On the number of divisors of Mersenne numbers

TL;DR

This paper investigates the growth of the sum of divisors of Mersenne numbers, showing that the ratio of these sums for powers of two is unbounded and providing conditional results and numerical evidence for divergence.

Contribution

It proves that the sequence of ratios of divisor sums of Mersenne numbers is unbounded and offers conditional results and numerical tests supporting divergence.

Findings

The ratio f(2n)/f(n) is unbounded.

Conditional divergence results are established.

Numerical evidence supports the conjecture of divergence.

Abstract

Denote $f(n):=\sum_{1\le k\le n} \tau(2^k-1)$, where $\tau$ is the number of divisors function. Motivated by a question of Paul Erd\H{o}s, we show that the sequence of ratios $f(2n)/f(n)$ is unbounded. We also present conditional results on the divergence of this sequence to infinity. Finally, we test numerically both the conjecture $f(2n)/f(n)\to\infty$ and our sufficient conditions for it to hold.