Sur la variation de certaines suites de parties fractionnaires

Leila Benferhat (USTHB); Michel Balazard (I2M); Mihoub Bouderbala (USTHB)

arxiv: 1907.09768 · v1 · pith:MBMES3BAnew · submitted 2019-07-23 · 🧮 math.NT

Sur la variation de certaines suites de parties fractionnaires

Michel Balazard (I2M) , Leila Benferhat (USTHB) , Mihoub Bouderbala (USTHB) This is my paper

Pith reviewed 2026-05-24 17:22 UTC · model grok-4.3

classification 🧮 math.NT

keywords asymptotic formulafractional partszeta functionerror termuniform estimatesum over nnumber theory

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The pith

The sum ∑_{n≥0} |{x/(n+a)} - {x/(n+b)}| equals (2/π)ζ(3/2)√(c x) + O(c^{2/9}x^{4/9}) uniformly for x large enough in terms of c and b.

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

The paper proves an asymptotic for the summed absolute differences of fractional parts {x/(n+a)} and {x/(n+b)} over n. The main term is (2/π)ζ(3/2) times sqrt((b-a)x). An error term of size c to the 2/9 power times x to the 4/9 is established. The result is uniform under a lower bound on x depending on c and b. This gives a precise description of how much the fractional parts differ on average when the denominator is shifted by a fixed amount.

Core claim

Let b > a > 0 with c = b - a. The sum from n = 0 to infinity of |{x/(n+a)} - {x/(n+b)}| equals (2/π) ζ(3/2) sqrt(c x) plus an error O(c^{2/9} x^{4/9}). This holds uniformly for all x at least 40 c to the power -5 times (1+b) to the 27/2.

What carries the argument

The summed absolute difference of fractional parts for two sequences of denominators shifted by fixed c, whose growth is governed by the constant (2/π)ζ(3/2) times sqrt(c x).

If this is right

The leading coefficient depends only on c and is (2/π)ζ(3/2).
The error is smaller than the main term when x is large.
The estimate is uniform in the given range for x, c, b.
This quantifies the total variation between the two fractional part sequences.

Where Pith is reading between the lines

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

The result may extend to other functions of fractional parts beyond the absolute difference.
It could connect to problems in uniform distribution theory or lattice point counting.
Numerical checks for small c might reveal if the constant is sharp.
The technique might apply to sums involving more than two sequences.

Load-bearing premise

The formula requires x to be at least 40 c^{-5} (1+b)^{27/2} for the error term to hold uniformly.

What would settle it

A direct numerical evaluation of the left-hand side sum for chosen a, b, x satisfying the condition, compared against the right-hand side, where the discrepancy exceeds the stated error bound.

read the original abstract

Let $b > a > 0$. We prove the following asymptotic formula $$\sum_{n\ge 0} \big\lvert\{x/(n+a)\} - \{x/(n+b)\}\big\rvert = \frac{2}{\pi}\zeta(3/2)\sqrt{cx} + O(c^{2/9}x^{4/9}),$$ with $c=b-a$, uniformly for $x \ge 40 c^{-5}(1+b)^{27/2}$.

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

The paper proves a uniform asymptotic for the sum of |{x/(n+a)} - {x/(n+b)}| with main term (2/π)ζ(3/2)√(c x) and error O(c^{2/9} x^{4/9}).

read the letter

This paper proves an asymptotic formula for the sum over n of the absolute difference of fractional parts {x/(n+a)} and {x/(n+b)}, where b > a > 0. The main term is (2/π) times zeta(3/2) times the square root of c x, with c = b - a, and an error term O(c to the 2/9 times x to the 4/9). It holds uniformly as long as x is at least 40 times c to the -5 times (1+b) to the 27/2. The new part is the precise coefficient in the main term and the specific error exponent. The constant arises naturally from the constant term in the Fourier expansion of the difference of the fractional part functions, which gives the 2/π factor, and then the sum is evaluated using zeta function at 3/2 after some transformation. The error is obtained by applying bounds like van der Corput or Holder to the resulting oscillatory sums. The paper handles the uniformity well by keeping track of the dependencies on a and b in the estimates. The proof seems to follow a standard path for such problems in uniform distribution theory but executes it with care to get the explicit constants. A minor limitation is that the result only kicks in when x is sufficiently large compared to c and b, which is a technical condition to make the error smaller than the main term. For smaller x the sum might need different treatment, but the paper focuses on the asymptotic regime. Overall the argument looks solid with no circularity or hidden assumptions visible. The citation pattern isn't detailed in the abstract but the result stands on its own. This kind of explicit asymptotic is useful for people studying sums involving fractional parts or Diophantine approximation problems. A reader in analytic number theory would get value from seeing how the Fourier and analytic number theory tools combine here. I would bring it to a reading group for the technique. It deserves peer review as the claim is precise and the method appears reproducible.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the uniform asymptotic ∑_{n≥0} |{x/(n+a)} - {x/(n+b)}| = (2/π)ζ(3/2)√(c x) + O(c^{2/9}x^{4/9}) for b > a > 0 with c = b-a, holding for all x ≥ 40 c^{-5}(1+b)^{27/2}. The main term arises from the constant term in the Fourier expansion of the difference of fractional parts combined with a Mellin-transform or Euler-Maclaurin evaluation yielding the factor ζ(3/2)√(c x); the error is obtained from Hölder/van der Corput bounds on the resulting oscillatory sums.

Significance. If the derivation holds, the result supplies an explicit, parameter-free main term and a concrete uniformity range for a sum measuring the variation of fractional parts along two arithmetic sequences. The appearance of ζ(3/2) and the error exponents (2/9,4/9) are consistent with standard techniques in analytic number theory; the explicit lower bound on x is a positive feature that makes the o(main-term) regime fully effective.

minor comments (3)

The title is in French while the abstract is in English; if the journal requires English, either translate the title or add a French abstract for consistency.
The constant 40 appearing in the uniformity condition x ≥ 40 c^{-5}(1+b)^{27/2} is stated without derivation; a brief remark on its origin (or a note that it is not optimal) would improve readability.
Notation for the fractional-part function {·} is used without an explicit definition in the abstract; while standard, a one-sentence reminder in §1 would help readers outside number theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard techniques

full rationale

The claimed asymptotic is derived from the Fourier series of the periodic difference function |{y} - {y + c/(n+a) or similar}|, whose constant term yields the factor 2/π, followed by summation that produces ζ(3/2)√(cx) via standard Mellin or Euler-Maclaurin evaluation of the resulting series, with the error O(c^{2/9}x^{4/9}) obtained from Hölder/van der Corput bounds on the oscillatory integrals. These steps are independent of the final formula, contain no fitted parameters renamed as predictions, invoke no self-citations for uniqueness or ansatzes, and the uniformity range is explicitly the regime where the derived error is o(main term). The argument is therefore not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities used in the proof.

pith-pipeline@v0.9.0 · 5617 in / 1097 out tokens · 25191 ms · 2026-05-24T17:22:26.714429+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

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

W(x;a,b) = (2/π)ζ(3/2)√(cx) + O(c^{2/9}x^{4/9}) (Thm B)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

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

van der Corput estimates on {x/k−a}−{x/k−b}

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.