The reciprocal sum of divisors of Mersenne numbers

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