2020/06/03 by Engberg, Zebediah, Pollack, Paul
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2006.02373
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers 2n-1. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine maxn≤ x ∑p | 2n-1 1/p to within o(1) and maxn≤ x ∑d| 2n-11/d to within a factor of 1+o(1), as x→∞. This refines, conditionally, earlier estimates of Erdős and Erdős-Kiss-Pomerance. Conditionally (only) on GRH, we also determine ∑ 1/d to within a factor of 1+o(1) where d runs over all numbers dividing 2n-1 for some n≤ x. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both ∑p| 2n-1 1/p and ∑d| 2n-11/d admit continuous distribution functions in the sense of probabilistic number theory.