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

Details On the behaviour of $γ\Log p$ modulo 1 On the behaviour of $γ\Log p$ modulo 1

2012-03-08

arxiv.org/abs/1203.1759v2

Mathematics

Number Theory

This paper has been withdrawn by the author due to a serious mistake that is being investigated

Scientific paper

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.

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