On divisor sums due to Erd\H{o}s and Ramanujan

TL;DR

This paper investigates the asymptotic behavior of a hybrid divisor sum involving the iterated divisor function, extending classical results by combining methods from Erdős, Ramanujan, and recent divisor sum studies.

Contribution

It establishes the asymptotic order of the sum of reciprocals of the iterated divisor function, combining classical and recent techniques in analytic number theory.

Findings

Proves that _{n \u2264 x} 1/d(d(n)) \u2248 x / { ext{log log x}}.

Uses Golomb's estimate for powerful numbers and Ture1n's Hardy-Ramanujan theorem.

Extends classical divisor sum asymptotics to a more complex hybrid sum.

Abstract

Let $d(n)$ denote the number of divisors of a positive integer $n$. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum $\sum_{n \leq x} \frac{1}{d(n)}$, with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant $A_1$ for the leading term $A_1 \frac{x}{\sqrt{\log x}}$. Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning $\sum_{p\leq x} d(p-1)$, proving that $$\sum_{p\leq x} \frac{1}{d(p-1)} \asymp \frac{x}{(\log x)^{3/2}}.$$ This result, together with Erd\H{o}s's asymptotic formula $\sum_{n \leq x} d(d(n)) \sim c \, x \log \log x $ for a constant $c \in (0, \infty)$, lead us to consider the hybrid $\sum_{n \leq x} \frac{1}{d(d(n))}$ of the Erd\H{o}s and Ramanujan divisor sums. The presence of the reciprocal significantly complicates the analysis, as it amplifies the contribution of integers for which $d(d(n))$ is exceptionally small. In this paper, we prove that $$\sum_{n \leq x} \frac{1}{d(d(n))} \asymp \frac{x}{ \log \log x}, $$ through a combined application of Golomb's estimate for powerful numbers and Tur\'an's quantitative form of the Hardy-Ramanujan theorem.