The reciprocal sum of divisors of Mersenne numbers
release_qxudtuzqarhpfl7mwuf4zhe4s4
by
Zebediah Engberg, Paul Pollack
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.
In text/plain
format
Archived Files and Locations
application/pdf 220.6 kB
file_i5jrc3pzifgcxkecsvthdepe4a
|
arxiv.org (repository) web.archive.org (webarchive) |
2006.02373v1
access all versions, variants, and formats of this works (eg, pre-prints)