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

2023/08/14 by Terence Tao, Tao, Terence
Mathematics · #11N05 #40A05 #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2308.07205

openalex publication_date 2023/08/14 · openalex created_date 2023/08/17 · openalex updated_date 2026/07/28

Abstract

It is an open question of Erdős as to whether the alternating series ∑_n=1^∞ ((-1)ⁿ n)/(p_n) is (conditionally) convergent, where p_n denotes the n^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.

Related

• On the frequency of small gaps between the primes
• Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture
• On a conjecture of Erdős concerning primitive sequences
• Generalising the Hardy-Littlewood Method for Primes
• Pointwise ergodic theorem along primes of the form x² + ny²