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

1. Problem Statement

The paper investigates the asymptotic behavior of a specific hybrid divisor sum that combines elements from two classical problems in analytic number theory:

• The Ramanujan Divisor Sum: Concerns the sum of reciprocals of the divisor function, $\sum_{n \le x} \frac{1}{d(n)}$. Ramanujan (1915) established that this sum behaves asymptotically as $\frac{x}{\sqrt{\log x}}$.
• The Erdős Divisor Sum: Concerns the sum of the iterated divisor function, $\sum_{n \le x} d(d(n))$. Erdős (1950s) proved this behaves as $c_1 x \log \log x$.
• The Titchmarsh Divisor Problem: Concerns sums over primes, specifically $\sum_{p \le x} d(p-1)$. Recent work by Gabdullin et al. (2020s) analyzed the reciprocal version $\sum_{p \le x} \frac{1}{d(p-1)}$, proving it is of order $\frac{x}{(\log x)^{3/2}}$.

The Core Problem:
The author studies the "hybrid" sum formed by taking the reciprocal of the Erdős iterated function:
$$S(x) = \sum_{n \le x} \frac{1}{d(d(n))}$$
The presence of the reciprocal significantly complicates the analysis because it amplifies the contribution of integers $n$ where $d(d(n))$ is exceptionally small. The paper aims to determine the order of magnitude of $S(x)$.

2. Methodology

The proof relies on decomposing the set of integers $\{n \le x\}$ into "typical" and "exceptional" sets, bounding the contribution of the exceptional sets, and estimating the main term using probabilistic number theory tools.

Key Tools and Definitions:

• Decomposition of $n$: Integers are split into a squarefree part $a(n)$ and a powerful part $b(n)$ (where $n = a(n)b(n)$). The divisor function is expressed as $d(n) = 2^{\omega(a(n))} d(b(n))$, where $\omega(n)$ is the number of distinct prime factors.
• Golomb's Estimate: Used to bound the density of powerful numbers. The number of powerful numbers $\le x$ is $O(x^{1/2})$.
• Turán's Quantitative Hardy–Ramanujan Theorem: Utilizes the Turán–Kubilius inequality to show that $\omega(n)$ is normally distributed around $\log \log n$. Specifically, the variance is small: $\sum_{n \le N} (\omega(n) - \log \log N)^2 \ll N \log \log N$.
• Abel Summation: Applied to sum over powerful numbers to derive bounds.

Proof Strategy (Lemma Decomposition):
The author partitions the summation range $n \le x$ into three disjoint sets to bound the sum from above and below:

1. Set $B_1(x)$ (Large Powerful Part): Integers where the powerful part $b(n) > (\log \log x)^2$.

   • Method: Using Golomb's estimate and Abel summation, the author proves the count of such integers is small ($\ll \frac{x}{\log \log x}$). Since $\frac{1}{d(d(n))} \le 1$, their total contribution is negligible.

2. Set $B_2(x)$ (Atypical Prime Factors): Integers where $\omega(n)$ deviates significantly from the mean $\log \log x$ (specifically outside $[\frac{1}{2}\log \log x, 2\log \log x]$).

   • Method: Using Turán's variance estimate (Eq. 15), the author shows the number of such integers is $\ll \frac{x}{\log \log x}$. Their contribution is also negligible.

3. Set $B_3(x)$ (Typical Squarefree Integers): Integers that are squarefree and have $\omega(n)$ within the normal range.

   • Method: For squarefree $n$, $d(n) = 2^{\omega(n)}$. Consequently, $d(d(n)) = d(2^{\omega(n)}) = \omega(n) + 1$.
   • Since $\omega(n) \approx \log \log x$, the term $\frac{1}{d(d(n))} \approx \frac{1}{\log \log x}$.
   • The density of squarefree numbers is $\frac{6}{\pi^2}x$. The author proves that the sum over this set provides the dominant lower bound.

3. Key Contributions and Results

Main Theorem:
The paper establishes the order of magnitude for the hybrid sum:
$$\sum_{n \le x} \frac{1}{d(d(n))} \asymp \frac{x}{\log \log x}$$
This means there exist positive constants $c_{min}$ and $c_{max}$ such that for sufficiently large $x$:
$$c_{min} \frac{x}{\log \log x} \le \sum_{n \le x} \frac{1}{d(d(n))} \le c_{max} \frac{x}{\log \log x}$$

Conjecture:
By analogy with the conjectured asymptotic equivalence for the Titchmarsh-Ramanujan hybrid (Gabdullin et al.), the author conjectures a precise asymptotic formula:
$$\sum_{n \le x} \frac{1}{d(d(n))} \sim c_3 \frac{x}{\log \log x}$$
for some constant $c_3 \in (0, \infty)$. The author notes that proving the existence of this specific constant is likely a difficult problem, similar to the difficulty of the Titchmarsh-Ramanujan conjecture.

4. Significance

• Bridging Classical Problems: The work successfully unifies the study of the Erdős iterated divisor sum and the Ramanujan reciprocal divisor sum, creating a new "hybrid" problem in analytic number theory.
• Handling Reciprocals: It demonstrates how to handle the analytical difficulties introduced by reciprocals in divisor sums, specifically how the "small divisor" cases (integers with few divisors) dominate the sum's behavior.
• Methodological Application: The paper effectively combines Golomb's bounds on powerful numbers with Turán's probabilistic estimates on the distribution of prime factors to solve a problem where standard multiplicative techniques (like Dirichlet's hyperbola method) fail due to the non-multiplicative nature of $d(d(n))$.
• Future Directions: The paper sets the stage for future research into the precise constant $c_3$ and the potential application of similar hybrid techniques to other iterated arithmetic functions.

In summary, Campbell proves that the sum of the reciprocals of the iterated divisor function grows as $\frac{x}{\log \log x}$, a result that is slower than the linear growth of the non-reciprocal Erdős sum but faster than the $\frac{x}{\sqrt{\log x}}$ growth of the standard Ramanujan reciprocal sum.