The reciprocal sum of divisors of Mersenne numbers
Volume 197 / 2021
Abstract
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditionally 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.
