The reciprocal sum of divisors of Mersenne numbers

TL;DR

This paper studies the sum of reciprocals of divisors and prime divisors of Mersenne numbers, providing conditional asymptotic results under major conjectures and confirming related conjectures and questions in number theory.

Contribution

It refines previous estimates on reciprocal sums of divisors of Mersenne numbers under the Elliott-Halberstam and GRH conjectures, and establishes distributional properties.

Findings

Conditional maximum of prime divisor reciprocals is determined within o(1).

Sum over all divisors of Mersenne numbers is approximated within a factor of 1+o(1).

Distribution functions for these sums are shown to exist.

Abstract

We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and $\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $x\to\infty$. This refines, conditionally, earlier estimates of Erd\H{o}s and Erd\H{o}s-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $n\le x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $\sum_{p\mid 2^n-1} 1/p$ and $\sum_{d\mid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.