On the behaviour of $\gamma\Log p$ modulo 1

TL;DR

This paper establishes lower bounds for sums involving the logarithm of primes and a periodic function, and applies these results to improve bounds on the Riemann zeta-function and its zero-free region.

Contribution

It introduces new lower bounds for sums over primes involving periodic functions and applies these to extend the zero-free region of the Riemann zeta-function.

Findings

Proves non-trivial lower bounds for sums over primes with periodic functions.

Shows that +it and its inverse are bounded by  Log Log (|t|).

Extends the zero-free region of the Riemann zeta-function.

Abstract

We prove non-trivial lower bounds for sums of type $\sum_{p\sim P}g(\gamma\Log p)$, where $g$ is a non-negative $2\pi$-periodical function and $\gamma$ is a given parameter. As an application we prove that $\zeta(1+it)^{\pm1}\ll\Log\Log (9+|t|)$ and extend the zero-free region of the Riemann zeta-function.