The reciprocal sum of divisors of Mersenne numbers release_qxudtuzqarhpfl7mwuf4zhe4s4

by Zebediah Engberg, Paul Pollack

Released as a article .

2020  

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< x∑_p | 2^n-1 1/p to within o(1) and max_n< x∑_d| 2^n-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 2^n-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| 2^n-1 1/p and ∑_d| 2^n-11/d admit continuous distribution functions in the sense of probabilistic number theory.
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Date   2020-06-03
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arXiv  2006.02373v1
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