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

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 $.