On the Sum of Reciprocals of Primes

TL;DR

The paper studies the divergence of reciprocal sums over primes constrained by a cosine phase condition $\cos(y\ln p+\alpha)$. It partitions primes into $P^+(y,\alpha,K)$ and $P^-(y,\alpha,K)$ and proves that both $\sum_{p\in P^+} 1/p$ and $\sum_{p\in P^-} 1/p$ diverge for any $y>0$, $0\le \alpha<2\pi$, $0<K<1$. The proof relies on the prime number theorem and a decomposition of primes into logarithmic intervals $A_n$ and $B_n$, establishing lower bounds for the corresponding partial sums and showing the divergence of the total sums. These results support the author's broader work related to the Riemann hypothesis by illustrating how primes distribute across oscillatory phase windows. The methods highlight the robustness of divergence phenomena even when primes are filtered by trigonometric phase constraints.

Abstract

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

Theorems & Definitions (5)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Definition 2.2