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.
Categories: math.NT