On the Sum of Reciprocals of Primes

Suppose that $y>0$, $0\le \alpha <2\pi$, and $0<K<1$. Let $P^{+}$ be the set of primes $p$ such that $\cos (y\ln p+\alpha )>K$ and $P^{-}$ the set of primes $p$ such that $\cos (y\ln p+\alpha )<-K$. In this paper, we prove ${\sum }_{p\in P^{+}}\frac{1}{p}=\mathrm{\infty }$ and ${\sum }_{p\in P^{-}}\frac{1}{p}=\mathrm{\infty }$.