Odd moments and adding fractions

Thomas F. Bloom; Vivian Kuperberg

arxiv: 2312.09021 · v2 · submitted 2023-12-14 · 🧮 math.NT

Odd moments and adding fractions

Thomas F. Bloom , Vivian Kuperberg This is my paper

Pith reviewed 2026-05-24 05:36 UTC · model grok-4.3

classification 🧮 math.NT

keywords odd momentscoprime residuesshort intervalsMontgomery-Vaughan conjecturesingular seriesHardy-Littlewood conjectureprime k-tuplesDiophantine equations

0 comments

The pith

Near-optimal upper bounds are proved for the odd moments of coprime residues in short intervals, confirming the Montgomery-Vaughan conjecture.

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

The paper establishes near-optimal upper bounds on the odd moments of the distribution of coprime residues in short intervals. This directly confirms a long-standing conjecture of Montgomery and Vaughan. The proof introduces a new bound on the number of ways an integer can be written as a sum of an odd number of reduced fractions a_i/q_i with controlled denominators and numerators. This ingredient is then used to bound averages of refined singular series appearing in the Hardy-Littlewood prime tuple conjectures when the number of primes k is odd.

Core claim

We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime k-tuples for k odd. The main new ingredient is a near-optimal upper bound for the number of solutions to sum a_i/q_i in Z when k is odd, with (a_i,q_i)=1 and restrictions on the size of the numerators and denominators, that is of independent interest.

What carries the argument

The near-optimal upper bound on the number of solutions to the equation summing an odd number of reduced fractions equaling an integer, under size restrictions on numerators and denominators.

If this is right

The Montgomery and Vaughan conjecture on these odd moments is confirmed.
Near-optimal upper bounds hold for the average of the refined singular series in Hardy-Littlewood conjectures for odd k.
The new bound on the additive Diophantine equation stands on its own as a result of interest.
These bounds are near-optimal, meaning they are close to what is expected from random models.

Where Pith is reading between the lines

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

The techniques may apply to bounding moments in other short-interval problems involving arithmetic functions.
Similar bounds could be sought for even moments, though the odd case exploits parity in the additive equation.
Applications to sieve methods for primes in short intervals or constellations might follow from the singular series bounds.

Load-bearing premise

The new upper bound on the number of solutions to the sum equation holds specifically when the number of terms k is odd.

What would settle it

An explicit construction or computation showing that for some odd k and size parameters the number of solutions to the sum equation exceeds the claimed near-optimal bound.

read the original abstract

We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime $k$-tuples for $k$ odd. The main new ingredient is a near-optimal upper bound for the number of solutions to $\sum_{1\leq i\leq k}\frac{a_i}{q_i}\in \mathbb{Z}$ when $k$ is odd, with $(a_i,q_i)=1$ and restrictions on the size of the numerators and denominators, that is of independent interest.

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

Bloom and Kuperberg confirm the Montgomery-Vaughan conjecture on odd moments by proving a new bound on solutions to the odd-length sum-of-fractions equation.

read the letter

The main thing here is that the paper settles the Montgomery-Vaughan conjecture on the odd moments of coprime residues in short intervals. It does this by establishing a new upper bound on the number of solutions to sum a_i/q_i equaling an integer when the number of terms k is odd, with the usual coprimality and size restrictions on numerators and denominators. The same bound then yields near-optimal estimates for the average of the refined singular series in the Hardy-Littlewood k-tuple problem for odd k. The fractional-sum count is presented as the central new ingredient and is flagged as independently useful. The reduction from moments to this counting problem follows the standard route in the area, so the work stands or falls on the quality of that new bound. The authors appear to have handled the odd case cleanly where earlier approaches stalled. The application to the singular series is straightforward once the moment bound is in hand and shows the result has some reach beyond the conjecture itself. One soft spot is the repeated use of “near-optimal” without an explicit comparison in the abstract to the conjectured main term or to any matching lower bound; it is not clear how close the constants are. The argument structure itself shows no circularity or hidden fitting. This is squarely for analytic number theorists working on short-interval statistics or prime tuples. A reader in that circle would find the Diophantine bound worth knowing even if the moments are not their main interest. The paper deserves a serious referee because it resolves a stated conjecture with a reusable technical step rather than a broad reorganization of the field.

Referee Report

0 major / 0 minor

Summary. The manuscript proves near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application, it establishes near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures for prime k-tuples when k is odd. The central new ingredient is a near-optimal upper bound on the number of solutions to ∑_{1≤i≤k} a_i/q_i ∈ ℤ (with (a_i,q_i)=1 and size restrictions on numerators and denominators) that holds for odd k.

Significance. If the proof is correct, the result confirms an important conjecture in analytic number theory on moments of coprime residues and supplies a new Diophantine estimate of independent interest with potential further applications to prime-tuple problems. The manuscript supplies a complete proof of its central claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, careful reading of the manuscript, and recommendation to accept. We are pleased that the central results and the new Diophantine estimate are viewed as significant.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper reduces the odd-moment bounds to a new upper bound on the number of solutions to the indicated Diophantine equation (for odd k), which is explicitly presented as the main new independent result. No equations or steps reduce by construction to fitted inputs, self-definitions, or prior self-citations. The Montgomery-Vaughan conjecture is external, and the argument structure (moment reduction followed by the new estimate) contains no load-bearing self-referential steps. This is the standard honest finding for a self-contained analytic-number-theory proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard arithmetic properties of integers and fractions together with the new combinatorial bound; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of the integers, coprimality, and the definition of the fractional sum equation hold.
Invoked throughout the statement of the main new ingredient and the moment bounds.

pith-pipeline@v0.9.0 · 5636 in / 1147 out tokens · 23872 ms · 2026-05-24T05:36:05.505028+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Blomer and J

V. Blomer and J. Br¨ udern. The density of rational points on a certain threefold , in Contributions in analytic and algebraic number theory, Spr inger Proc. Math. 9 (2012), 1–15

work page 2012
[2]

Blomer, J

V. Blomer, J. Br¨ udern, and P. Salberger. On a certain senary cubic form , Proc. Lond. Math. Soc. (3) 108 (2014), no. 4, 911–964

work page 2014
[3]

T. F. Bloom and J. Maynard, A new upper bound for sets with no square diﬀerences , Compos. Math. 158 (2022), no. 8, 1777–1798

work page 2022
[4]

Bourgain, S

J. Bourgain, S. J. Dilworth, K. Ford, S. V. Konyagin, and D . Kutzarova. Breaking the k2 barrier for explicit RIP matrices , STOC’11 – Proceedings of the 43rd ACM Symposium on Theory of Computing (2011), 637–644

work page 2011
[5]

de la Bret` eche

R. de la Bret` eche. Sur le nombre de matrices al´ eatoires ` a coeﬃcients rationnels, Q. J. Math. 68 (2017), no. 3, 935–955

work page 2017
[6]

Dedekind, Gesammelte mathematische Werke, Band 2, Hrsg

R. Dedekind, Gesammelte mathematische Werke, Band 2, Hrsg. v. Robert Fri cke, Emmy Noether u. Oeystein Ore., ¨Uber Zerlegungen von Zahlen durch ihren gr¨ oßten gemeinsamen Teiler, (Festscrhift der Universit¨ at Braunschweig, 1897 , Braunschweig: Friedr. Vieweg and Sohn A.-G., 1931

work page 1931
[7]

Destagnol, Manin ’s conjecture for a family of varieties of higher dimen sion, Math

K. Destagnol, Manin ’s conjecture for a family of varieties of higher dimen sion, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 3, 433-486

work page 2019
[8]

Elsholtz, Sums of k Unit Fractions, PhD thesis, 1998

C. Elsholtz, Sums of k Unit Fractions, PhD thesis, 1998

work page 1998
[9]

P. X. Gallagher, On the distribution of primes in short intervals , Mathematika 23 (1976), no. 1, 4–9. MR 409385

work page 1976
[10]

Gorodetsky, A

O. Gorodetsky, A. Mangerel, and B. Rodgers, Squarefrees are Gaussian in short intervals, Journal f¨ ur die reine und angewandte Mathematik (2023), 1 –44

work page 2023
[11]

G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes , Acta Math. 44 (1923), no. 1, 1–70

work page 1923
[12]

D. R. Heath-Brown. The density of rational points on Cayley’s cubic surface , Proceed- ings of the Session in Analytic Number Theory and Diophantin e Equations, Bonner Math. Schriften 360 (2003), Univ. Bonn, Bonn

work page 2003
[13]

R. J. Lemke Oliver and K. Soundararajan. Unexpected biases in the distribution of consecutive primes, Proc. Natl. Acad. Sci. USA 113 (2016), no. 31, E4446–E4454

work page 2016
[14]

H. L. Montgomery and K. Soundararajan. Primes in short intervals , Comm. Math. Phys. 252 (2004), no. 1-3, 589–617

work page 2004
[15]

H. L. Montgomery and R. C. Vaughan. On the distribution of reduced residues , Ann. of Math. (2) 123 (1986), no. 2, 311–333

work page 1986
[16]

I. E. Shparlinski. Linear equations with rational fractions of bounded height and sto- chastic matrices , Q. J. Math. 69 (2018), no. 2, 487–499

work page 2018
[17]

Kuperberg

V. Kuperberg. Odd moments in the distribution of primes , arXiv 2109.03767

work page arXiv

[1] [1]

Blomer and J

V. Blomer and J. Br¨ udern. The density of rational points on a certain threefold , in Contributions in analytic and algebraic number theory, Spr inger Proc. Math. 9 (2012), 1–15

work page 2012

[2] [2]

Blomer, J

V. Blomer, J. Br¨ udern, and P. Salberger. On a certain senary cubic form , Proc. Lond. Math. Soc. (3) 108 (2014), no. 4, 911–964

work page 2014

[3] [3]

T. F. Bloom and J. Maynard, A new upper bound for sets with no square diﬀerences , Compos. Math. 158 (2022), no. 8, 1777–1798

work page 2022

[4] [4]

Bourgain, S

J. Bourgain, S. J. Dilworth, K. Ford, S. V. Konyagin, and D . Kutzarova. Breaking the k2 barrier for explicit RIP matrices , STOC’11 – Proceedings of the 43rd ACM Symposium on Theory of Computing (2011), 637–644

work page 2011

[5] [5]

de la Bret` eche

R. de la Bret` eche. Sur le nombre de matrices al´ eatoires ` a coeﬃcients rationnels, Q. J. Math. 68 (2017), no. 3, 935–955

work page 2017

[6] [6]

Dedekind, Gesammelte mathematische Werke, Band 2, Hrsg

R. Dedekind, Gesammelte mathematische Werke, Band 2, Hrsg. v. Robert Fri cke, Emmy Noether u. Oeystein Ore., ¨Uber Zerlegungen von Zahlen durch ihren gr¨ oßten gemeinsamen Teiler, (Festscrhift der Universit¨ at Braunschweig, 1897 , Braunschweig: Friedr. Vieweg and Sohn A.-G., 1931

work page 1931

[7] [7]

Destagnol, Manin ’s conjecture for a family of varieties of higher dimen sion, Math

K. Destagnol, Manin ’s conjecture for a family of varieties of higher dimen sion, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 3, 433-486

work page 2019

[8] [8]

Elsholtz, Sums of k Unit Fractions, PhD thesis, 1998

C. Elsholtz, Sums of k Unit Fractions, PhD thesis, 1998

work page 1998

[9] [9]

P. X. Gallagher, On the distribution of primes in short intervals , Mathematika 23 (1976), no. 1, 4–9. MR 409385

work page 1976

[10] [10]

Gorodetsky, A

O. Gorodetsky, A. Mangerel, and B. Rodgers, Squarefrees are Gaussian in short intervals, Journal f¨ ur die reine und angewandte Mathematik (2023), 1 –44

work page 2023

[11] [11]

G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes , Acta Math. 44 (1923), no. 1, 1–70

work page 1923

[12] [12]

D. R. Heath-Brown. The density of rational points on Cayley’s cubic surface , Proceed- ings of the Session in Analytic Number Theory and Diophantin e Equations, Bonner Math. Schriften 360 (2003), Univ. Bonn, Bonn

work page 2003

[13] [13]

R. J. Lemke Oliver and K. Soundararajan. Unexpected biases in the distribution of consecutive primes, Proc. Natl. Acad. Sci. USA 113 (2016), no. 31, E4446–E4454

work page 2016

[14] [14]

H. L. Montgomery and K. Soundararajan. Primes in short intervals , Comm. Math. Phys. 252 (2004), no. 1-3, 589–617

work page 2004

[15] [15]

H. L. Montgomery and R. C. Vaughan. On the distribution of reduced residues , Ann. of Math. (2) 123 (1986), no. 2, 311–333

work page 1986

[16] [16]

I. E. Shparlinski. Linear equations with rational fractions of bounded height and sto- chastic matrices , Q. J. Math. 69 (2018), no. 2, 487–499

work page 2018

[17] [17]

Kuperberg

V. Kuperberg. Odd moments in the distribution of primes , arXiv 2109.03767

work page arXiv