The convergence of an alternating series of Erdős, assuming the Hardy--Littlewood prime tuples conjecture

Abstract

It is an open question of Erdős 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.

16 pages, no figures. A generalization to the main result added, as well as some more references and open questions