On a Smoothed Dirichlet Divisor Problem

Florian Daval; Olivier Bordell\`es

REVIEW 3 major objections 3 minor 1 cited by

On a Smoothed Dirichlet Divisor Problem

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that the difference between the summatory function of the divisor function and its logarithmic version equals exactly 1/4 plus an error term that is O(log x / x^{1/4}) for x ≥ 300, with explicit constants, and derives a po

desk verdict A genuinely explicit and mostly sound proof of a real conjecture, with a small constant typo in the second branch of Theorem 1 that is trivial to fix; worth refereeing. read the letter →

arxiv 2601.01905 v3 pith:E2N42L3L submitted 2026-01-05 math.NT

Olivier Bordellès , Florian Daval This is my paper

classification math.NT MSC 11A2511L07

keywords DirichletdivisorproblemhyperbolaprincipleexponentialsumsvanderCorputestimatesChowla–WalumBernoullipolynomialsexplicitboundsfractionalparts

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that the difference between the divisor summatory function and its logarithmic version is exactly 1/4 plus an error term that stays bounded for all real x ≥ 1 and shrinks like O(x^{-1/4} log x) once x ≥ 300. That is far smaller than the error term in the classical Dirichlet divisor problem, whose order is believed to be at least x^{1/4}. The result is proved with explicit constants, using the hyperbola principle, Euler–Maclaurin estimates, and new explicit bounds for weighted sums of second Bernoulli polynomials. As a consequence, the integral of the divisor error term from x to infinity is shown to be non-negative for every x ≥ 1, confirming a conjecture. The closeness of the two error terms means that results about one can be converted to results about the other with almost no loss.

What carries the argument

The identity Δ(x) − xδ(x) = 1/4 + x R_1(√x)^2 − 2√x R_1(√x) ψ(√x) − 2x (R_2 − R_1)(√x) − 2x ∑_{k≤√x} R_1(x/k)/k, obtained from the hyperbola principle, expresses the difference as a sum of small error terms. The decisive new estimate is an explicit bound for the generalized Chowla–Walum sum ∑_{n≤√x} n B_2({x/n}) < x^{3/4} log x for x ≥ 300, proved by splitting the sum into dyadic intervals and applying an explicit van der Corput bound to the resulting exponential sums. This converts an O(1) term into O(x^{-1/4} log x), which is the key to the sharper error term.

What would settle it

Compute the sum ∑_{n≤√x} n B_2({x/n}) numerically for a few large values of x (e.g., 10^4, 10^6, 10^8) and verify it stays below x^{3/4} log x; a violation would falsify the stronger error bound. Alternatively, evaluate Δ(x) − xδ(x) − 1/4 at those x and check that it satisfies the claimed inequalities, or search numerically for an x ≥ 1 where ∫_x^∞ Δ(u)/u^2 du is negative.

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Extended reading notes

Core claim

The central claim is that Δ(x) − x δ(x) = 1/4 + r(x), where Δ(x) is the usual sum of τ(n) up to x and δ(x) is the corresponding sum of τ(n)/n with its main term removed. The paper proves fully explicit bounds on r(x): |r(x)| ≤ 1/8 + 0.316/√x + 1/(64x) for all x ≥ 1, and the sharper |r(x)| ≤ (log x)/x^{1/4} + 0.236/√x + 1/(64x) for x ≥ 300. The constant 1/4 is ζ(0)^2, the residue of the Dirichlet series of τ(s)=ζ(s)^2. The proof is elementary in structure and yields the corollary that ∫_x^∞ Δ(u)/u^2 du ≥ 0 for all x ≥ 1.

Load-bearing premise

The sharper error term O(x^{-1/4} log x) in the theorem depends on the explicit bound |∑_{n≤√x} n B_2({x/n})| < x^{3/4} log x for x ≥ 300, proved with van der Corput exponential-sum estimates; if that bound were to fail, the x^{-1/4} log x branch would collapse, although the constant-order bound for all x ≥ 1 would remain intact.

Editorial extensions

If this is right

The integral ∫_x^∞ Δ(u)/u^2 du is non-negative for every real x ≥ 1, confirming a conjecture about the divisor error term.
The classical result that the divisor error term is not o(x^{1/4}) transfers to the logarithmic version, so its error term is also not o(x^{1/4}).
Bounds on one error term convert to bounds on the other with almost no loss; the paper shows a bound of order x^{1/2} on Δ implies a bound of order x^{-1/2} on δ, improving earlier explicit results by a factor of 2.
For large x, the two error terms differ from 1/4 by O(x^{-1/4} log x), so they are interchangeable in asymptotic formulas up to that precision.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The appearance of ζ(0)^2 as the constant suggests a general principle: for arithmetic functions whose Dirichlet series has a pole of order k at s = 1, the difference between the summatory function and its logarithmic analogue may equal the square of the residue of the zeta factor, possibly with oscillations when nontrivial zeros exist, as seen in the Möbius analogue in the paper.
The explicit exponential-sum technique used here may generalize to higher Bernoulli polynomials or other weights, producing effective estimates for other smoothed divisor problems that are currently conditional.
A natural test is whether the logarithmic factor in the error term can be removed via a more refined exponential-sum estimate, which would sharpen the bound to O(x^{-1/4}) with explicit constants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper is a legitimate explicit result in a crowded area, and it does what it says. The new contribution is the fully explicit, unconditional estimate Δ(x) − xδ(x) = 1/4 + O(x^{−1/4} log x), with constants valid for all x ≥ 1, and the consequent proof of the Berkane–Bordellès–Ramaré positivity conjecture for ∫_x^∞ Δ(u)/u^2 du. The asymptotic version was already claimed by Mahatab and Mukhopadhyay, but the authors cite that only through a personal communication [14]; the explicit treatment here is their own.

Structurally the proof is clean. Lemmas 1 and 2 give Euler–Maclaurin bounds with explicit constants. Proposition 1 is a general explicit bound for generalized Chowla–Walum sums, proved with a dyadic van der Corput argument; Corollary 2 is the concrete effective bound. Lemma 10 converts that into the required O(x^{−1/4} log x) contribution. Lemma 3 is borrowed from [2], a published paper, so the self-citation is not circular. The final assembly in Theorem 1 is straightforward and the main identities check out. This is honest, careful work.

The one real flaw is the constant in the second branch of Theorem 1: the text prints 0.236, but the proof at that point gives coefficient α + 2β + 0.047 with α = 1/8, β ≈ 0.033, so about 0.2368–0.238. The printed 0.236 is slightly smaller than the displayed inequalities justify. That is a minor numerical correction; replace it with 0.237 or 0.238 and the theorem stands. Corollary 1 is unaffected.

A smaller citation concern: the claim that Mahatab–Mukhopadhyay settled the conjecture asymptotically rests on [14], 'Personal Communication.' That is unverifiable. A referee should ask the authors to either locate a public version or soften the wording. It does not affect the mathematics of this paper.

Who is this for: people working on explicit estimates for the divisor problem. It is not a breakthrough but it is a solid, complete result, and the constant-order branch is strong enough to be usable. I would send it to a serious referee and expect acceptance after the constant fix and a reference cleanup.

Referee Report

3 major / 3 minor

Summary. The paper studies the difference between the Dirichlet divisor summatory error term Δ(x) and its logarithmic analogue xδ(x). The main result, Theorem 1, asserts the explicit bound Δ(x)−xδ(x)=1/4+r(x) with |r(x)|≤1/8+0.316/√x+1/(64x) for all x≥1 and |r(x)|≤log x/x^{1/4}+0.236/√x+1/(64x) for x≥300. From this the authors derive Corollary 1, namely ∫_x^∞ Δ(u)/u^2 du≥0 for all x≥1, settling a conjecture of Berkane, Bordellès, and Ramaré. The proof combines Euler–Maclaurin estimates for harmonic-type sums, an explicit version of a generalized Chowla–Walum sum, and a van der Corput argument; all constants are claimed explicitly.

Significance. If the proof is fully correct, this is a substantial contribution: it gives an unconditional, explicit O(x^{-1/4} log x) estimate for a smoothed divisor problem, resolves a conjecture in the literature, and provides a useful conversion between Δ and δ. The paper is careful in proving most auxiliary lemmas and in making constants explicit. The main structural idea — that the smoothing eliminates the difficult fractional-part sum — is elegant and likely correct. However, as discussed below, several load-bearing points in the present manuscript need to be repaired before the stated results are fully established.

major comments (3)

[Theorem 1, §2.4] The second branch of Theorem 1 states the coefficient 0.236, but the proof gives only 0.238. From the displayed inequality |r(x)|≤log x/x^{1/4}+(α+2β+0.047)/√x+α²/x and Lemmas 1–2 and 10, one has α≤0.125, β≤0.033, so α+2β+0.047≤0.238. The printed value 0.236 is smaller than anything justified by the displayed inequalities. Replacing 0.236 by 0.238 (or sharpening one of the constituent bounds) is necessary.
[Proof of Corollary 1, §2.4] The claimed identity Δ(x)/x−δ(x)=∫_x^∞ Δ(u)/u^2 du is not a direct consequence of summation by parts. Writing A(x)=∑_{n≤x}τ(n), summation by parts gives ∑τ(n)/n = A(x)/x + ∫_1^x A(u)/u^2 du, which leads to Δ/x−δ = −∫_1^x Δ(u)/u^2 du + γ²−2γ−2γ1+1 before one knows ∫_1^∞ Δ(u)/u^2 du = γ²−2γ−2γ1+1 (equivalently δ(x)→0). This nontrivial evaluation must be supplied, or the proof of Corollary 1 is incomplete. The corollary is load-bearing for the paper's main application.
[§2.3.3, Eq. (8)] The passage from Lemma 8 to Eq. (8) is not correct as written. With λ2=mx/(4N^3) and c2=8, the bound supplied by Lemma 8 has an additional factor 1/√π, and the displayed intermediate term √(mx)/N should be √(mx)/√N in order to match the subsequent x^{1/2}N^{β−1/2} term. As written, the inequality chain is invalid, so Proposition 1 and Corollary 2 are not established. Since Lemma 10 and the second branch of Theorem 1 use Corollary 2, the authors need to reconcile Lemma 8 with its application and provide a correct explicit constant.

minor comments (3)

[§2.3.3, after Eq. (8)] The second exponent in the displayed bound after reporting (8) is printed identical to the first; it should be β/α + 3/(2α) − 1/2, not β/α − 1/(2α) + 1/2.
[Lemma 8 and its application] The statement of Lemma 8 appears to contain a typo: the proof gives 4/√π, while the displayed statement reads 4√π in the typeset text. This ambiguity should be corrected, as it affects the verification of the subsequent estimates.
[Lemma 4 proof] The notation R_1(t) is reused for R_1(t)−ψ(t)/t, which is confusing. A separate symbol or a clear definition would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the target error term results from exact cancellation of identical fractional-part sums, with the remaining bounds proved internally by Euler-Maclaurin and van der Corput estimates.

full rationale

The derivation is self-contained rather than circular. Theorem 1 is obtained by subtracting the exact identity of Lemma 4 from Lemma 3 ([2, Lemma 3.1]); the troublesome sums Σ_{n≤√x} ψ(x/n) appear identically in both Δ(x) and xδ(x) and cancel, leaving only R1, R2, and R1-error terms. The constant 1/4 arises from those exact Euler-Maclaurin identities, not from fitting. The only nontrivial input, Corollary 2's generalized Chowla-Walum bound, is proved internally in Sections 2.3.2–2.3.4 via an explicit van der Corput inequality from external standard sources; it is not equivalent to the theorem. The one self-citation, [2, Lemma 3.1], is an elementary hyperbola/Euler-Maclaurin formula with independent published provenance and is not the target statement; no fitted parameter is renamed as a prediction. In line with the reviewing rule, I flag the skeptic's constant discrepancy in Theorem 1 / end of §2.4: the displayed bounds α≤1/8, β≤0.033, and √300 f(300)≤0.047 give about 0.238 (or about 0.2368 with sharp values), so the printed 0.236 appears unsupported by the displayed inequalities. This is a correctness/rounding issue, not a circularity issue, and it does not affect Corollary 1, which relies only on the first branch.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests on standard analytic number theory tools and one prior published lemma [2, Lemma 3.1]. No free parameters are fitted; all constants are derived from explicit bounding. No invented entities (forces, particles, or new axioms) are introduced. The reliance on [2] is not circular since it is an external published result.

assumptions (8)

standard math Euler-Maclaurin summation formula
Used to derive the explicit bounds for R1 and R2 in Lemmas 1-2 (§2.2.1).
standard math Kusmin-Landau inequality
Used as Lemma 6 to bound exponential sums in the proof of Proposition 1 (§2.3.2).
standard math Van der Corput inequality (explicit form)
Derived as Lemma 8 and applied with f(n)=mx/n to bound exponential sums (§2.3.3).
standard math Uniform Fourier expansion of periodic Bernoulli functions
Used in the proof of Proposition 1 to expand B_j({t}) (§2.3.3).
standard math Second mean value theorem
Used in Lemmas 2 and 10 to bound integrals of B2({t}) g(t) (§2.2.1, §2.4).
standard math Partial summation identity Δ(x)/x - δ(x) = ∫_x^∞ Δ(u)/u^2 du
Used to derive Corollary 1 from Theorem 1 (§2.4).
domain assumption [2, Lemma 3.1]: explicit hyperbola-principle decomposition of Σ_{n≤x}τ(n) with constant 1/4
Prior published result by Berkane-Bordellès-Ramaré, cited and used as a starting point for Δ(x); not derived in this paper.
standard math Lehmer's bound for odd periodic Bernoulli polynomials
Used in Lemma 9 to bound the sup norm of B_j({x}).

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Pith. "Pith review of On a Smoothed Dirichlet Divisor Problem." pith.science (2026). https://pith.science/paper/E2N42L3L

@misc{pith2026260101905,
  author       = {Pith},
  title        = {Pith review of: On a Smoothed Dirichlet Divisor Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E2N42L3L}},
  note         = {Machine review of arXiv:2601.01905}
}

abstract

Hardy showed that $\sum_{n \ioe x}\tau(n)-x(\log x +2\gamma -1)$ is not $o(x^{1/4})$. In this article, we prove that $\sum_{n \ioe x}\tau(n)(1-\frac{x}{n})-xP(\log x)=\frac{1}{4}+O \left( \frac{\log x}{x^{1/4}} \right)$, where $P$ is a polynomial of degree 2. As a corollary, this estimate enables us to settle a conjecture surmised by Berkane, Bordell\`{e}s, and Ramar\'{e} dealing with the positivity of an integral of the error term in the Dirichlet divisor problem. All results are entirely explicit and allow us to study the proximity between the remainder of the Dirichlet divisor problem and its logarithmic version.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On a Smoothed Walfisz Divisor Problem
math.NT 2026-07 unverdicted novelty 4.0 of 10

Explicit asymptotic formula proved for smoothed sum of σ(n), yielding convergence of integral for Walfisz error term.

Reference graph

Works this paper leans on

14 extracted references · 1 linked inside Pith · cited by 1 Pith paper

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