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

Let $E(s, Q)$ be the Epstein zeta function attached to a positive definite quadratic form of discriminant $D<0$, such that $h(D)\geq 2$, where $h(D)$ is the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{D})$. We denote by $N_E(\sigma_1, \sigma_2, T)$ the number of zeros of $E(s, Q)$ in the rectangle $\sigma_1 <\text{Re}(s)\leq \sigma_2$ and $T\leq \text{Im}(s)\leq 2T$, where $1/2<\sigma_1<\sigma_2<1$ are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for $N_E(\sigma_1, \sigma_2, T)$, obtaining a saving of a power of $\log T$ in the error term.

Zeros of the Epstein zeta function to the right of the critical line

On the zeros of Epstein zeta functions near the critical line

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 Radziwill'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 $.

Yoonbok Lee

Papers

Zero-density estimates for epstein zeta functions of class numbers 2 or 3