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arxiv: 2604.20400 · v1 · submitted 2026-04-22 · 🧮 math.NT

On the Hyperbolic Fractional Sum of the Divisor Function

Ling Li This is my paper

Pith reviewed 2026-05-09 23:33 UTC · model grok-4.3

classification 🧮 math.NT
keywords divisor functionhyperbolic sumexponential sumsasymptotic formulaerror termanalytic number theory
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The pith

New estimates on three-dimensional exponential sums improve the asymptotic for the hyperbolic fractional sum T(x) to O(x^{17/30+ε}).

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

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.

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.

read the original 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 $1/4+\varepsilon$.

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 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)
  1. [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.
  2. [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)
  1. [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.
  2. [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

  1. 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

  2. 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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.0 · 5473 in / 1245 out tokens · 133123 ms · 2026-05-09T23:33:22.488831+00:00 · methodology

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Reference graph

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