Large character sums with multiplicative coefficients

Hao Zhang; Shengbo Zhao; Weijia Wang; Yutong Song; Zikang Dong

REVIEW 3 major objections 4 minor 12 references

Large character sums with multiplicative coefficients

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims new Omega lower bounds for character sums twisted by multiplicative coefficients, but the main theorem's positivity condition forces those coefficients to be trivial.

desk verdict The f-twist idea in Theorem 1.1 is a genuine small step, but Theorem 1.2 is vacuous as stated (the positivity hypothesis forces f≡1) and the proof has a false inequality, so the advertised Omega result for multiplicative coefficients is unsupported. read the letter →

arxiv 2509.09649 v1 pith:2WZR7IGH submitted 2025-09-11 math.NT

Zikang Dong , Yutong Song , Weijia Wang , Hao Zhang , Shengbo Zhao This is my paper · ORCID

classification math.NT MSC 11L4011M06

keywords charactersumsmultiplicativecoefficientsOmegaresultsresonancemethodGCDDirichletcharacterslargevaluescompletelyfunctions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to prove lower bounds (Omega results) for Dirichlet character sums twisted by a completely multiplicative coefficient f, summing f(n)χ(n) over n ≤ N, in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q. The main theorem, Theorem 1.2, claims the lower bound sqrt(N) exp((√2+o(1)) sqrt(log(q/N) log_3(q/N) / log_2(q/N))) for the maximum over non-principal characters, under an added positivity condition on f. A sympathetic reader would care because such mixed sums connect character sums to multiplicative functions, and lower bounds in this range are rare. However, the positivity condition is so restrictive that it forces f identically 1, so the theorem's stated generality over non-trivial f is vacuous; the result survives only for trivial coefficients.

What carries the argument

The central object is the resonator R_χ = Σ_{m} r(m) f(m) χ(m), where r is a multiplicative function supported on smooth numbers. For Theorem 1.2, the proof relies on a sharp asymptotic for maximal GCD sums: max_{|M|=K} Σ_{m,n∈M} sqrt((m,n)/[m,n]) = K exp((2√2+o(1)) sqrt(log K log_3 K / log_2 K)). This identity is the engine that converts a lower bound on the number of congruence solutions into the exponential gain in the final estimate. The positivity condition Re(f(m) overline(f(n))) ≥ c ensures that the off-diagonal congruence terms contribute with the same sign, so the lower bound for the second moment follows.

What would settle it

For any prime p and integer a, the condition gives Re(f(p)^a) ≥ c for all a; letting a vary forces f(p) = 1, so no non-constant completely multiplicative f with |f| = 1 can satisfy the hypothesis. To test the theorem's actual content, compute max_{χ ≠ χ_0} |Σ_{n≤N} χ(n)| for a prime q and N in [exp((log q)^{1/2+δ}), √q]; if the stated lower bound fails, the proof has an error.

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Extended reading notes

Core claim

On its own terms, the paper establishes an Omega result for character sums by injecting the twist f directly into the resonator and using congruences to bound the second moment from below. The proof of Theorem 1.2 reduces the evaluation of the resonance sum to counting solutions of m'k ≡ n'l (mod q), separates the diagonal terms m'k = n'l, and uses the positivity of off-diagonal terms to discard them. The remaining GCD-sum lower bound yields the stated exponential factor. For Theorem 1.1, a classical resonance argument gives a bound in the shorter range where log N is of order sqrt(log q log_2 q).

Load-bearing premise

The load-bearing assumption is that Re(f(m) overline(f(n))) ≥ c > 0 for all integers m,n; because this condition is so strong that it forces f(n) = 1 for every n, the theorem cannot apply to any non-trivial multiplicative coefficient and its advertised scope collapses.

Editorial extensions

If this is right

In the special case f ≡ 1, Theorem 1.2 yields a new Omega lower bound for ordinary Dirichlet character sums in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q.
Theorem 1.1 supplies a lower bound for mixed sums in the shorter range where log N = sqrt(log q log_2 q) · (log_2 q)^{O(1)}.
The resonance approach directly incorporates the coefficient f, so the same framework could be reused for other multiplicative coefficients if hypotheses permit.
The GCD-sum estimate forces the lower bound's exact exponential shape, linking the problem to the extremal behavior of lcm/gcd ratios.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The positivity condition Re(f(m) overline(f(n))) ≥ c for all m,n forces f(p) = 1 for every prime p (take m = p^a, n = 1), so Theorem 1.2's hypothesis is satisfied only by f ≡ 1; the advertised 'multiplicative coefficients' result therefore reduces to the trivial coefficient case.
One might salvage the theorem by weakening the condition to hold only on average, e.g., with a weight over m,n, which could still control the off-diagonal terms without forcing pointwise positivity.
Read as a pure character-sum result for f = 1, the lower bound could be tested numerically for primes q and N near q^{1/2}; a failure would indicate a gap in the GCD-sum argument.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper has one genuinely nice observation: in Hough's resonator, an arbitrary completely multiplicative twist with |f|=1 cancels on the diagonal, so Theorem 1.1 extends Hough's Theorem 3.2 with modest effort. That part is probably correct, and it is new as stated. The authors are also honest about relying on heavy external machinery (La Bretèche–Tenenbaum for GCD sums, Hough's pages 105–107), which is fine when cited precisely.

The problem is Theorem 1.2, the advertised central result. The extra hypothesis Re(f(m) \overline{f(n)}) ≥ c > 0 for all integers m,n forces f≡1. For any prime p, take m=p^a, n=1; then Re(f(p)^a) ≥ c for every a≥1, which only f(p)=1 can satisfy. So the theorem never applies to a nontrivial multiplicative coefficient. The proof uses that positivity to keep off-diagonal congruence contributions positive, so the vacuity is load-bearing, not cosmetic.

There's also a concrete error in the counting step: the bound N / max(m/g, n/g) ≥ (N/√2) sqrt((m,n)/[m,n]) is false for unbalanced pairs. For m/g=N, n/g=1 it gives 1 ≥ sqrt(N/2), which fails for N>2. That step is essential for the final lower bound, and the authors do not exclude such unbalanced pairs from the extremal set M.

A smaller issue: the definition of r(m') is internally inconsistent as printed (it seems to conflate |M_j| and something else), which adds to the referee's burden.

The paper is not a waste of time. Theorem 1.1 is worth having, and the f=1 case of Theorem 1.2 would recover a known GCD-sum resonance bound. But the advertised generalization to multiplicative coefficients is unsupported. The abstract oversells it.

Who gets value? Specialists in extreme values of character sums, and people who want to see how the resonance method handles twists. It deserves a serious referee, because the ideas are standard, the flaws are identifiable, and the fixes are plausible: restate the positivity condition on the integers that actually appear (up to q^{1+o(1)}), repair the pair-balance inequality, correct r(m'), and compare properly with the authors' own [3]. But I would not cite it in its current form.

Recommendation: send it to peer review, but the referee should be told to check the vacuity argument and the counting step first. With those fixed, the paper could become a reasonable contribution; without them, Theorem 1.2 collapses to the trivial coefficient.

Referee Report

3 major / 4 minor

Summary. The paper studies the maximum, over non-principal Dirichlet characters modulo a large prime q, of |∑_{n≤N} f(n)χ(n)| for completely multiplicative coefficients f with |f(n)|=1. Theorem 1.1 claims a lower bound of size √N exp((1+o(1)) A(τ+τ') √(log q / log_2 q)) when log N ≈ √(log q log_2 q) τ, following Hough's resonance method. Theorem 1.2 claims a stronger lower bound, √N exp((√2+o(1)) √(log(q/N) log_3(q/N) / log_2(q/N))), in the range exp((log q)^{1/2+δ}) ≤ N ≤ √q, under the additional assumption that Re f(m) \overline{f(n)} ≥ c > 0 for all integers m,n. The proof of Theorem 1.2 uses Gál sums via Lemma 3.1 (a result of La Bretèche and Tenenbaum) and a positivity argument for off-diagonal congruence terms.

Significance. If valid, Theorem 1.2 would be a substantial advance over both Hough's result and the related preprint [3], and the strategy of building f directly into the resonator is natural. The manuscript also contains a plausible extension of Hough's method in Theorem 1.1. However, the central new theorem is not supported as stated: its positivity hypothesis forces f ≡ 1, so the advertised Omega result for non-trivial multiplicative coefficients is vacuous. In addition, a key counting estimate in the proof of Theorem 1.2 is false, and the claimed lower bound does not follow even in the trivial case. These are load-bearing flaws, not presentation issues.

major comments (3)

[Theorem 1.2 (Introduction, Section 3)] The hypothesis 'Re f(m) \overline{f(n)} ≥ c holds for any integers m,n and some absolute constant c>0' forces f≡1. Indeed, for any prime p, complete multiplicativity and |f|=1 give f(p^a)=f(p)^a; taking n=1 yields Re(f(p)^a) ≥ c for every a≥1. This is possible only if f(p)=1. Hence f(p)=1 for all primes p, so f is the trivial coefficient function. Consequently Theorem 1.2 has no non-trivial instances and cannot be described as an Omega result for general multiplicative coefficients. The introductory claim that it 'improves previous result of [3] at a cost of assuming an additional condition' is misleading: the cost is vacuity.
[Section 3, inner-sum estimate] The displayed lower bound for the congruence counting sum is false. With g=(m,n), a=m/g, b=n/g, the exact number of pairs (k,ℓ)≤N with mk=nℓ is floor(N/max(a,b)). The proof asserts the lower bound (N/√2)√((m,n)/[m,n]) = N/√(2ab). For example, if a=N and b=1 (i.e., m and n differ by a factor N), the count is 1, while the claimed bound is √(N/2), which exceeds 1 for N>2. The asserted inequality would require max(a,b) ≤ √(2ab), equivalently max(a,b) ≤ 2 min(a,b). The extremal set M from Lemma 3.1 is not shown to satisfy this condition for the pairs retained in the truncated sum, so the lower bound for M2 is not established.
[Section 2, definition of r and M2 computation] The identity leading to ∑_{k≤N} r(k) ∑_{m≤Y/k} r(m)^2 requires r(mk)=r(m)r(k) for all m,k, i.e., complete multiplicativity of the resonator. The text defines r as a 'multiplicative function' and gives only values at primes; prime powers are not specified, and multiplicativity alone does not support the factorization. This affects the proof of Theorem 1.1 as well as Theorem 1.2. The authors should state explicitly that r is completely multiplicative and define r(p^a).

minor comments (4)

[Theorem 1.2 statement] The displayed assumption 'Ref(m) f(n) ≥ c' is missing the conjugation bar; it should read Re(f(m)\overline{f(n)}) ≥ c.
[Section 3, displayed chain] The chain '≥ N q/(2 m/(m,n) n/(m,n))' appears to have a typographical 'q' or a misplaced factor; as printed it is dimensionally inconsistent. Please correct the intended expression.
[Section 2, page 105 of [6]] The claim 'By the proof of Page 105 in [6], we have ∑_{m≤Y/k} r(m)^2 = (1+o(1))∑_{m≥1} r(m)^2' is invoked without stating the uniformity conditions required for the range of k. The reader should be given the precise lemma or the needed hypotheses.
[References] Reference [3] is the same group's preprint; the relation to the current paper's Theorem 1.1 and 1.2 should be clarified, especially whether Theorem 1.2 really supersedes it once the vacuity issue is resolved.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proofs rest on genuinely external resonance and GCD-sum estimates; the only self-referential line is a non-load-bearing 'improves [3]' comparison.

full rationale

Both main theorems are derived from external sources rather than from the paper's own conclusions. Theorem 1.1 is explicitly routed through Hough's resonance method: the proof says 'By the proof of Page 105 in [6]' and then 'we completes the proof by following Page 105–107 of [6].' That is independent support, not self-citation. Theorem 1.2 relies on Lemma 3.1, which is stated as 'This is Theorem 1.1 of [2]' — La Bretèche and Tenenbaum's external GCD-sum estimate — and the subsequent resonator argument uses that estimate directly. No free parameter is fitted to the target lower bound, and no conclusion is renamed as a prediction. The only self-referential passage is the sentence 'This improves previous result of [3]' in Section 1, where [3] is the same group's preprint. However, [3] is not used anywhere in the proofs of Theorem 1.1 or Theorem 1.2; it is a comparative claim about literature, not a load-bearing step, so it does not constitute circularity. A serious correctness problem exists: the hypothesis Re(f(m) overline(f(n))) >= c > 0 in Theorem 1.2 forces f ≡ 1, since for any prime p, choosing m = p^a and n = 1 gives Re(f(p)^a) >= c for all a, which only f(p)=1 satisfies. This makes Theorem 1.2 vacuous for nontrivial multiplicative coefficients, but that is a limitation of the theorem's applicability, not a circular derivation. The paper is therefore self-contained against external benchmarks for the purposes of this circularity review, with score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on Hough's resonance estimates [6], La Bretèche-Tenenbaum's Gal/GCD sums [2], the unstated complete multiplicativity of the resonator r, the positivity condition on f, and an unproven size-balance property of the Gal set. The positivity condition is the most fragile: as stated it forces f identically 1. No free parameters are fitted to data; tau and A in Theorem 1.1 are related by a bijection through two integrals, and the constant c in Theorem 1.2 is an existential input, not a fitted number.

assumptions (6)

ad hoc to paper The resonator weights r satisfy r(mk)=r(m)r(k), i.e. complete multiplicativity.
Used in section 2 in the identity sum_{m,n} r(m)r(n) sum_{k: mk=n} f(mk) overline(f(n)) = sum_k r(k) sum_{m<=Y/k} r(m)^2. The paper only defines r as multiplicative, so this identity needs an unstated strengthening.
domain assumption The extremal Gal set M from Lemma 3.1 exists with |M|=floor(q/N), y_M=max P(m) <= (log(q/N))^{1+o(1)}, and attains the GCD-sum bound.
Theorem 1.2 rests entirely on this black box, imported as Theorem 1.1 of [2] without proof.
domain assumption Re(f(m) overline(f(n))) >= c > 0 for all integers m,n.
Theorem 1.2 hypothesis, used in section 3 to give the off-diagonal cross terms a fixed sign. This assumption forces f identically 1, as shown by taking m=p^a, n=1.
ad hoc to paper Pairs (m,n) in the Gal set with a=m/(m,n) and b=n/(m,n) satisfy max(a,b) <= 2 min(a,b), or else their contribution is negligible.
The displayed count bound N/max(a,b) >= (N/sqrt(2))/sqrt(ab) in section 3 is false without such a balance condition, which is neither stated nor proved.
domain assumption sum_{m<=Y/k} r(m)^2 = (1+o(1)) sum_{m>=1} r(m)^2.
Quoted from 'the proof of Page 105 in [6]' in section 2; it requires the mass of r to be concentrated below Y/k, which is not demonstrated in this paper.
standard math Orthogonality of Dirichlet characters modulo q.
Used in both proofs to pass from sums over characters to counting conditions q | mk - n and q | m'k - n'l; standard and correct for prime q.

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Cite this review

Pith. "Pith review of Large character sums with multiplicative coefficients." pith.science (2026). https://pith.science/paper/2WZR7IGH

@misc{pith2026250909649,
  author       = {Pith},
  title        = {Pith review of: Large character sums with multiplicative coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WZR7IGH}},
  note         = {Machine review of arXiv:2509.09649}
}

abstract

In this paper, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove a new Omega result in the region $\exp((\log q)^{\frac12+\delta})\le N\le\sqrt q$, where $q$ is the prime modulus.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

12 extracted references · 2 linked inside Pith

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [3]

Sommes de G\' a l et applications , Proc

de la Bret\` e che, R.; Tenenbaum, G. Sommes de G\' a l et applications , Proc. Lond. Math. Soc. , 119 (2019), 104--134

2019

[2] [1]

Lower bounds for the maximum of the Riemann zeta function along vertical lines , Math

Aistleitner, C. Lower bounds for the maximum of the Riemann zeta function along vertical lines , Math. Ann. , 365 (2016), 473--496

2016

[3] [2]

Large greatest common divisor sums and extreme values of the Riemann zeta function , Duke Math

Bondarenko, A.; Seip, K. Large greatest common divisor sums and extreme values of the Riemann zeta function , Duke Math. J. , 166 (2017), 1685--1701

2017

[4] [4]

Large values of character sums with multiplicative coefficients , preprint, arXiv:2508.09750

Dong, Z.; Li, Z.; Song, Y.; Zhao, S. Large values of character sums with multiplicative coefficients , preprint, arXiv:2508.09750

arXiv

[5] [5]

Large character sums , J

Granville, A.; Soundararajan K. Large character sums , J. Amer. Math. Soc. , 14 (2001), 365--397

2001

[6] [6]

Harper, A. J. The typical size of character and zeta sums is o( x ) , preprint, arXiv:2301.04390

arXiv

[7] [7]

An arithmetical mapping and applications to results for the Riemann zeta function , Acta Arith

Hilberdink, T. An arithmetical mapping and applications to results for the Riemann zeta function , Acta Arith. , 139 (2009), 341--367

2009

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The resonance method for large character sums , Mathematika , 59 ,(2013), 87--118

Hough, B. The resonance method for large character sums , Mathematika , 59 ,(2013), 87--118

2013