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

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.