The convergence of an alternating series of Erd\H{o}s, assuming the Hardy--Littlewood prime tuples conjecture

TL;DR

This paper investigates the convergence of a specific alternating series involving primes, demonstrating that it converges conditionally under a strong form of the Hardy--Littlewood prime tuples conjecture using a probabilistic prime model.

Contribution

It proves the conditional convergence of an Erdős question assuming a strong Hardy--Littlewood prime tuples conjecture, employing a novel probabilistic prime model.

Findings

Conditional convergence of the series under the conjecture

Application of a random sifted model of primes

Extension of Gallagher's calculation to this context

Abstract

It is an open question of Erd\H{o}s as to whether the alternating series $\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$ is (conditionally) convergent, where $p_n$ denotes the $n^{\mathrm{th}}$ prime. By using a random sifted model of the primes recently introduced by Banks, Ford, and the author, as well as variants of a well known calculation of Gallagher, we show that the answer to this question is affirmative assuming a suitably strong version of the Hardy--Littlewood prime tuples conjecture.