On the distribution of values of the argument of the Riemann zeta-function

TL;DR

This paper investigates the distribution of the argument of the Riemann zeta-function along the critical line, providing asymptotic formulas for the measure of sets where the argument is positive or negative, based on the distribution of a related Dirichlet polynomial.

Contribution

It establishes a new asymptotic formula for the measure of the argument's sign distribution, using a uniform distribution approach and analysis of a Dirichlet polynomial approximation.

Findings

Measure of t where S(t)>0 is approximately H/2 with explicit error bounds.

Similar measure estimate holds for S(t)<0, symmetric to the positive case.

Distribution of S(t) is shown to be uniform in relevant parameters.

Abstract

Let $S(t) \;:=\; \frac{\displaystyle 1}{\displaystyle \pi}\arg \zeta(\frac{1}{2} + it)$. We prove that, for $T^{\,27/82+\varepsilon} \le H \le T$, we have $$ {\rm mes}\Bigl\{t\in [T, T+H]\;:\; S(t)>0\Bigr\} = \frac{H}{2} + O\left(\frac{H\log_3T}{\varepsilon\sqrt{\log_2T}}\right), $$ where the $O$-constant is absolute. A similar formula holds for the measure of the set with $S(t)<0$, where $\log_kT = \log(\log_{k-1}T)$. This result is derived from an asymptotic formula for the distribution of values of $S(t)$, which is uniform in the relevant parameters, and this is of crucial importance. This in fact depends on the distribution of values of the Dirichlet polynomial which approximates $S(t)$, namely ($p$ denotes primes) $$V_{y}(t)\,=\,\sum\limits_{p\le y}\frac{\sin{(t\log{p})}}{\sqrt{p}}.$$