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arXiv · 2308.07205

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

Terence Tao
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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.

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Terence Tao. 2023-08-14. The convergence of an alternating series of Erd\H{o}s, assuming the Hardy--Littlewood prime tuples conjecture. https://arxiv.org/abs/2308.07205

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