The reciprocal sum of divisors of Mersenne numbers

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 Erdős and Erdős-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.

15 pages. Made several clarifications and corrections. Added new section on generalizations