The $a$-values of the Riemann zeta function near the critical line

TL;DR

This paper investigates the distribution of the Riemann zeta function's values near the critical line, deriving asymptotic formulas and extending recent discrepancy results to better understand its behavior.

Contribution

It provides an asymptotic formula for the number of a-values near the critical line and extends the range of discrepancy results for the zeta function's distribution.

Findings

Asymptotic formula for a-values in a specified rectangle near the critical line

Extension of discrepancy results for the zeta function's distribution

Sharper estimates for the secondary main term in Selberg's CLT

Abstract

We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $, $T \leq \Im s \leq 2T$ for fixed $h_1, h_2>0$ and $ 0 < \theta <1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwi\l\l's recent results on the discrepancy between the distribution of $\zeta(s)$ and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line $\Re s = 1/2 + 1/(\log T)^\theta $.