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.

Locations

• Communications of the American Mathematical Society - View - PDF
• Communications of the American Mathematical Society - View - PDF
• Communications of the American Mathematical Society - View - PDF