On the behaviour of $\gamma \text{Log}p$ modulo 1

We prove non-trivial lower bounds for sums of type $\sum_{p\sim P}g(\gamma\Log p)$,where$g$isanon-negative$2\pi$-periodicalfunctionand$\gamma$isagivenparameter\mathrm{.}Asanapplicationweprovethat$\zeta(1+it)^{\pm1}\ll\Log\Log (9+|t|)$ and extend the zero-free region of the Riemann zeta-function.