On the Sum of Reciprocals of Primes

Abstract

Suppose that $y>0$, $0\leq\alpha<2\pi$, 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 $\sum_{p\in P^+}\frac{1}{p}=\infty$ and $\sum_{p\in P^-}\frac{1}{p}=\infty$.