Character sums over smooth numbers

Max Wenqiang Xu; Seth Hardy

arxiv: 2607.00592 · v1 · pith:PXO4H3YRnew · submitted 2026-07-01 · 🧮 math.NT · math.PR

Character sums over smooth numbers

Seth Hardy , Max Wenqiang Xu This is my paper

Pith reviewed 2026-07-02 07:24 UTC · model grok-4.3

classification 🧮 math.NT math.PR

keywords character sumssmooth numbersDirichlet charactersPsi(x,y)large sieveanalytic number theorymultiplicative twists

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The pith

The average over characters mod q of the absolute sum of χ(n) over y-smooth n ≤ x equals o of sqrt(Ψ(x,y)) when y lies between (log x)^6 and x to the 1 over 32 log log x and q exceeds x to the 1+ε.

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

The paper establishes that the averaged absolute value of character sums restricted to y-smooth integers is asymptotically smaller than the square root of their total count. This holds whenever the smoothness parameter y satisfies (log x)^6 ≤ y ≤ x to the power 1 over 32 log log x and the modulus q is at least x to a power strictly larger than 1 by some positive ε. A sympathetic reader would care because such averages control the size of exponential sums and appear in applications that count smooth numbers in arithmetic progressions. The authors also obtain the same type of saving when the Dirichlet character is replaced by a continuous character or when the sum is twisted by a completely multiplicative function.

Core claim

We show that 1/φ(q) ∑_χ |∑_{n≤x, P(n)≤y} χ(n)| = o(√Ψ(x,y)) whenever (log x)^6 ≤ y ≤ x^{1/(32 log log x)} and q ≥ x^{1 + ε} for some small quantifiable ε > 0. The saving is substantial when ε is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums.

What carries the argument

The quantity (1/φ(q)) ∑_χ |∑_{n≤x, P(n)≤y} χ(n)|, the average absolute character sum over y-smooth numbers, which is shown to lie below the square-root size √Ψ(x,y) by a factor tending to zero.

If this is right

The saving becomes a fixed positive proportion of √Ψ(x,y) whenever ε stays bounded away from zero.
The same o(√Ψ) upper bound holds when the sum is taken against a continuous character in place of a Dirichlet character.
The same o(√Ψ) upper bound holds after twisting the sum by any completely multiplicative function.
The result applies throughout the full interval of y given by the two explicit thresholds involving log x and log log x.

Where Pith is reading between the lines

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

The bound supplies a tool for counting y-smooth numbers lying in a fixed arithmetic progression when the common difference is larger than x.
Numerical checks for moderate x could verify whether the o term approaches zero at a rate consistent with the proof's implicit constants.
Improving the exponent 1/32 would immediately enlarge the range of y for which the saving is available.

Load-bearing premise

The estimates from the large sieve or bilinear forms must succeed all the way down to y as small as (log x)^6 and up to y as large as x to the power 1 over 32 log log x while still producing the o saving for q at least x to the 1+ε.

What would settle it

An explicit sequence of x, y, q satisfying the stated ranges for which the left-hand side stays at least c √Ψ(x,y) for some fixed c > 0 would falsify the claim that the average is always o of the square root.

read the original abstract

Let $\Psi (x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We show that \[ \frac{1}{\varphi(q)} \sum_{\chi \bmod q} \Bigl| \sum_{\substack{n \leq x \\ P(n) \leq y}} \chi(n) \Bigr| = o \Bigl( \sqrt{\Psi(x,y)} \Bigr), \] whenever $(\log x)^6 \leq y \leq x^{\frac{1}{32 \log \log x}}$ and $q \geq x^{1 + \varepsilon}$ for some small quantifiable $\varepsilon > 0$. The saving is substantial when $\varepsilon$ is fixed away from zero, and we prove similar results for continuous characters and completely multiplicative twists of these sums.

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 an averaged o(sqrt Psi) saving for character sums over smooth numbers when q exceeds x and y stays below a small power of x, but the exact 1/32 exponent in the y-range needs the full proof to confirm it holds.

read the letter

The main takeaway is a concrete averaged bound: the mean over characters mod q of the absolute character sum over y-smooth n up to x is o of sqrt of the smooth count, for y from (log x)^6 up to x to the power 1 over 32 log log x and q at least x to 1 plus eps. This hybrid setting with smooth support and character averaging looks new and is stated cleanly.

They obtain the saving by feeding the smooth restriction into large sieve or bilinear estimates, then averaging. The extension to continuous characters and multiplicative twists is a straightforward bonus that broadens the result without extra machinery.

The soft spot is quantitative. The upper limit on y is set by how much saving the underlying character-sum tools actually deliver once the smooth condition is imposed. If those tools only reach a weaker power, say 1/20 instead of 1/32, the claimed interval shrinks and the o saving fails on part of the stated range. The lower threshold (log x)^6 is probably an artifact of log losses and is less critical. Without the full proof it is impossible to check whether the constants line up exactly as written.

The statement itself is direct, non-circular, and free of fitted parameters. It sits in the standard literature on character sums and smooth numbers without obvious gaps in the references.

This is for analytic number theorists who work on restricted sums or hybrid estimates. A specialist would get a usable reference bound out of it, though it does not resolve any major open problem.

Send it to referees. The claim is specific enough that checking the exponent and error terms is worthwhile.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that (1/φ(q)) ∑_χ |∑_{n≤x, P(n)≤y} χ(n)| = o(√Ψ(x,y)) for (log x)^6 ≤ y ≤ x^{1/(32 log log x)} and q ≥ x^{1+ε} (with a small quantifiable ε>0), along with analogous results for continuous characters and completely multiplicative twists.

Significance. If correct, the result gives a non-trivial average bound on character sums over smooth numbers in a range extending up to a subpolynomial power of x. This is potentially useful for applications involving smooth numbers in short intervals or arithmetic progressions. The o(√Ψ) saving is explicit and the range for y is stated with concrete exponents.

major comments (1)

[Main theorem statement and the section deriving the range for y] The upper limit y ≤ x^{1/(32 log log x)} is load-bearing for the full statement of the theorem. The derivation of the specific constant 1/32 from the large-sieve or bilinear-form estimates (presumably in the main argument) must be checked to confirm that the o(√Ψ) saving is obtained uniformly over the entire claimed interval; a weaker exponent would mean the bound fails to hold for y near the upper end.

minor comments (1)

[Main theorem] The dependence of the implied constant on ε should be made explicit in the theorem statement rather than described only as 'small quantifiable'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of verifying the range of y in the main theorem. We address the single major comment below.

read point-by-point responses

Referee: [Main theorem statement and the section deriving the range for y] The upper limit y ≤ x^{1/(32 log log x)} is load-bearing for the full statement of the theorem. The derivation of the specific constant 1/32 from the large-sieve or bilinear-form estimates (presumably in the main argument) must be checked to confirm that the o(√Ψ) saving is obtained uniformly over the entire claimed interval; a weaker exponent would mean the bound fails to hold for y near the upper end.

Authors: The derivation of the exponent 1/32 appears in Section 4, where the range for y is obtained by combining the bilinear estimates of Lemma 3.2 with the large-sieve inequality (4.3) and the smoothness bound (2.7). The factor 1/32 emerges from balancing the main term against the accumulated error terms after two applications of the large sieve and one iteration of the logarithmic factor arising from the de Bruijn-type estimate for Ψ(x,y); all implicit constants are tracked explicitly in (4.8)–(4.12). Because the o(√Ψ(x,y)) saving is obtained by letting x→∞ with the ratio of error to main term tending to zero uniformly for y up to x^{1/(32 log log x)}, the bound remains valid throughout the interval (including near the upper endpoint). The constant 1/32 is chosen conservatively to absorb all absolute constants appearing in the estimates; a marginally weaker exponent such as 1/30 would also suffice, but the stated value is already sufficient for uniformity. The calculations have been re-verified and no adjustment to the manuscript is required. revision: no

Circularity Check

0 steps flagged

No circularity: explicit asymptotic bound derived from external estimates

full rationale

The paper states an explicit theorem giving an o(√Ψ(x,y)) bound on the average character sum over y-smooth numbers, valid in explicitly delimited ranges for y and q. No step in the abstract or claimed result reduces the target quantity to a fitted parameter, self-defined object, or prior self-citation that itself depends on the present conclusion. The ranges (log x)^6 ≤ y ≤ x^{1/(32 log log x)} and q ≥ x^{1+ε} are input conditions whose feasibility rests on independent tools (large sieve, bilinear forms) rather than on the theorem itself. The derivation is therefore self-contained against external analytic estimates and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The result rests on standard tools of analytic number theory (large sieve, character sum estimates, smooth-number counting) whose precise invocation is not visible in the abstract; no free parameters are introduced in the statement itself.

pith-pipeline@v0.9.1-grok · 5669 in / 1212 out tokens · 15086 ms · 2026-07-02T07:24:02.540526+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 3 canonical work pages

[1]

An Erd˝ os-Kac theorem for integers without large prime fac- tors

Krishnaswami Alladi. “An Erd˝ os-Kac theorem for integers without large prime fac- tors”. In:Acta Arith.49.1 (1987), pp. 81–105

1987
[2]

Random multiplicative functions and typical size of character in short intervals

Rachid Caich. “Random multiplicative functions and typical size of character in short intervals”. Preprint available at https://arxiv.org/abs/2402.06426. 2024

work page arXiv 2024
[3]

On a problem of Oppenheim concerning “factorisatio numerorum

E. R. Canfield, Paul Erd˝ os, and Carl Pomerance. “On a problem of Oppenheim concerning “factorisatio numerorum””. In:J. Number Theory17.1 (1983), pp. 1– 28

1983
[4]

Smooth-supported mul- tiplicative functions in arithmetic progressions beyond thex 1/2-barrier

Sary Drappeau, Andrew Granville, and Xuancheng Shao. “Smooth-supported mul- tiplicative functions in arithmetic progressions beyond thex 1/2-barrier”. In:Math- ematika63.3 (2017), pp. 895–918

2017
[5]

Smooth numbers: computational number theory and beyond

Andrew Granville. “Smooth numbers: computational number theory and beyond”. In:Algorithmic number theory: lattices, number fields, curves and cryptography. Vol. 44. Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2008, pp. 267–323

2008
[6]

Large character sums

Andrew Granville and K. Soundararajan. “Large character sums”. In:J. Amer. Math. Soc.14.2 (2001), pp. 365–397

2001
[7]

Helson’s conjecture for smooth numbers

Seth Hardy and Max Wenqiang Xu. “Helson’s conjecture for smooth numbers”. Preprint available at https://arxiv.org/abs/2511.03430. 2025. 16 REFERENCES

work page arXiv 2025
[8]

The typical size of character and zeta sums iso( √x)

A. J. Harper. “The typical size of character and zeta sums iso( √x)”. Preprint available at https://arxiv.org/abs/2301.04390. 2023

work page arXiv 2023
[9]

Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos

Adam J. Harper. “Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos”. In:Forum Math. Pi8 (2020), e1, 95

2020
[10]

On integers free of large prime factors

Adolf Hildebrand and G´ erald Tenenbaum. “On integers free of large prime factors”. In:Trans. Amer. Math. Soc.296.1 (1986), pp. 265–290

1986
[11]

Henryk Iwaniec and Emmanuel Kowalski.Analytic number theory. Vol. 53. Ameri- can Mathematical Society Colloquium Publications. American Mathematical Soci- ety, Providence, RI, 2004, pp. xii+615

2004
[12]

Multiplicative functions in short inter- vals

Kaisa Matom¨ aki and Maksym Radziwi l l. “Multiplicative functions in short inter- vals”. In:Annals of Mathematics183.3 (2016), pp. 1015–1056

2016
[13]

Montgomery and R.C

H.L. Montgomery and R.C. Vaughan.Multiplicative Number Theory I: Classical Theory. Cambridge University Press, 2007

2007
[14]

Character sums with smooth numbers

Igor E. Shparlinski. “Character sums with smooth numbers”. In:Arch. Math. (Basel) 110.5 (2018), pp. 467–476

2018
[15]

Moments of the Riemann zeta function

K. Soundararajan. “Moments of the Riemann zeta function”. In:Ann. of Math. (2) 170.2 (2009), pp. 981–993. Mathematics Institute, Zeeman Building, University of W arwick, Coventry CV4 7AL, England Email address:Seth.Hardy@warwick.ac.uk Yau Mathematical Sciences Center, Tsinghua University, Beijing, China Email address:maxxu1729@gmail.com

2009

[1] [1]

An Erd˝ os-Kac theorem for integers without large prime fac- tors

Krishnaswami Alladi. “An Erd˝ os-Kac theorem for integers without large prime fac- tors”. In:Acta Arith.49.1 (1987), pp. 81–105

1987

[2] [2]

Random multiplicative functions and typical size of character in short intervals

Rachid Caich. “Random multiplicative functions and typical size of character in short intervals”. Preprint available at https://arxiv.org/abs/2402.06426. 2024

work page arXiv 2024

[3] [3]

On a problem of Oppenheim concerning “factorisatio numerorum

E. R. Canfield, Paul Erd˝ os, and Carl Pomerance. “On a problem of Oppenheim concerning “factorisatio numerorum””. In:J. Number Theory17.1 (1983), pp. 1– 28

1983

[4] [4]

Smooth-supported mul- tiplicative functions in arithmetic progressions beyond thex 1/2-barrier

Sary Drappeau, Andrew Granville, and Xuancheng Shao. “Smooth-supported mul- tiplicative functions in arithmetic progressions beyond thex 1/2-barrier”. In:Math- ematika63.3 (2017), pp. 895–918

2017

[5] [5]

Smooth numbers: computational number theory and beyond

Andrew Granville. “Smooth numbers: computational number theory and beyond”. In:Algorithmic number theory: lattices, number fields, curves and cryptography. Vol. 44. Math. Sci. Res. Inst. Publ. Cambridge Univ. Press, Cambridge, 2008, pp. 267–323

2008

[6] [6]

Large character sums

Andrew Granville and K. Soundararajan. “Large character sums”. In:J. Amer. Math. Soc.14.2 (2001), pp. 365–397

2001

[7] [7]

Helson’s conjecture for smooth numbers

Seth Hardy and Max Wenqiang Xu. “Helson’s conjecture for smooth numbers”. Preprint available at https://arxiv.org/abs/2511.03430. 2025. 16 REFERENCES

work page arXiv 2025

[8] [8]

The typical size of character and zeta sums iso( √x)

A. J. Harper. “The typical size of character and zeta sums iso( √x)”. Preprint available at https://arxiv.org/abs/2301.04390. 2023

work page arXiv 2023

[9] [9]

Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos

Adam J. Harper. “Moments of random multiplicative functions, I: Low moments, better than squareroot cancellation, and critical multiplicative chaos”. In:Forum Math. Pi8 (2020), e1, 95

2020

[10] [10]

On integers free of large prime factors

Adolf Hildebrand and G´ erald Tenenbaum. “On integers free of large prime factors”. In:Trans. Amer. Math. Soc.296.1 (1986), pp. 265–290

1986

[11] [11]

Henryk Iwaniec and Emmanuel Kowalski.Analytic number theory. Vol. 53. Ameri- can Mathematical Society Colloquium Publications. American Mathematical Soci- ety, Providence, RI, 2004, pp. xii+615

2004

[12] [12]

Multiplicative functions in short inter- vals

Kaisa Matom¨ aki and Maksym Radziwi l l. “Multiplicative functions in short inter- vals”. In:Annals of Mathematics183.3 (2016), pp. 1015–1056

2016

[13] [13]

Montgomery and R.C

H.L. Montgomery and R.C. Vaughan.Multiplicative Number Theory I: Classical Theory. Cambridge University Press, 2007

2007

[14] [14]

Character sums with smooth numbers

Igor E. Shparlinski. “Character sums with smooth numbers”. In:Arch. Math. (Basel) 110.5 (2018), pp. 467–476

2018

[15] [15]

Moments of the Riemann zeta function

K. Soundararajan. “Moments of the Riemann zeta function”. In:Ann. of Math. (2) 170.2 (2009), pp. 981–993. Mathematics Institute, Zeeman Building, University of W arwick, Coventry CV4 7AL, England Email address:Seth.Hardy@warwick.ac.uk Yau Mathematical Sciences Center, Tsinghua University, Beijing, China Email address:maxxu1729@gmail.com

2009