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arxiv: 2605.00695 · v1 · submitted 2026-05-01 · 🧮 math.NT

On divisor sums due to ErdH{o}s and Ramanujan

John M. Campbell This is my paper

Pith reviewed 2026-05-09 18:28 UTC · model grok-4.3

classification 🧮 math.NT
keywords divisor functioniterated divisorsasymptotic estimatespowerful numbersHardy-Ramanujan theoremErdős divisor sumRamanujan divisor sum
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The pith

The sum ∑_{n≤x} 1/d(d(n)) is asymptotically x / log log x.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the growth rate of the sum over n up to x of the reciprocal of d applied twice to n. This combines the reciprocal structure from Ramanujan's divisor sum with the iterated divisor count from Erdős's work. The reciprocal emphasizes cases where d(d(n)) is small, so the analysis must show these cases do not overwhelm the total. If the result holds, the typical size of d(d(n)) still determines the order of the sum despite the inversion.

Core claim

We prove that ∑_{n ≤ x} 1/d(d(n)) ≍ 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.

What carries the argument

The joint application of Golomb's estimate on the number of powerful integers and Turán's quantitative Hardy-Ramanujan theorem, which together control the contribution from n where d(d(n)) is unusually small.

If this is right

  • The sum is of order x / log log x with implied constants independent of x.
  • The reciprocal form reduces the order by roughly a factor of (log log x)^2 relative to the non-reciprocal Erdős sum.
  • The exceptional set where d(d(n)) is small does not alter the main asymptotic term.
  • The same bounding technique applies to other hybrid divisor sums that mix reciprocals with iterated divisor counts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same order may hold for sums involving one further iterate, such as 1/d(d(d(n))).
  • The average value of 1/d(d(n)) for n up to x is therefore on the order of 1 / log log x.
  • Obtaining a precise leading constant in the asymptotic may be feasible with a more refined version of the same estimates.

Load-bearing premise

The integers n for which d(d(n)) is exceptionally small contribute only a negligible portion that can be bounded using Golomb's and Turán's estimates without changing the leading term.

What would settle it

Direct numerical evaluation of the partial sum for a large explicit value of x such as 10^12, checking whether the result remains between two positive constant multiples of x / log log x.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that ∑_{n≤x} 1/d(d(n)) ≍ x / log log x. The lower bound follows from Erdős's asymptotic for the average order of d(d(n)) combined with Jensen's inequality applied to the convex function 1/t. The upper bound is obtained by showing that the contribution from the exceptional set where d(d(n)) is o(log log x) is negligible, using Golomb's estimate on the count of powerful numbers together with Turán's quantitative form of the Hardy-Ramanujan theorem.

Significance. The result supplies a natural hybrid between Ramanujan's asymptotic for ∑ 1/d(n) and Erdős's asymptotic for ∑ d(d(n)), clarifying the effect of taking reciprocals on the iterated divisor function. The proof strategy rests on two classical, independent estimates rather than self-referential fitting, which is a methodological strength. If the error-term bookkeeping is completed, the claim would be a clean addition to the literature on arithmetic functions with small values.

major comments (2)
  1. [Upper bound section] The upper-bound argument (outlined after the statement of the main theorem) must exhibit an explicit error term showing that the contribution over the set where d(d(n)) = o(log log x) is O(x / log log x) or smaller; the abstract asserts this follows from Golomb and Turán but does not display the quantitative estimates or the resulting o-term.
  2. [Lower bound paragraph] The application of Jensen's inequality for the lower bound requires a brief justification that the average order of d(d(n)) is sufficiently concentrated or that the convexity inequality passes to the partial sum without losing the main term; this step is asserted but not derived in the provided outline.
minor comments (2)
  1. [Abstract and introduction] Define the symbol ≍ explicitly in the introduction (both ≪ and ≫ directions) rather than relying on the reader to infer it from context.
  2. [Preliminaries] Supply precise statements (or at least equation numbers) of the forms of Golomb's theorem and Turán's theorem that are invoked, together with the exact references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation, the recommendation of minor revision, and the precise comments on the upper- and lower-bound arguments. The suggestions will improve the explicitness of the error-term bookkeeping. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Upper bound section] The upper-bound argument (outlined after the statement of the main theorem) must exhibit an explicit error term showing that the contribution over the set where d(d(n)) = o(log log x) is O(x / log log x) or smaller; the abstract asserts this follows from Golomb and Turán but does not display the quantitative estimates or the resulting o-term.

    Authors: We agree that the abstract and outline summarize the strategy at a high level without displaying the quantitative estimates. In the full manuscript the upper bound proceeds by splitting the sum at the threshold d(d(n)) ≫ log log x. The contribution of the exceptional set is controlled by combining Golomb’s O(x^{1/2}) bound on the count of powerful numbers with Turán’s quantitative form of the Hardy–Ramanujan theorem, which limits the measure of integers n for which d(n) is abnormally small. We will insert the explicit calculation showing that this exceptional contribution is O(x / (log log x)^2), which is o(x / log log x) and therefore negligible for the claimed upper bound. The revised upper-bound section will contain these estimates in full. revision: yes

  2. Referee: [Lower bound paragraph] The application of Jensen's inequality for the lower bound requires a brief justification that the average order of d(d(n)) is sufficiently concentrated or that the convexity inequality passes to the partial sum without losing the main term; this step is asserted but not derived in the provided outline.

    Authors: We accept that a short derivation is desirable. Because t ↦ 1/t is convex for t > 0, Jensen’s inequality applied directly to the sums yields (1/x) ∑_{n≤x} 1/d(d(n)) ≥ 1 / ((1/x) ∑_{n≤x} d(d(n))). Erdős’s theorem supplies ∑_{n≤x} d(d(n)) ∼ c x log log x with an error o(x log log x). Consequently the right-hand side is asymptotically 1/(c log log x) and the inequality produces the lower bound ∑_{n≤x} 1/d(d(n)) ≫ x / log log x without requiring additional concentration. We will add this one-paragraph justification immediately after the statement of Erdős’s result in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies independent external theorems

full rationale

The claimed asymptotic is obtained by applying the pre-existing Erdős asymptotic for ∑ d(d(n)), Jensen's inequality on the convex function 1/t to obtain the matching lower bound, and the classical Golomb estimate on powerful numbers together with Turán's quantitative Hardy-Ramanujan theorem to control the measure of the exceptional set where d(d(n)) is unusually small. None of these ingredients is derived, fitted, or redefined inside the paper; each is cited as an established external result. The derivation chain therefore contains no self-definitional step, no fitted-input prediction, and no load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two external classical results without new free parameters or invented entities.

axioms (2)
  • domain assumption Golomb's estimate for powerful numbers
    Invoked to bound the contribution of powerful numbers where d(d(n)) may be small.
  • domain assumption Turán's quantitative form of the Hardy-Ramanujan theorem
    Used to describe the typical size of d(n) and control the main contribution.

pith-pipeline@v0.9.0 · 5604 in / 1184 out tokens · 38155 ms · 2026-05-09T18:28:16.965746+00:00 · methodology

discussion (0)

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Reference graph

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