The reciprocal sum of divisors of Mersenne numbers
Tom 197 / 2021
Streszczenie
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers . 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.