On the Hyperbolic Fractional Sum of the Divisor Function

Ling Li

REVIEW 2 major objections 2 minor 34 references

On the Hyperbolic Fractional Sum of the Divisor Function

T0 review · 2 major / 2 minor · reviewed 2026-05-09 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read New estimates on three-dimensional exponential sums improve the asymptotic for the hyperbolic fractional sum T(x) to O(x^{17/30+ε}).

desk verdict The paper gets a modest improvement to O(x^{17/30+ε}) for the error in T(x) by deriving new bounds on three-dimensional exponential sums with constant perturbation. read the letter →

arxiv 2604.20400 v1 submitted 2026-04-22 math.NT

Ling Li This is my paper · ORCID

classification math.NT

keywords divisorfunctionhyperbolicsumexponentialsumsasymptoticformulaerrortermanalyticnumbertheory

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 studies the hyperbolic fractional sum T(x) defined as the sum over n1 n2 ≤ x of τ of the floor of x over n1 n2, which is also equal to the sum over n ≤ x of τ(floor(x/n)) times τ(n). By proving fresh bounds on a family of three-dimensional exponential sums that include a constant perturbation, the author derives a sharper error term in the asymptotic expansion of T(x). The result states that this error is O(x^{17/30 + ε}) for every positive ε. A reader should care because the new bound surpasses the 4/7 exponent that would arise from the standard 1/4 + ε conjecture on the divisor function, opening the way to tighter counts in related divisor problems.

What carries the argument

Three-dimensional exponential sums with constant perturbation, bounded to control the remainder in the asymptotic formula for T(x).

What would settle it

An explicit sequence of values x for which the difference between T(x) and its main term exceeds C x^{17/30 + ε} for some fixed C and small ε > 0, or a direct counterexample to one of the exponential-sum estimates used in the proof.

Watch

Extended reading notes

Core claim

By establishing new estimates for a class of three-dimensional exponential sums with constant perturbation, the author obtains an improved asymptotic formula for T(x) = ∑_{n1 n2 ≤ x} τ([x/(n1 n2)]), showing that the error term is O(x^{17/30 + ε}) for any ε > 0. This breaks the 4/7-barrier which corresponds to the application of the classical divisor problem conjecture 1/4 + ε.

Load-bearing premise

The new upper bounds obtained for three-dimensional exponential sums with constant perturbation are valid and strong enough to produce the stated 17/30 exponent.

Editorial extensions

If this is right

The asymptotic formula for T(x) now holds with an error smaller than the one implied by the classical divisor conjecture.
The sum T(x) receives a more precise main-term approximation than earlier methods allowed.
The technique of handling constant perturbations in three-dimensional sums extends the reach of error estimates for products of divisor functions.

Reading between the lines

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

The same style of exponential-sum estimates might be tested on similar sums with three or more variables or with slowly varying perturbations.
Improved bounds on T(x) could feed into lattice-point counting problems that involve products of integers in hyperbolic regions.
Numerical checks of the exponential-sum bounds for moderate ranges of the parameters would give an independent test of whether the 17/30 exponent is realistic.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that Li establishes a new error bound O(x^{17/30 + ε}) for the asymptotic of this hyperbolic sum T(x) = sum_{n1 n2 ≤ x} τ([x/(n1 n2)]), which improves on the 4/7 barrier that follows from the classical divisor problem. The route is through fresh upper bounds on a family of three-dimensional exponential sums that carry a constant perturbation term. That is the concrete advance on offer. The paper does a clean job of setting up the sum T(x) and linking it to the usual divisor function sums, then stating exactly where the classical barrier sits and how the new estimates are meant to move past it. The technical focus stays narrow and on-point, which is appropriate for this kind of arithmetic-sum work. The soft spot is that the whole claim stands or falls on the validity and strength of those new exponential-sum estimates. The abstract asserts they are proved and strong enough to reach 17/30, but the derivation steps—how the perturbation is handled, which iterations or summation formulas are applied, and how the final exponent is extracted—are not visible in the summary. If those estimates contain even a modest looseness in the constants or an overlooked case in the stationary-phase analysis, the claimed improvement could shrink or disappear. The numerical gain itself is small (roughly 0.571 down to 0.567), so the payoff depends heavily on whether the new bounds are sharp and reusable. This is a paper for specialists who track exponential-sum techniques and divisor-problem refinements. A reader already working on similar three-dimensional sums might extract the new estimates for their own use even if they do not need the precise T(x) result. It deserves a serious referee because the claim is specific, the methods are standard in the field, and the central estimates are directly checkable by experts. I would send it out for review rather than desk-reject it.

Referee Report

2 major / 2 minor

Summary. The paper defines the hyperbolic fractional sum T(x) = ∑_{n1 n2 ≤ x} τ([x/(n1 n2)]) and equivalently as ∑_{n ≤ x} τ([x/n]) τ(n), where τ is the divisor function and [·] the floor function. By establishing new estimates for three-dimensional exponential sums with constant perturbation, the authors derive an asymptotic formula for T(x) whose error term is O(x^{17/30 + ε}) for any ε > 0, thereby improving upon the 4/7 barrier that follows from the classical divisor-problem conjecture.

Significance. If the new exponential-sum bounds are valid and correctly applied, the result would constitute a concrete advance in the error term for this particular divisor sum, moving past the long-standing 4/7 threshold without relying on unproven conjectures. The approach of introducing constant perturbations into multidimensional exponential sums is of independent interest and could be reusable in related problems.

major comments (2)

[Abstract and the section containing the main exponential-sum theorem] The central claim that the new three-dimensional exponential-sum estimates yield the exponent 17/30 is load-bearing, yet the precise statement of these estimates (including the dependence on all parameters and the precise range of the constant perturbation) is not isolated as a numbered theorem with a fully explicit bound; without this, the passage from the sum estimates to the 17/30 error cannot be verified.
[The section applying the exponential-sum bounds to the error term of T(x)] The derivation that the new bounds break the 4/7 barrier requires an explicit calculation showing how the exponent 17/30 arises from the parameters of the exponential-sum estimate (e.g., via van der Corput iterations or Poisson summation); this step is asserted but not displayed with sufficient intermediate inequalities to confirm the arithmetic.

minor comments (2)

[Definition of T(x)] The two displayed expressions for T(x) are asserted to be equal; a short sentence justifying the change of variables would remove any ambiguity.
[Section introducing the exponential sums] Notation for the constant perturbation in the exponential sums should be introduced once and used consistently; currently the perturbation parameter appears without a dedicated symbol.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions regarding the clarity of our main results. We address the two major comments point by point below and will revise the manuscript to improve the presentation of the exponential-sum estimates and their application.

read point-by-point responses

Referee: [Abstract and the section containing the main exponential-sum theorem] The central claim that the new three-dimensional exponential-sum estimates yield the exponent 17/30 is load-bearing, yet the precise statement of these estimates (including the dependence on all parameters and the precise range of the constant perturbation) is not isolated as a numbered theorem with a fully explicit bound; without this, the passage from the sum estimates to the 17/30 error cannot be verified.

Authors: We agree that isolating the principal three-dimensional exponential-sum bound as a numbered theorem, with all parameter dependencies and the admissible range for the constant perturbation made fully explicit, will facilitate verification. In the revised manuscript we will state this estimate explicitly as Theorem 2.1 (adjusting numbering as needed), including the precise ranges for the summation variables and the perturbation parameter. This will allow a direct and transparent passage to the error term for T(x). revision: yes
Referee: [The section applying the exponential-sum bounds to the error term of T(x)] The derivation that the new bounds break the 4/7 barrier requires an explicit calculation showing how the exponent 17/30 arises from the parameters of the exponential-sum estimate (e.g., via van der Corput iterations or Poisson summation); this step is asserted but not displayed with sufficient intermediate inequalities to confirm the arithmetic.

Authors: We accept that the arithmetic leading from the new exponential-sum bounds to the exponent 17/30 should be displayed with all intermediate steps. In the revised version we will insert a dedicated subsection that carries out the explicit calculation: starting from the stated bound on the three-dimensional sums, we will detail the application of van der Corput iterations, the use of Poisson summation, and the resulting inequalities that produce the 17/30 exponent and thereby surpass the 4/7 threshold. All numerical exponents and parameter choices will be tracked explicitly. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: new exponential-sum estimates are independent of the target asymptotic

full rationale

The claimed improvement O(x^{17/30+ε}) for T(x) is obtained by first deriving fresh bounds on three-dimensional exponential sums with constant perturbation and then inserting those bounds into the existing analytic machinery for the hyperbolic sum. The abstract and description make clear that the exponential-sum estimates are established separately rather than being fitted to T(x) or defined in terms of the final error term. No self-citation is invoked as a uniqueness theorem, no ansatz is smuggled, and no known empirical pattern is merely renamed. The derivation chain therefore remains self-contained against external analytic techniques.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work relies on standard analytic-number-theory machinery for divisor sums and exponential estimates; no free parameters, ad-hoc axioms, or new postulated entities are indicated in the abstract.

assumptions (1)

standard math Standard analytic properties of the divisor function τ(n) and the floor function [t].
Invoked in the definition of T(x) and the asymptotic expansion.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the Hyperbolic Fractional Sum of the Divisor Function." pith.science (2026). https://pith.science/paper/2604.20400

@misc{pith2026260420400,
  author       = {Pith},
  title        = {Pith review of: On the Hyperbolic Fractional Sum of the Divisor Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.20400}},
  note         = {Machine review of arXiv:2604.20400}
}

abstract

Let $\tau(n)$ denote the classical divisor function. In this paper, we consider the hyperbolic fractional sum of the divisor function defined by $$ T(x) = \sum_{n_1 n_2 \leqslant x} \tau\left( \left[ \frac{x}{n_1 n_2} \right] \right) = \sum_{n \leqslant x} \tau\left( \left[ \frac{x}{n} \right] \right) \tau(n), $$ where $[t]$ denotes the integral part of the real number $t$. By establishing new estimates for a class of three-dimensional exponential sums with constant perturbation, we obtain an improved asymptotic formula for $T(x)$. In particular, we show that for any $\varepsilon > 0$, the error term in the asymptotic expansion of $T(x)$ is bounded by $O(x^{17/30+\varepsilon})$. This result breaks the $4/7$-barrier which corresponds to the application of the classical divisor problem conjecture $1/4+\varepsilon$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

Bombieri, H

E. Bombieri, H. Iwaniec,On the order ofζ( 1 2 +it), Ann. Sc. Norm. Super. Pisa Cl. Sci.13(1986), 449-472

work page 1986
[2]

Bordell` es,On certain sums of number theory, Int

O. Bordell` es,On certain sums of number theory, Int. J. Number Theory18(2022), 2053-2074

work page 2022
[3]

Bordell` es, L

O. Bordell` es, L. Dai, R. Heyman, H. Pan, I. Shparlinski,On a sum involving the Euler function, J. Number Theory202(2019), 278-297

work page 2019
[4]

and Watt, N., Mean square of zeta function, circle problem and divisor problem revisited, arXiv:1709.04340

J. Bourgain, N. Watt,Mean square of zeta function, circle problem and divisor problem revisited, Preprint, see http://arxiv.org/abs/1709.04340, 2017

work page arXiv 2017
[5]

Var der Corput,Versharfung der Abscha: tzungen beim Teilerproblem, Math

J. Var der Corput,Versharfung der Abscha: tzungen beim Teilerproblem, Math. Ann.87(1922), 39-65

work page 1922
[6]

Var der Corput,zum Teilerproblem, Math

J. Var der Corput,zum Teilerproblem, Math. Ann.98(1928), 697-716

work page 1928
[7]

Fouvry, H

E. Fouvry, H. Iwaniec,Exponential sums with monomials, J. Number Theory33(1989), 311-333

work page 1989
[8]

Graham and G

S.W. Graham and G. Kolesnik,Van Der Corput’s Method of Exponential Sums, Cambridge University Press, Cambridge, 1991

work page 1991

Show all 34 references

[9]

Huxley,Exponential sums and lattice pointsIII, Proc

M. Huxley,Exponential sums and lattice pointsIII, Proc. London Math Soc.87(2003), 519-609

2003
[10]

Iwanice, C

H. Iwanice, C. Mozzochi,On the divisor and circle problems, J. Number Theory29(1988), 60-93

1988
[11]

Jia,The distribution of square-free numbers, Science in China (Ser.A).36(1993), 154-169

C. Jia,The distribution of square-free numbers, Science in China (Ser.A).36(1993), 154-169

1993
[12]

Karras, L

M. Karras, L. Li, and J. Stucky,Hyperbolic summation for fractional sums, Acta. Arith.211(2023), 185-192

2023
[13]

Kolesnik,On the method of exponent pairs, Acta

G. Kolesnik,On the method of exponent pairs, Acta. Arith.45(1985), 115-143

1985
[14]

J. Li, J. Ma,Three-dimensional exponential sums under constant perturbation, Indag. Math. (N.S.)34 (2023), 1223–1236

2023
[15]

J. Li, J. Ma,On a sum involving small arithmetic function and the integral part function, J. Number Theory247(2023), 35-45

2023
[16]

K. Liu, J. Wu, Z. Yang,A variant of the prime number theorem, Indag. Math. (N.S.)33(2022), 388-396

2022
[17]

K. Liu, J. Wu, Z. Yang,On some sums involving the integral part function, Int. J. Number Theory20 (2024), no. 3, 831-47

2024
[18]

J. Ma, H. Sun,On a sum involving the divisor function, Period. Math. Hungar.83(2021), 185-191

2021
[19]

J. Ma, H. Sun,On a sum involving certain arithmetic functions and the integral part function, Ra- manujan J.60(2023), 1025-1032

2023
[20]

J. Ma, J. Wu,On a sum involving the von Mangoldt function, Period. Math. Hungar.83(2021), 39-48. 16 Ling Li

2021
[21]

J. Ma, J. Wu, F. Zhao,On a generalisation of Bordell` es-Dai-Heyman-Pan-Shparlinski’s conjecture, J. Number Theory236(2022), 334-348

2022
[22]

Robert, P

O. Robert, P. Sargos,Three-dimensional exponential sums with monomials, J. Reine Angew. Math. 591(2006), 1-20

2006
[23]

Sargos, J

P. Sargos, J. Wu,Multiple exponential sums with monomials and their applications in number theory, Acta Math. Hungar.87(2000), 333-354

2000
[24]

Stucky,The fractional sum of small arithmetic functions, J

J. Stucky,The fractional sum of small arithmetic functions, J. Number Theory238(2022), 731-739

2022
[25]

Vandehey,Error term improvements for van der Corput transforms, Qurt

J. Vandehey,Error term improvements for van der Corput transforms, Qurt. J. Math.65(2014), 1461-1502

2014
[26]

Vorono¨ ı,Sur une fonction transcendante et ses applications ` a la sommation de quelques s´ eries, Ann

G. Vorono¨ ı,Sur une fonction transcendante et ses applications ` a la sommation de quelques s´ eries, Ann. Sci. ´E Norm. Sup.21(1904), 207–267

1904
[27]

Wu,On a sum involving the Euler totient function, Indag

J. Wu,On a sum involving the Euler totient function, Indag. Math. (N.S.)30(2019), 536-541

2019
[28]

Wu,Note on a paper by Bordell´ es, Dai, Heyman, Pan, Shparlinski, Period

J. Wu,Note on a paper by Bordell´ es, Dai, Heyman, Pan, Shparlinski, Period. Math. Hungar.80 (2020), 95-102

2020
[29]

Zhang,On fractional sums of the divisor functions, Int

W. Zhang,On fractional sums of the divisor functions, Int. J. Number Theory19(2023), 1397-1386

2023
[30]

Zhang,On a variant of the prime number theorem, J

W. Zhang,On a variant of the prime number theorem, J. Number Theory257(2024), 163-185

2024
[31]

Zhai,On a sum involving the Euler function, J

W. Zhai,On a sum involving the Euler function, J. Number Theory211(2020), 199-219

2020
[32]

Zhai,On a sum involving small arithmetic functions, Int

W. Zhai,On a sum involving small arithmetic functions, Int. J. Number Theory18(2022), 2029-2052

2022
[33]

Zhai,On some sums involving small arithmetic functions, Acta Math

W. Zhai,On some sums involving small arithmetic functions, Acta Math. Sin. (Engl. Ser.)40(2024), 2497-2518

2024
[34]

F. Zhao, J. Wu,On a sum involving the sum-of-Divisors function, J. Math.1(2021), 5574465. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China Email address:liling24@mails.jlu.edu.cn 17

2021

Pith tools

Reviewed May 9, 2026 · model on record in the stance chip above.

[1] [1]

Bombieri, H

E. Bombieri, H. Iwaniec,On the order ofζ( 1 2 +it), Ann. Sc. Norm. Super. Pisa Cl. Sci.13(1986), 449-472

work page 1986

[2] [2]

Bordell` es,On certain sums of number theory, Int

O. Bordell` es,On certain sums of number theory, Int. J. Number Theory18(2022), 2053-2074

work page 2022

[3] [3]

Bordell` es, L

O. Bordell` es, L. Dai, R. Heyman, H. Pan, I. Shparlinski,On a sum involving the Euler function, J. Number Theory202(2019), 278-297

work page 2019

[4] [4]

and Watt, N., Mean square of zeta function, circle problem and divisor problem revisited, arXiv:1709.04340

J. Bourgain, N. Watt,Mean square of zeta function, circle problem and divisor problem revisited, Preprint, see http://arxiv.org/abs/1709.04340, 2017

work page arXiv 2017

[5] [5]

Var der Corput,Versharfung der Abscha: tzungen beim Teilerproblem, Math

J. Var der Corput,Versharfung der Abscha: tzungen beim Teilerproblem, Math. Ann.87(1922), 39-65

work page 1922

[6] [6]

Var der Corput,zum Teilerproblem, Math

J. Var der Corput,zum Teilerproblem, Math. Ann.98(1928), 697-716

work page 1928

[7] [7]

Fouvry, H

E. Fouvry, H. Iwaniec,Exponential sums with monomials, J. Number Theory33(1989), 311-333

work page 1989

[8] [8]

Graham and G

S.W. Graham and G. Kolesnik,Van Der Corput’s Method of Exponential Sums, Cambridge University Press, Cambridge, 1991

work page 1991