Equivalents of the Riemann hypothesis involving the Gram points

Abstract

The Riemann hypothesis (RH) on zeros of the zeta-function $\zeta(s)$, $s=\sigma +it$, states that all zeros of $\zeta(s)$ in the strip $0 < \sigma < 1$ lie on the line $\sigma =1/2$. Several equivalents of RH are known. In the paper, we obtain equivalents of RH in terms of self-approximation of $\zeta(s)$ by shifts $\zeta(s+iht_k)$, $k\in \mathbb{N}$, where $\{t_k, k\in \mathbb{N}\}$ is the sequence of the Gram points.