On the number of divisors of Mersenne numbers

TL;DR

This paper investigates the growth of the divisor-sum over Mersenne numbers, $f(n)=\sum_{1\le k\le n} \tau(2^k-1)$. It proves that the doubling ratio $f(2n)/f(n)$ is unbounded in the sense that $\limsup_{n\to\infty} f(2n)/f(n)=\infty$, and it achieves this via a comparison to the function $f'(n)=\sum_{1\le k\le n} 2^{\tau(k)}$ together with highly--composite Mersenne indices. The authors also formulate two conditional results: if either Conjecture 1 (existence of indices of highly--composite Mersenne numbers implying growth of $\tau(2^N+1)$) or Conjecture 2 (a logarithmic bound on $\omega(\Phi_d(2))$) holds, then $f(2n)/f(n)\to\infty$. Extensive numerical data, along with a partially heuristic model based on Gillies' approximation, corroborate the observed large fluctuations in $\tau(2^n-1)$ and support the proposed divergence, providing insight into the complex divisor structure of Mersenne numbers and its connections to cyclotomic polynomials and highly--composite phenomena.

Abstract

Denote $f(n):=\sum_{1\le k\le n} τ(2^k-1)$, where $τ$ is the number of divisors function. Motivated by a question of Paul Erdős, 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.

Figures (5)

• Figure 1: Values of $\tau(2^n-1)$ for $1\leqslant n\leqslant 100$.
• Figure 2: Numerical values of the ratios $f(2n)/f(n)$ for $1\leqslant n\leqslant 50$, $1\leqslant n\leqslant 200$, and $1\leqslant n\leqslant 603$, respectively.
• Figure 3: Values of $\omega(\Phi_d(2))$ for $1\leqslant d\leqslant 400$ and $1\leqslant d\leqslant 1206$, respectively.
• Figure 4: Base $10$ logarithms of the ratios between the approximate and the true values of $\tau(2^n-1)$ for $1001\leqslant n\leqslant 1100$.
• Figure 5: Numerical values of the approximately evaluated ratios $f(2n)/f(n)$ for $1\leqslant n\leqslant 1000$ (left) and the last $100$ points zoomed (right).

Theorems & Definitions (9)

• Theorem 1
• Proposition 2
• Conjecture 1
• Conjecture 2
• Theorem 3
• proof : Proof of Proposition \ref{['prop']}
• proof : Proof of Theorem \ref{['thm:limsup']}
• proof : Proof of Theorem \ref{['thm:cond']} assuming Conjecture \ref{['conj1']}
• proof : Proof of Theorem \ref{['thm:cond']} assuming Conjecture \ref{['conj2']}