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On a Smoothed Walfisz Divisor Problem

T0 review · 2 major / 2 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read A fully explicit asymptotic for the weighted sum of σ(n) removes the main error term in its average order.

desk verdict This paper gives an explicit smoothed asymptotic for the weighted sum of σ(n) plus a convergence corollary for the Walfisz integral, extending the authors' prior τ work, but the appendix constants are the part that needs checking. read the letter →

arxiv 2607.01956 v1 pith:YTSEZFDE submitted 2026-07-02 math.NT

Olivier Bordellès , Florian Daval This is my paper
classification math.NT
keywords sum-of-divisorsfunctionsmoothedsumsWalfiszdivisorproblemexplicitasymptoticsEuler-Maclaurinformulameanvaluetheoremerrorterms
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

The paper establishes a completely explicit asymptotic formula for the sum of σ(n) twisted by the smoothing weight 1 - x/n. This formula lets the authors bypass the difficult error term that normally appears when estimating the average order of the sum-of-divisors function. As a corollary they obtain the convergence of a certain integral that encodes the error term in the Walfisz divisor problem. The derivation rests on explicit bounds supplied in an appendix that applies the mean value theorem and the Euler-Maclaurin summation formula.

What carries the argument

The explicit asymptotic formula for the sum of σ(n) weighted by 1-x/n, obtained from the mean value theorem and Euler-Maclaurin summation.

What would settle it

Numerical evaluation of the integral over successively larger intervals to test whether the partial integrals remain bounded, or direct comparison of the stated asymptotic against computed values of the weighted sum for large explicit x.

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Extended reading notes

Core claim

We prove a totally explicit asymptotic formula for the sum of σ(n) twisted by the weight 1-x/n, which enables us to eliminate the difficult part in the classical average order of σ(n). As a corollary, we deduce the convergence of an integral dealing with the error term in the Walfisz divisor problem.

Load-bearing premise

The explicit constants obtained from the mean value theorem and Euler-Maclaurin summation formula in the appendix are sufficiently accurate to justify both the main asymptotic formula and the convergence of the integral.

Editorial extensions

If this is right

  • The difficult part in the classical average order of σ(n) is eliminated by the explicit formula.
  • Convergence is established for the integral that controls the error term in the Walfisz divisor problem.
  • The appendix supplies explicit constants that support both the main formula and the convergence claim.

Reading between the lines

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

  • The same smoothing technique might be applied to obtain explicit formulas for other divisor functions such as τ(n).
  • Convergence of the integral implies improved integrability properties for the error term in related divisor problems.
  • The method could be tested on higher-order moments of σ(n) to check whether similar explicit asymptotics hold.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper proves a totally explicit asymptotic formula for the sum of σ(n) twisted by the weight 1-x/n. This is used to eliminate the difficult remainder term in the classical average order of σ(n). As a corollary, the authors deduce the convergence of an integral involving the error term from the Walfisz divisor problem. The proofs rely on an appendix supplying explicit estimates obtained via the mean value theorem and the Euler-Maclaurin summation formula.

Significance. If the explicit constants derived in the appendix are sufficiently sharp, the result supplies a concrete, parameter-free handle on a smoothed sum-of-divisors sum and resolves a convergence question tied to the Walfisz error term. Such explicit control is valuable in analytic number theory when one wishes to pass from smoothed to unsmoothed statements or to justify integral representations of remainder terms.

major comments (2)
  1. [Appendix] Appendix (explicit bounds via MVT and Euler-Maclaurin): the manuscript asserts that the derived constants suffice both to absorb the difficult remainder into the main asymptotic for the weighted sum of σ(n) and to guarantee absolute convergence of the integral in the Walfisz corollary. No numerical check or comparison with known sharper bounds is supplied; if any constant is understated by a modest factor, both the elimination step and the corollary fail. A concrete verification (e.g., explicit numerical evaluation of the leading error term for a moderate x) is required.
  2. [Main theorem] Main theorem (asymptotic for the weighted sum): the claim that the explicit formula eliminates the difficult part of the classical average order of σ(n) rests entirely on the error term being smaller than the main term after the constants from the appendix are inserted. The manuscript does not display the numerical size of this error term relative to the main term for any concrete range of x, leaving the load-bearing comparison unverified.
minor comments (2)
  1. [Introduction] Notation for the weight function 1-x/n should be introduced with a displayed equation and a clear statement of the range of summation.
  2. [Introduction] The abstract and introduction both refer to “the difficult part in the classical average order of σ(n)” without a precise citation to the classical formula or the precise remainder that is being removed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the bounds. We respond point by point to the major comments.

read point-by-point responses

  1. Referee: [Appendix] Appendix (explicit bounds via MVT and Euler-Maclaurin): the manuscript asserts that the derived constants suffice both to absorb the difficult remainder into the main asymptotic for the weighted sum of σ(n) and to guarantee absolute convergence of the integral in the Walfisz corollary. No numerical check or comparison with known sharper bounds is supplied; if any constant is understated by a modest factor, both the elimination step and the corollary fail. A concrete verification (e.g., explicit numerical evaluation of the leading error term for a moderate x) is required.

    Authors: The constants are obtained by applying the mean value theorem and Euler-Maclaurin formula while retaining explicit remainder terms at every step; the subsequent proofs then insert these constants directly into the estimates for the weighted sum and the Walfisz integral, showing analytically that the required inequalities hold. We acknowledge that an independent numerical check would increase , and we will add such a verification (evaluation of the leading error term at x=10^4 together with the bound) to the revised appendix. revision: partial

  2. Referee: [Main theorem] Main theorem (asymptotic for the weighted sum): the claim that the explicit formula eliminates the difficult part of the classical average order of σ(n) rests entirely on the error term being smaller than the main term after the constants from the appendix are inserted. The manuscript does not display the numerical size of this error term relative to the main term for any concrete range of x, leaving the load-bearing comparison unverified.

    Authors: The explicit formula supplies a concrete error term whose size relative to the main term is controlled by the appendix constants; the absorption is therefore a direct (if tedious) consequence of those inequalities. To make the comparison transparent we will insert, in the revised main theorem section, a short table or paragraph displaying the numerical ratio of error to main term for several moderate values of x (e.g., 10^3 to 10^5). revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; explicit bounds derived from standard MVT and Euler-Maclaurin

full rationale

The paper establishes its main asymptotic for the weighted sum of σ(n) directly via the mean value theorem and Euler-Maclaurin formula, with all explicit constants computed in the appendix. No equation reduces a claimed prediction or result to a fitted parameter or prior self-result by construction. The reference to prior work on the τ case is contextual and not load-bearing for the σ derivation or the integral convergence corollary. The chain is therefore self-contained against external analytic tools.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper rests on standard analytic-number-theory tools whose explicit versions are supplied in the appendix; no free parameters or new entities are introduced.

assumptions (2)
  • standard math Mean value theorem
    Invoked in appendix to obtain explicit estimates.
  • standard math Euler-Maclaurin summation formula
    Used in appendix to derive explicit remainder terms.

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

@misc{pith2026260701956,
  author       = {Pith},
  title        = {Pith review of: On a Smoothed Walfisz Divisor Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YTSEZFDE}},
  note         = {Machine review of arXiv:2607.01956}
}
abstract

This work is in the spirit of our previous investigation on a smooth Dirichlet divisor problem, where we now replace the Dirichlet divisor function $\tau$ by the sum-of-divisors function $\sigma$. We prove a totally explicit asymptotic formula for the sum of $\sigma(n)$ twisted by the weight $1-x/n$, which enables us to eliminate the difficult part in the classical average order of $\sigma(n)$. As a corollary, we deduce the convergence of an integral dealing with the error term in the Walfisz divisor problem. We also provide an appendix containing the necessary explicit results derived from the mean value theorem and the Euler-Maclaurin summation formula.

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Reference graph

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