On divisor sums due to Erdős and Ramanujan

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 ${\sum }_{n\le x}\frac{1}{d(n)}$, 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 ${\sum }_{p\le x}d(p-1)$, proving that $$\sum _{p\le x}\frac{1}{d(p-1)}\asymp \frac{x}{(\log x{)}^{3\mathrm{/}2}}\mathrm{.}$$ This result, together with Erdős’s asymptotic formula ${\sum }_{n\le x}d(d(n))\sim cx\log \log x$ for a constant $c\in (0,\mathrm{\infty })$, lead us to consider the hybrid ${\sum }_{n\le x}\frac{1}{d(d(n))}$ of the Erdős 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 $$\sum _{n\le x}\frac{1}{d(d(n))}\asymp \frac{x}{\log \log x},$$ through a combined application of Golomb’s estimate for powerful numbers and Turán’s quantitative form of the Hardy-Ramanujan theorem.