TITLE: Best error terms for functions related to square free numbers
QUESTION [0 upvotes]: Let $\mu$ denote the Möbius function, and let's define the functions $\mu_{-}$ and $\mu_{+}$ as follows: $\mu_{-}(n):=\frac{\mu(n)^{2}-\mu(n)}{2}$ and $\mu_{+}(n):=\frac{\mu(n)^{2}+\mu(n)}{2}$. Let $M_{-}$ be the summatory function of $\mu_{-}$ and $M_{+}$ the summatory function of $\mu_{+}$. One has $M(x)=M_{+}(x)-M_{-}(x)$, where $M$ is the summatory fonction of $\mu$, and $\hat{M}(x):=M_{+}(x)+M_{-}(x)$ is the number of squarefree integers below $x$. One has $\hat{M}(x)=\frac{1}{\zeta(2)}x+O(\sqrt{x})$. Let $\alpha$ and $\beta$ be such that $M_{-}(x)=\frac{1}{2\zeta(2)}x+O(x^{\alpha})$ and $M_{+}(x)=\frac{1}{2\zeta(2)}x+O(x^{\beta})$. If one can have $\alpha=\beta$ then these two quantities can be taken equal to $\frac{1}{2}$, and thus one would have $M(x)\ll\sqrt{x}$ which would imply RH.
What are the best known (unconditional) error terms for $M_{+}$ and $M_{-}$? Is there any piece of evidence aside RH that they should be the same up to a multiplicative bounded quantity?
Thanks in advance.

REPLY [4 votes]: As I say in the comment, the asymptotics for $M_{+}$ and $M_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M_{-}(x) = \frac{1}{2 \zeta(2)} x - \frac{1}{2} M(x) + O(x^{1/2})$. These error terms will have the same order of magnitude unless $M(x)$ is small.
If $M(x) \ll \sqrt{x}$ for all $x$, then Gonek's conjecture mentioned in the paper of Ng is false. However,
there will probably be infinitely many $x$ for which $M(x)$ is small, and for
these $x$ it will be less clear what the size of the error terms of $M_{+}$ and $M_{-}$ will be. (Corollary 1 of Ng's paper, which is very conditional, implies among other things that the logarithmic density of $x$ for which $M(x) \leq \epsilon \sqrt{x}$ tends to zero as $\epsilon \to 0$.)
Finally, the best unconditional bound on $M(x)$ was given by Ivic in his book "The Riemann Zeta-Function" and states that
$$
M(x) = O\left(x \exp\left(-c_{1} (\log^{3/5} x) (\log \log x)^{-1/5}\right)\right).
$$
(The best conditional bound was given by Soundararajan in 2009.)


TITLE: How to show optimal value of linear program is unbounded?
QUESTION [2 upvotes]: The problem is given as
\begin{equation*}
\text{minimize} \hspace{.8em} -x_1-3x_2\\
\text{subject to} \hspace{.8em} x_1-2x_2 \le 4\\
\hspace{2 cm} -x_1+x_2 \le 3 \\ \hspace{2 cm} x_1,x_2\geq 0.
\end{equation*}
Graphically it's clear that the solution is unbounded but how do i show this analytically? i have converted it into the form
\begin{equation*}
\text{minimize} \hspace{.8em} C^{\top}x\\
\text{subject to} \hspace{.8em} Ax=b\\
\hspace{2 cm} x_1,x_2 \ge 0
\end{equation*}
Where $C^{\top}=[-1~-3~~ 0~ ~0~ ~0], b=[4~~3]^{\top}$ and $A=\begin{pmatrix}1&-2&1&0\\-1&1&0&1\end{pmatrix}$ , Now i think i have to find a vector $d$ s.t cost is unbounded for $x_{*}+\alpha d$, so how do i choose $d$ and $x_{*}$ ? or there's any other way ?

REPLY [1 votes]: If your plot out the diagram,

you can pick $x_*=(4,0)^T$ and pick $d=(2,1)^T$.  Note that $(2,1)$ is orthogonal to $(1,-2)$ which is the normal direction of the first inequality.
$$\lim_{\alpha \to \infty}-(4+2\alpha)-3\alpha \to -\infty$$
$$(4+2\alpha)-2(\alpha)\le 4$$
$$-(4+2\alpha)+\alpha=-4-\alpha\le -4\le -3$$
$$4+2\alpha \ge 0, \alpha \ge 0$$


TITLE: Why do objects move back after collision?
QUESTION [0 upvotes]: When a ball collides with a wall, the ball moves in the opposite direction, why does this happen if Newton's third law operates. I mean the ball collided with the wall and imparted a force on it and simultaneously the wall imparted a force on the ball making the net force on the ball zero. So why is there still acceleration (motion) in the opposite direction?

REPLY [1 votes]: I mean the ball collided with the wall and imparted a force on it [the wall].

Here you are writing about the force on the wall.

and simultaneously the wall imparted a force on the ball.

Here you are writing about the force on the ball and this force is the Newton’s third law pair to the force on the wall.

making the net force on the ball zero.

This cannot be correct as you have only identified one force on the ball.
A point to remember is that Newton’s third law forces always act on different objects.
The force on object A due to object B is equal in magnitude and opposite in direction to the force on object B due to object A.

REPLY [1 votes]: When the ball collided with the wall, it also collided with whatever the wall was attached to. In this case, perhaps you can see the collision with the wall as a collision with the earth.
You can see using this simulation that if something of low mass collides at high velocity with a stationary object of high mass, it is reflected in the opposite direction with roughly the same velocity (provided the collision is elastic): https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html. In this case the mass of the ball is insignificant compared to that of the earth so its velocity following the collision can be approximated as you have mentioned. On the other hand, the earth and the wall moves in the opposite direction with a really small velocity.
If you assume that both momentum and kinetic energy due to movement remain constant in the collision, then the outcome of the collision must involve the ball moving in the other direction.


TITLE: Energy in nuclear reactions
QUESTION [1 upvotes]: I have two questions, been trying to get definite and intuitive answers to them for some time so hopefully you can help me:
1) I understand both the strong force and binding energy but what is the relationship between the two?
2) What actually causes energy to be released when nuclei fuse or split? In my high school textbook it says changing the average binging energy between nucleons causes energy to be released... could you please make this clearer?

REPLY [3 votes]: Well, there are only two types of energy: the kinetic energy associated with motion and the potential energy associated with an interaction (i.e. a force). For example, if the force is gravity, the potential energy is the gravitational potential energy. In the case of the strong force in the nuclei, the associated energy is called the binding energy.
Let say we have the following nuclear reaction  $$_{Z_1}^{A_1}X_1+_{Z_2}^{A_2}X_2\longrightarrow _{Z'_1}^{A'_1}Y_1+_{Z'_2}^{A'_2}Y_2+_{Z'_3}^{A'_3}Y_3+_0^0\gamma$$
then the difference of masses between the Xs and the Ys is converted to kinetic energy of the Ys and to energy of $\gamma$ (this also happens in chemical reactions).


TITLE: Ring Inside an Algebraic Field Extension
QUESTION [8 upvotes]: Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?

REPLY [3 votes]: A more general argument can be given: let $A\subset B$ be two integral domains. An element $b\in B$  is said to be integral over $A$ if there exists a monic polynomial $p(t)\in A[t]$ with $p(b)=0$. The ring $B$ is said to be integral over $A$ is every element in $B$ is integral over $A$.
Theorem: Let $A\subset B$ be I.D.'s , with $B$ integral over $A$. Then, $A$ is a field iff $B$ is a field.