  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.
