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 $1/2+h_1/(\log T{)}^{\theta }\le \Re s\le 1/2+h_2/(\log T{)}^{\theta }$, $T\le \Im s\le 2T$ 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 Radziwiłł'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 $\Re s=1/2+1/(\log T{)}^{\theta }$.