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Back ground

I studied the proof of "$KT$ zero theorem" and "$KT\log T$" theorem in Edwards book. And I'm looking for other kind of evaluation of the number of zeros on the line.

Take $$ \sigma<\min\lbrace \Re\rho|\frac{1}{2}<\Re\rho<1,0\leqq\Im\rho\leqq T,\zeta(\rho)=0\rbrace $$

Define

$N_0(T)$ the number of zeros on the critical line

$S_0(T)$ the argument of $\zeta(s)$ measured along the path

$\sigma\rightarrow \sigma+iT\rightarrow \frac{1}{2}+iT$

so that

$$ N_0(T)=\frac{T}{2\pi}\log\frac{T}{2\pi}-\frac{T}{2\pi}+S_0(T)+O(1) $$

holds.

How can I evaluate $S_0(T)$ to get a result about the zeros.

In particular, Can we show that $S_0(T)$　change its sign infinitely as the argument $S(T)$ along the path $2\rightarrow 2+iT\rightarrow \frac{1}{2}+iT$.

More precisely, can I apply the technique of Selberg in the paper "Contributions to the theory of the Rieman zeta function" to evaluate $S_0(T)$ here?

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    $\begingroup$ The formula you have given here is the zero counting formula for all nontrivial zeros with positive imaginary part $\le T$ instead of the one that only counts zeros on the critical line. $\endgroup$
    – TravorLZH
    Commented Sep 6, 2023 at 4:40
  • $\begingroup$ This formula holds also for this $N_0(T)$ and $S_0(T)$. You can prove in the same way as usual one. Note that $S_0(T)$ is define along the path $\sigma\rightarrow\sigma+Ti\rightarrow 1/2+Ti$. $\endgroup$
    – George
    Commented Sep 6, 2023 at 5:00

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