1}\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\Psi_{k,\beta}(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m\; dt\\
    &=\alpha_{k,\beta}\gamma_{k,\beta}\left(\log\tfrac{T}{2\pi}\right)^{k^2\beta^2-k+1}\left(1+O\left(\log^{-1}\tfrac{T}{2\pi}\right)\right),
  \end{align}
  where  $\alpha_{k,\beta}\coloneqq A_{k\beta}(0,\dots,0)$ and
  \begin{equation*}
    \gamma_{k,\beta}\coloneqq\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2(2\pi i)^{2k\beta}}\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\Psi_{k,\beta}(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m.
  \end{equation*}

\end{proof}

\begin{proof}[Proof of lemma~\ref{positivity_coefficient}]

  We recall the statement of lemma~\ref{positivity_coefficient}.

  \begin{lemmum}
    The coefficient $\alpha_{k,\beta}\gamma_{k,\beta}$ appearing in the statement of lemma~\ref{correct_size} is non-zero.
  \end{lemmum}

  From lemma~\ref{correct_size}, we have that $\alpha_{k,\beta}=A_{k\beta}(0,\dots,0)$. The value of $A_n(0,\dots,0)$ was calculated by Conrey et al.~\cite{cfkrs2}. Hence, using their formula, we have

  \begin{equation}\label{arithmetic_term}
    A_{k\beta}(0,\dots,0)=\prod_p\left(1-\frac{1}{p}\right)^{(k\beta-1)^2}\sum_{m=0}^{k\beta-1}\binom{k\beta-1}{m}^2p^{-m}.
  \end{equation}

  Thus, all that remains to justify is that $\gamma_{k,\beta}\neq 0$.  To do this we appeal to the residue theorem\footnote{The initial part of the proof borrows calculations found in the proof of lemma 3.7 in~\cite{baikea19}.}.  Fix a choice of $l_1,\dots,l_{k-1}$ and notice that
  \[\frac{c_{\underline{l}}(k,\beta)}{(k\beta)!^2}>0.\]

  To conclude, we compare to the random matrix case.  In previous work~\cite{baikea19}, recall (cf. \eqref{eq:baikea}) that the moments of the moments of unitary characteristic polynomials
  \begin{equation}
    \mom_{U(N)}(k,\beta)\coloneqq\int_{U(N)}\left(\frac{1}{2\pi}\int_0^{2\pi}|P_N(A,\theta)|^{2\beta}d\theta\right)^kdA
  \end{equation}
  were shown to satisfy
  \begin{equation}
    \mom_{U(N)}(k,\beta)=c_{k,\beta}N^{k^2\beta^2-k+1}(1+O(\tfrac{1}{N})),
  \end{equation}
  and $c_{k,\beta}$ is non-zero. The exact form\footnote{In~\cite{baikea19}, the multiple contour integral that they treat differs only from the one we consider here by a simple transformation.  For simplicity, the results presented henceforth have been translated so to be consistent with the notation within this article.} of $c_{k,\beta}$ is found by evaluating
  \begin{align*}
    c_{k,\beta}=\sum_{l_1,\dots,l_{k-1}=0}^{2\beta}\frac{c_{\underline{l}}(k,\beta)}{((k\beta)!)^2(2\pi i)^{2k\beta}}\int_{\Gamma_0}\cdots\int_{\Gamma_0}f(\underline{v};\underline{l})\Omega_{k,\beta}(\underline{v};\underline{l})\prod_{m=1}^{2k\beta}dv_m,
  \end{align*}
  where $c_{\underline{l}}(k,\beta)$ and $f(\underline{v};\underline{l})$ are the same functions used throughout the present work (see~\eqref{binom_prod} and~\eqref{none_h_terms}),
  \begin{align}
    \Omega_{k,\beta}(\underline{v};\underline{l})=\lim_{N\rightarrow\infty}\frac{1}{(2\pi)^kN^{\cT_{k,\beta;\underline{l}}-k+1}}\int_0^{2\pi}\cdots\int_0^{2\pi}&\frac{e^{iN\sum_{j=1}^{k-1}(l_j-\beta)(\theta_j-\theta_k)}}{\prod_{(m,n)\in \cT_{k,\beta;\underline{l}}}(1-e^{\frac{v_n-v_m}{N}}e^{i(\mu_n-\mu_m)})}d\theta_1\cdots d\theta_k\\
    =\lim_{N\rightarrow\infty}\frac{1}{(2\pi)^kN^{\cT_{k,\beta;\underline{l}}-k+1}}\int_0^{2\pi}\cdots\int_0^{2\pi}&\frac{e^{iN\sum_{j=1}^{k-1}(l_j-\beta)(\theta_j-\theta_k)}}{\prod_{1\leq \sigma<\tau\leq k}\prod_{(m,n)\in \cV^+_{\sigma,\tau}}\left(1-e^{\frac{v_n-v_m}{N}}e^{i(\theta_\tau-\theta_\sigma)}\right)}\nonumber\\
    &\times \frac{d\theta_1\cdots d\theta_k}{\prod_{(m,n)\in \cV^-_{\sigma,\tau}}\left(1-e^{\frac{v_m-v_n}{N}}e^{i(\theta_\sigma-\theta_\tau)}\right)}.
  \end{align}
  The sets $\cT_{k,\beta;\underline{l}}$, $\cV^+_{\sigma,\tau}$, and $\cV^-_{\sigma,\tau}$ are as defined in \eqref{T_set}, \eqref{V_subset1}, and \eqref{V_subset2} respectively and $\mu$ matches the definition given by \eqref{mu_vector}.

  In~\cite{baikea19}, the denominator in the integrand of $\Omega_{k,\beta}$ was expanded as a geometric series, and then the integral was explicitly calculated.  However, one could proceed in a similar way to the proof of lemma~\ref{correct_size}, i.e. noticing that the integrand is simply a function of the differences $\delta_j\coloneqq \theta_j-\theta_k$ and then use the Laurent expansion of the denominator to pull out the power of $N$. Proceeding in this way we find
  \begin{alignat}{2}
    \Omega_{k,\beta}(\underline{v};\underline{l}) &\sim \frac{1}{(2\pi)^k}\int_0^{\infty}\cdots\int_0^{\infty} &&\frac{e^{i\sum_{j=1}^{k-1}(l_j-\beta)\delta_j}}{\prod_{1\leq \sigma<\tau\leq k}\prod_{(m,n)\in \cV^+_{\sigma,\tau}}\left({v_n-v_m+i(\delta_\tau-\delta_\sigma)}\right)}\nonumber\\
    &&&\times \frac{d\delta_1\cdots d\delta_{k-1}}{\prod_{(m,n)\in \cV^-_{\sigma,\tau}}\left(v_n-v_m-i(\delta_\tau-\delta_\sigma)\right)}\\

    &\sim \frac{(-1)^{|\cT_{k,\beta;\underline{l}}|}}{(2\pi)^k}\int_0^{\infty}\cdots\int_0^{\infty}&&\frac{e^{i\sum_{j=1}^{k-1}(l_j-\beta)\delta_j}}{\prod_{1\leq \sigma<\tau\leq k}\prod_{(m,n)\in \cV^+_{\sigma,\tau}}\left({v_m-v_n+i(\delta_\sigma-\delta_\tau)}\right)}\nonumber\\
    &&&\times \frac{d\delta_1\cdots d\delta_{k-1}}{\prod_{(m,n)\in \cV^-_{\sigma,\tau}}\left(v_m-v_n-i(\delta_\sigma-\delta_\tau)\right)}.
  \end{alignat}

  Thus, since $|\cS_{k,\beta;\underline{l}}|+|\cT_{k,\beta;\underline{l}}|=k^2\beta^2$, and $|\cS_{k,\beta;\underline{l}}|=\sum_{j=1}^kl_j(2\beta-l_j)$, we have
  \[(-1)^{|\cT_{k,\beta;\underline{l}}|}=(-1)^{k\beta(k\beta-1)}=1.\]
  So $\Omega_{k,\beta}(\underline{v};\underline{l})\sim \kappa \Psi_{k,\beta}(\underline{v};\underline{l})$ for some positive constant $\kappa$. Hence $\gamma_{k,\beta}\neq 0$ by comparison with $c_{k,\beta}$.
\end{proof}

\section{Acknowledgements}
We would like to thank Valeriya Kovaleva for helpful comments and discussions. We are grateful to the anonymous referee for their careful reading of the paper and suggestions. ECB would like to thank the Heilbronn Institute for Mathematical Research for support.   JPK is pleased to acknowledge support from ERC Advanced Grant 740900 (LogCorRM).


text:
{\bf Problem.} Let $x$, $y$, and $z$ be positive real numbers such that $(x \cdot y) + z = (x + z) \cdot (y + z)$. What is the maximum possible value of $xyz$?
{\bf Level.} Level 3
{\bf Type.} Intermediate Algebra
{\bf Solution.} The condition is equivalent to $z^2+(x+y-1)z=0$. Since $z$ is positive, $z=1-x-y$, so $x+y+z=1$. By the AM-GM inequality, $$xyz \leq \left(\frac{x+y+z}{3}\right)^3 = \boxed{\frac{1}{27}},$$with equality when $x=y=z=\frac{1}{3}$.

set_name: MATH


TITLE: How to find image
QUESTION [0 upvotes]: I am little confused with the idea of images of a function
How do you find the image of a particular interval with respect to a function?
For example : How do you find the image of the interval $[-1,3]$, under the mapping $f:\Bbb R\to \Bbb R$ given by $f(x)=4x^2-12x$?

REPLY [3 votes]: General advice when trying to solve a math problem is to try to visualize it. So when we want to find the image of a function on a certain interval, the first thing we might try is to plot that function. For your example $$f(x) = 4x^2-12x$$ we get something like this

Now we can conjecture what the answer might be, namely the interval $[-9,16]$. From the plot we can also get some idea of what our strategy to prove this claim might be. We can, for instance, use the concept of the derivative to prove that the minimum of $f$ is attained at $x=\frac{3}{2}$ and is equal to $f\left(\frac{3}{2}\right)=-9$. Now you might use symmetry arguments to show that the maximum on $[-1,3]$ must be attained in $x=-1$ and thus be equal to $f(-1)=16$. Then use continuity of the function to show that every value in between $-9$ and $16$ is attained as well.
Be aware that the plot of the function is useful for conjecturing, but can not be used to prove anything definitively. Every claim you make must be proved rigorously.

Another general strategy when tackling math problems is to try and break it up into smaller problems. The function $f$ can be seen as first sending $x$ to $x^2-3x$ and then multiplying the whole thing by $4$. If we let $g(t)=t^2-3t$ and $h(t)=4t$ we get
$$f(x) = h(g(x)).$$
This does not seem to make the problem much easier yet, but I claim that the functions $g$ and $h$ are more easy to handle than $f$. Let's focus on $g$ a little more. We have
$$
g(t)=t^2-3t=t(t-3).
$$
This expression seems horribly asymmetric, but we might make it symmetric if we could just shift the variable by $\frac{3}{2}$. To do this we split $g$ up into two parts again. First send a variable $x$ to $x-\frac{3}{2}$, call this $k$ such that $k(x)=x-\frac{3}{2}$. Then apply $t\mapsto (t+\frac{3}{2})(t-\frac{3}{2})$ and call this function $m$. We see that
$$
g(x) = m(k(x)).
$$
Now we can investigate what we have gained from this symmetry. If we multiply the terms in $m$ we get
$$
m(x) = x^2-\frac{9}{4}.
$$
This function is certainly easier to deal with than $f$. We can split it up one more time into first sending $x$ to $x^2$ and then subtracting $\frac{9}{4}$. If we have done our bookkeeping correctly we see that we have now split up $f$ into the following four maps
\begin{align*}
f_1:&\ x \mapsto x - \frac{3}{2}\\
f_2:&\ x \mapsto x^2 \\
f_3:&\ x \mapsto x - \frac{9}{4} \\
f_4:&\ x \mapsto 4x.
\end{align*}
Now we have
$$f(x)=f_4\left(f_3\left(f_2\left(f_1\left(x\right)\right)\right)\right)$$ and we can just follow the interval around. We get
$$
[-1,3]\stackrel{f_1}{\mapsto} \left[-\frac{5}{2},\frac{3}{2}\right] \stackrel{f_2}{\mapsto} \left[0,\frac{25}{4}\right] \stackrel{f_3}{\mapsto} \left[-\frac{9}{4},\frac{16}{4}\right] \stackrel{f_4}{\mapsto} [-9,16].
$$
The images shown in each step here are hopefully easier to find than the image of $f$, if not obvious.


TITLE: count of Ordered Pairs such that their product is less than a number
QUESTION [1 upvotes]: I need a mathematical formulation for count of total ordered sets s.t. the product of two elements is less than an number, say n..
count{(i,j)}, s.t.i*j<=n

REPLY [1 votes]: If $d(x)$ is the number of divisors of $x$, then the answer to your question is given by
$$
D(n)=\sum_{x\le n}d(x).
$$
Determining the behavior of this function is known as Dirichlet's divisor problem.  There is not a closed-form solution, but it is known to behave asymptotically as
$$
D(n) \sim n \log n+n(2\gamma-1)+O(\sqrt{n}),
$$
where $\gamma$ is the Euler-Mascheroni constant; the $O(\sqrt{n})$ error term can be tightened further.


TITLE: Gradient of logarithm of determinant of a matrix
QUESTION [0 upvotes]: Given that $A$ is a positive definite square matrix, is the following equality valid: $(A^{-1})^{T}=\nabla_{A} \ln(\det(A))$

REPLY [1 votes]: The formula works with the natural logarithm, just using the expansions of both the inverse and the determinant.
By the Laplace expansion, the determinant can be computed as:
$$
\det(A)=\sum_j (-1)^{i+j} A_{ij}\,M_{ij}
$$
where $M_{ij}$ is the $(i,j)$ minor of $A$, i.e. the determinant of the matrix obtained by removing row $i$ and column $j$. Then,
$$
\frac{\partial}{\partial A_{ij}}\det(A)= (-1)^{i+j} M_{ij}=C_{ij}
$$
where $C$ is the matrix of cofactors.
By using the chain rule,
$$
\frac{\partial}{\partial A_{ij}} \log(\det(A))=\frac{1}{\det(A)} C_{ij}
$$
so that we can write the gradient as a matrix $G$ where
$$
G_{ij}=\frac{\partial}{\partial A_{ij}} \log(\det(A))=\frac{1}{\det(A)} C_{ij}\implies G=\frac{1}{\det(A)}C
$$
Now, just recall that the inverse can be computed as:
$$
A^{-1}=\frac{1}{\det(A)} C^\top
$$
and you are done.