Discrete universality of the Riemann zeta-function and uniform distribution modulo 1

Abstract:

It is proved that a wide class of analytic functions can be approximated by shifts $\zeta (s+i\varphi (k))$, $k\geq k_0$, $k\in \mathbb {N}$, of the Riemann zeta-function. Here the function $\varphi (t)$ has a continuous nonvanishing derivative on $[k_0,\infty )$ satisfying the estimate $\varphi (2t) \max _{t\leq u \leq 2t} \left (\varphi ’(u)\right )^{-1}\ll t$, and the sequence $\{a\varphi (k) : k\geq k_0\}$ with every real $a\neq 0$ is uniformly distributed modulo 1. Examples of $\varphi (t)$ are given.