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

Abstract

Let $S(t):=\frac{1}{\pi }\arg \zeta (\frac{1}{2}+it)$. We prove that, for $T^{27/82+\epsilon }\le H\le T$, we have $$\mathrm{mes}\{t\in [T,T+H]:S(t)>0\}=\frac{H}{2}+O\left(\frac{H{\log }_{3}T}{\epsilon \sqrt{{\log }_{2}T}}\right),$$ where the $O$-constant is absolute. A similar formula holds for the measure of the set with $S(t)<0$, where ${\log }_{k}T=\log ({\log }_{k-1}T)$. 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) $$V_y(t)=\sum _{p\le y}\frac{\sin (t\log p)}{\sqrt{p}}.$$

28 pages