On general divisor functions over Piatetski-Shapiro sequences

Wei Zhang

arxiv: 2304.10119 · v2 · submitted 2023-04-20 · 🧮 math.NT

On general divisor functions over Piatetski-Shapiro sequences

Wei Zhang This is my paper

Pith reviewed 2026-05-24 09:48 UTC · model grok-4.3

classification 🧮 math.NT

keywords Piatetski-Shapiro sequencesdivisor functionsarithmetic progressionsgeneralized divisor problemanalytic estimateserror termsnumber theory

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The pith

Divisor problem estimates over Piatetski-Shapiro sequences carry over to a general class of functions f under a growth bound on g.

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

The paper seeks to generalize known results on the divisor problem summed along Piatetski-Shapiro sequences to a wider class of arithmetic functions. These functions f are defined by writing each n as a product n1 n2 and summing τ(n1) g(n2), where τ is the usual divisor function. The generalization requires only that f itself grows slower than any positive power of n and that the partial sums of |g| grow no faster than x to the 5/8 power plus an epsilon. A sympathetic reader would care because the same error terms then apply without new case-by-case analysis. The paper also proves the corresponding statements when the sums are restricted to terms lying in a fixed arithmetic progression.

Core claim

The central claim is that the divisor problem over Piatetski-Shapiro sequences extends to the function f(n), where f(n) ≪ n^ε, f(n) = ∑_{n=n1 n2} τ(n1) g(n2), and ∑_{1≤n≤x} |g(n)| ≪ x^{5/8+ε}. The bound on g is used to control its contribution inside the analytic estimates that were previously developed for the standard divisor function, so that the same main-term and error-term statements remain valid. The same extension is established when the sums are taken only over elements of the sequence that lie in a given arithmetic progression.

What carries the argument

The representation f(n) = ∑_{n=n1 n2} τ(n1) g(n2) together with the growth bound on the partial sums of g, which transfers the analytic estimates from the classical case while absorbing the extra factor coming from g.

If this is right

The same error bounds previously obtained for the ordinary divisor function apply directly to this general f summed over Piatetski-Shapiro numbers.
The asymptotic formulas remain valid when the sum is restricted to an arithmetic progression.
Any special divisor-like function whose auxiliary part g obeys the stated growth bound inherits the estimates without further work.
The method unifies a range of convolution-type divisor functions under a single set of hypotheses.

Where Pith is reading between the lines

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

The same transfer argument might apply to other floor-function sequences such as Beatty sequences with irrational ratios.
If a stronger bound on g can be established for a concrete choice, the overall error term could be improved beyond what the paper states.
The arithmetic-progression version suggests possible applications to counting problems in residue classes inside non-integer sequences.

Load-bearing premise

The assumption that the partial sums of |g(n)| up to x grow at most like x to the 5/8 power is enough to keep the error terms under control in the existing analytic machinery.

What would settle it

An explicit function g satisfying ∑ |g(n)| ≪ x^{5/8+ε} for which the sum of the corresponding f over the Piatetski-Shapiro sequence up to x deviates from the predicted main term plus error term by a larger amount would falsify the claim.

read the original abstract

In this paper, we consider the general divisor functions over Piatetski-Shapiro sequences. We can give some general results which contain some special divisor functions. Precisely, we extend the divisor problem over Piatetski-Shapiro sequences to the function $f(n),$ where $f(n)\ll n^{\varepsilon},$ $$f(n)=\sum_{n=n_{1}n_{2}} \tau(n_{1})g(n_{2}),$$ $\tau(n)$ is the number of representations of $n$ as product of two natural numbers and \[ \sum_{1\leq n\leq x}|g(n)|\ll x^{5/8+\varepsilon}. \] On the other hand, we also considered these arithmetic functions over Piatetski-Shapiro sequences in arithmetic progressions.

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.

Desk Editor's Note private letter to a colleague

This paper gives a modest technical extension of divisor estimates on Piatetski-Shapiro sequences to a convoluted form f = tau * g with a 5/8+ε bound on the partial sums of g.

read the letter

The central new content is the move from the plain divisor function to f(n) = sum_{n=n1 n2} tau(n1) g(n2) where f(n) ≪ n^ε and the partial sums of |g| are ≪ x^{5/8+ε}. The authors claim this is enough to carry earlier error terms over to sums of f over the sequences n^α (α irrational) and also in arithmetic progressions. That unification is the actual increment; it recovers some standard special cases when g is suitably chosen. The work is otherwise a direct application of existing analytic machinery for these sequences, with the growth condition on g used to absorb the second factor into the error estimates. No internal contradiction appears in the setup, and the f ≪ n^ε hypothesis is compatible with the convolution under the stated bound on g. The 5/8 exponent is presented without fresh derivation in the abstract, but it is plausible that it sits comfortably below the thresholds from prior papers on the same sequences. The main limitation is that the abstract supplies no proof outline or explicit comparison of exponents, so the precise range of applicability remains to be verified in the full text. This is narrow analytic number theory aimed at people already working on divisor problems or Piatetski-Shapiro sums. A reader who needs the general f version for a follow-up calculation will find the statement useful; others will see only incremental progress. The paper is coherent on its own terms and shows honest engagement with the literature, so it deserves referee time even if the final verdict is that the extension is routine.

Referee Report

1 major / 1 minor

Summary. The manuscript extends the divisor problem for Piatetski-Shapiro sequences to a general arithmetic function f(n) defined by the convolution f(n) = ∑_{n=n1 n2} τ(n1) g(n2), under the hypotheses f(n) ≪ n^ε and ∑_{n≤x} |g(n)| ≪ x^{5/8+ε}. It also treats the corresponding problems in arithmetic progressions.

Significance. If the extension holds, the result would unify several special cases of divisor functions over Piatetski-Shapiro sequences under a single set of growth conditions on the convolution factors, with the 5/8 exponent on the partial sums of |g| serving as the threshold that permits the earlier error-term estimates to carry over.

major comments (1)

Abstract: the central claim that the stated growth hypotheses on f and g suffice to extend the divisor estimates is presented without any proof outline, error-term derivation, or indication of how the 5/8 exponent arises or compares with prior exponents in the literature, rendering the main result unverifiable from the supplied text.

minor comments (1)

Abstract, final sentence: the phrasing 'we also considered' is in the past tense; consistent present tense ('we also consider') would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. Below we respond point-by-point to the single major comment.

read point-by-point responses

Referee: [—] Abstract: the central claim that the stated growth hypotheses on f and g suffice to extend the divisor estimates is presented without any proof outline, error-term derivation, or indication of how the 5/8 exponent arises or compares with prior exponents in the literature, rendering the main result unverifiable from the supplied text.

Authors: The abstract is a concise statement of the main theorem. The derivation of the error terms, the manner in which the hypothesis ∑|g(n)| ≪ x^{5/8+ε} combines with the standard estimates for the divisor function τ(n1) and the Piatetski-Shapiro sequence to recover the earlier error bounds, and the comparison with previous exponents appear in full in Sections 2–4 of the manuscript. We nevertheless accept that a brief indication of the origin of the 5/8 threshold would make the abstract more self-contained. We will therefore revise the abstract to include one sentence noting that the 5/8 exponent is the threshold that permits the earlier error-term estimates to carry over under the given convolution structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states an extension of prior divisor estimates to the general f(n) defined by the convolution f(n) = sum τ(n1)g(n2) under the explicit independent growth hypotheses f(n) ≪ n^ε and sum |g(n)| ≪ x^{5/8+ε}. These bounds are invoked as assumptions to control error terms rather than being derived from or equivalent to the target result by construction. No equations, fitted parameters renamed as predictions, self-citation chains, or ansatzes are exhibited that reduce the claimed extension to its inputs. The derivation chain remains self-contained against the stated external hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions of analytic number theory for handling exponential sums or Dirichlet series over Piatetski-Shapiro sequences; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard tools of analytic number theory (Dirichlet series, exponential sum estimates) apply to sums over Piatetski-Shapiro sequences.
The extension of prior divisor estimates presupposes that the same analytic machinery remains valid under the new growth conditions on f and g.

pith-pipeline@v0.9.0 · 5657 in / 1248 out tokens · 137618 ms · 2026-05-24T09:48:40.007171+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

f(n)=∑_{n=n1 n2} τ(n1)g(n2) with ∑|g(n)|≪x^{5/8+ε} and the resulting asymptotic for ∑ f([n^c]) via 3-dimensional exponential sums
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3 and its reduction to exponential sums over Piatetski-Shapiro sequences

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Graham and G

W. Graham and G. Kolesnik, Van der Corput’s Method of Exponential Sums. Cambridge Univ. Press, 1991

work page 1991
[2]

Perron, ¨Uber einen asymptotischen Ausdruck

O. Perron, ¨Uber einen asymptotischen Ausdruck. Acta Math 59 (1932) 89-97

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[3]

Rivat and J

J. Rivat and J. Wu, Prime numbers of the form [nc]. Glasg. Math. J. 43 (2001) 237-254

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Rivat and P

J. Rivat and P. Sargos, Nombres premiers de la forme [nc]. Canad. J. Math. 53 (2001) 414-433

work page 2001
[5]

Rivat and A

J. Rivat and A. S´ ark¨ ozy,A sequences analog of the Piatetski-Shapiro problem. Acta Math. Hungar. 74 (1997) 245-260

work page 1997
[6]

Robert and P

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

work page 2006
[7]

Vaaler, Some extremal functions in Fourier analysis

J.D. Vaaler, Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc. (N.S.) 12 (1985) 183-216. Wei Zhang, School of Mathematics, Henan University, Kaifen g 475004, Henan, China Email address : zhangweimath@126.com 8

work page 1985

[1] [1]

Graham and G

W. Graham and G. Kolesnik, Van der Corput’s Method of Exponential Sums. Cambridge Univ. Press, 1991

work page 1991

[2] [2]

Perron, ¨Uber einen asymptotischen Ausdruck

O. Perron, ¨Uber einen asymptotischen Ausdruck. Acta Math 59 (1932) 89-97

work page 1932

[3] [3]

Rivat and J

J. Rivat and J. Wu, Prime numbers of the form [nc]. Glasg. Math. J. 43 (2001) 237-254

work page 2001

[4] [4]

Rivat and P

J. Rivat and P. Sargos, Nombres premiers de la forme [nc]. Canad. J. Math. 53 (2001) 414-433

work page 2001

[5] [5]

Rivat and A

J. Rivat and A. S´ ark¨ ozy,A sequences analog of the Piatetski-Shapiro problem. Acta Math. Hungar. 74 (1997) 245-260

work page 1997

[6] [6]

Robert and P

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

work page 2006

[7] [7]

Vaaler, Some extremal functions in Fourier analysis

J.D. Vaaler, Some extremal functions in Fourier analysis. Bull. Amer. Math. Soc. (N.S.) 12 (1985) 183-216. Wei Zhang, School of Mathematics, Henan University, Kaifen g 475004, Henan, China Email address : zhangweimath@126.com 8

work page 1985