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

Olivier Ramaré

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

Number Theory 2024-12-17 v6

Authors: Olivier Ramaré

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.

Keywords

gamma log p modulo 1 lower bounds Riemann zeta function zero-free region prime numbers

Cite

@article{arxiv.1203.1759,
  title  = {On the behaviour of $\gamma\Log p$ modulo 1},
  author = {Olivier Ramaré},
  journal= {arXiv preprint arXiv:1203.1759},
  year   = {2024}
}

Comments

Fundamental error detected

Related papers

View all related →