PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

The reciprocal sum of divisors of Mersenne numbers

Volume 197 / 2021

Zebediah Engberg, Paul Pollack Acta Arithmetica 197 (2021), 421-440 MSC: Primary 11N37; Secondary 11A25, 11B37. DOI: 10.4064/aa200602-11-9 Published online: 1 December 2020

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.

Authors

  • Zebediah EngbergWasatch Academy
    Mt. Pleasant, UT 84647, U.S.A.
    e-mail
  • Paul PollackDepartment of Mathematics
    University of Georgia
    Athens, GA 30602, U.S.A.
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image