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Large values of character sums with multiplicative coefficients

T0 review · 1 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For every large prime modulus and any bounded multiplicative coefficient, some non-principal character makes the weighted sum $\sum_{n\le N} f(n)\chi(n)$ at least $\sqrt N\exp((1+o(1))\sqrt{\log(q/N)/\log_2(q/N)})$.

desk verdict The claimed generalization to arbitrary multiplicative f fails at Section 3: |f|=1 does not make the diagonal phases disappear; the theorem needs complete multiplicativity or a nontrivial defect estimate. read the letter →

arxiv 2508.09750 v1 pith:4JJ3AVUM submitted 2025-08-13 math.NT

Zikang Dong , Zhonghua Li , Yutong Song , Shengbo Zhao This is my paper · ORCID
classification math.NT MSC 11L4011M06
keywords charactersumsmultiplicativecoefficientslargevaluesresonancemethodDirichletcharactersOmegaresultssecondmomentprimemodulus
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 proves a lower bound for the largest possible size of a Dirichlet character sum whose terms carry an arbitrary multiplicative coefficient $f(n)$ with $|f(n)|=1$. The bound has the same shape as the unweighted case: for every sufficiently large prime $q$ and every $N$ with $\exp((\log q)^{1/2+\delta})\le N\le \sqrt q$, there is a non-principal character $\chi$ such that $|\sum_{n\le N} f(n)\chi(n)|\ge \sqrt N\exp((1+o(1))\sqrt{\log(q/N)/\log_2(q/N)})$. In other words, multiplying by a bounded multiplicative weight does not shrink the extreme values that character sums can attain. The proof uses the resonance method, choosing an auxiliary multiplicative weight so that a second-moment average over characters is dominated by the diagonal terms $an=bm$. The result extends the known $f(n)=1$ theorem to arbitrary bounded multiplicative coefficients and confirms a natural expectation in the range where character sums are hardest to pin down.

What carries the argument

The resonance method for character sums is the load-bearing device. An auxiliary multiplicative function $r(n)$ is chosen (squarefree-supported, with prime values set to a resonance profile on the interval $\lambda\le p\le\exp((\log\lambda)^2)$, $\lambda=\sqrt{\log X\log_2 X}$) so that the second moment of the product of the weighted character sum with its resonator concentrates on the diagonal. The key identity is the congruence reduction: because $NX\le q$, the condition $an\equiv bm\pmod q$ collapses to the equality $an=bm$, and character orthogonality then produces the main term $\phi(q)\sum_{an=bm}r(a)r(b)$. Lemma 2.1 supplies the lower bound for this diagonal sum, which is what ultimat

What would settle it

Compute, for a moderately large prime $q$ with $N$ near $q^{1/3}$, the actual maximum over all non-principal $\chi$ of $|\sum_{n\le N}\lambda(n)\chi(n)|$, where $\lambda$ is the Liouville function; if for some $q$ this is consistently below $\sqrt N\exp((1+o(1))\sqrt{\log(q/N)/\log_2(q/N)})$, the theorem's lower bound cannot hold as stated. A more direct check is to evaluate the diagonal sum in Section 3 for a non-completely multiplicative $f$ and see whether the missing phase factors alter the main term.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1. It asserts that for fixed $\delta\in(0,1/100)$, for all sufficiently large primes $q$, all $N$ in the stated interval, and all multiplicative $f$ with $|f(n)|=1$, the maximum over non-principal characters of $|\sum_{n\le N}f(n)\chi(n)|$ is at least $\sqrt N\exp((1+o(1))\sqrt{\log(q/N)/\log_2(q/N)})$. The proof sets $X=q/N$ and studies the ratio $M_2/M_1$, where $M_2$ is the second moment over $\chi\ne\chi_0$ of $|\sum_{n\le N}f(n)\chi(n)\sum_{a\le X}f(a)r(a)\chi(a)|^2$ and $r$ is a deliberately chosen multiplicative function. Character orthogonality and $NX\le q$ turn the second moment into a diagonal sum over $an=bm$, which is then evaluated with a

Load-bearing premise

The load-bearing premise is that, after character orthogonality, the diagonal contribution $\sum_{an=bm} f(n)f(a)r(a)r(b)$ collapses to $\sum_{an=bm} r(a)r(b)$, which requires $f$ to be completely multiplicative on those pairs while the theorem states only ordinary multiplicativity.

Editorial extensions

If this is right

  • For any sufficiently large prime $q$ and any $N$ in the range, the lower bound holds for every multiplicative $f$ with $|f|=1$, not just for the constant function.
  • The exponent gained over $\sqrt N$ is exactly $\sqrt{\log(q/N)/\log_2(q/N)}$, matching the unweighted extreme-value profile throughout the interval.
  • The proof's diagonal reduction works whenever $NX\le q$, so the whole range below $\sqrt q$ is covered uniformly.
  • The theorem establishes the same lower-bound shape as the unweighted case, which is what the conjectural picture predicts for weighted character sums.

Reading between the lines

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

  • Inference: the proof as written appears to use complete multiplicativity in the reduction to $\sum_{an=bm}r(a)r(b)$; a reader extending this to all multiplicative $f$ should check whether the phase defects $f(na)/(f(n)f(a))$ average to $1$ on the diagonal.
  • Inference: the same diagonal orthogonality argument is naturally portable to other arithmetic weights, such as $n^{it}$ or short-interval twists, where the equality $an=bm$ is replaced by a different additive condition.
  • Inference: the resonance interval $\lambda\le p\le\exp((\log\lambda)^2)$ controls the achievable exponent; varying its length is a plausible route to sharpening constants or extending the range below $\exp((\log q)^{1/2})$.
  • Inference: a numerical experiment with the Liouville function at $N=q^{1/3}$ would give a concrete check of whether the diagonal main term survives without complete multiplicativity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper claims an Omega lower bound for character sums with multiplicative coefficients: for a prime q, an integer N with exp((log q)^{1/2+δ}) ≤ N ≤ √q, and any multiplicative f with |f(n)| = 1, there exists a nonprincipal character χ mod q such that |Σ_{n≤N} f(n)χ(n)| ≥ √N exp((1+o(1))√(log(q/N)/log₂(q/N))). The proof uses the resonance method: it defines a resonator r supported on squarefree integers with r_f(n)=f(n)r(n), expands the second moment over nonprincipal characters, reduces the congruence an≡bm mod q to the equality an=bm, and then invokes Hough's counting lemma to evaluate the diagonal sum.

Significance. If correct, the result is a natural and valuable generalization of Hough's large-character-sum lower bound to arbitrary unit-valued multiplicative weights, matching the unweighted extreme-value profile over the whole range N ≤ √q. The paper relies on external, independently established inputs (Hough's resonance counting lemma, the Granville–Soundararajan framework), and the proof strategy is coherent. However, the main theorem as stated is not established because a key diagonal simplification requires complete multiplicativity, which is not assumed.

major comments (1)
  1. [Section 3, diagonal step after defining r_f(n):=f(n)r(n)] The step 'Since |f(n)|=1, we can obtain M2 = φ(q)Σ_{an=bm} r(a)r(b) + O(qN)Σ|r_f|²' is invalid for a merely multiplicative f. The actual diagonal summand is f(n)overline{f(m)}f(a)overline{f(b)}r(a)r(b). Collapsing this to r(a)r(b) requires f(n)f(a)=f(na) and f(m)f(b)=f(mb), i.e. complete multiplicativity; the condition |f(n)|=1 alone is insufficient when n and a, or m and b, share a prime. For example, choose two primes p,ℓ in the resonator support and define a multiplicative f by f(p)=1, f(p²)=-1, f(ℓ)=1, and all other prime-power values 1. Take n=p², a=ℓ, m=pℓ, b=p; then an=bm=p²ℓ, all four variables are within range for large q, yet the diagonal summand equals -r(ℓ)r(p) instead of r(ℓ)r(p). Thus Hough's lemma is applied to an expression that has not been shown to equal the diagonal sum of r(a)r(b). No averaging estimate for the phase defects is supplied. The theorem as stated for all
minor comments (5)
  1. [Theorem 1.1 and throughout] The manuscript text contains severe encoding/rendering corruption, e.g. 'Theorem 1.1. ��� δ ∈ ...'. The formulas need to be carefully regenerated before the paper can be assessed or published.
  2. [Lemma 2.1 and Section 3] The resonator weight is printed as r(p)=λ√q log p, and q is not introduced in Lemma 2.1. This is likely an OCR/typographical error (perhaps λ√p/log p); as printed, the dependence on q and the missing division are confusing and should be corrected.
  3. [Section 3, range condition] The assertion that the range exp((log q)^{1/2+δ}) ≤ N ≤ √q implies log N > 3λ log₂λ is made without derivation. A short justification would help the reader verify that Hough's lemma applies.
  4. [Notation] After defining r_f(n):=f(n)r(n), the notation switches back to r for the counting lemma; the distinction between r_f and r should be maintained to avoid confusion in the diagonal simplification.
  5. [Abstract and introduction] The abstract promises 'multiplicative coefficients', but the proof as written supports only completely multiplicative f. If the theorem is revised, the abstract and introduction should be aligned with the actual hypothesis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof imports Hough's resonance-counting lemma as external support and does not fit parameters to the target; the |f(n)|=1 diagonal step is a potential correctness gap, not a circular reduction.

full rationale

The paper's central derivation is not circular. Theorem 1.1 is obtained by a standard resonance-method argument that imports Lemma 2.1 from Hough's paper [5] as an external, independently established counting estimate. The resonator parameters lambda = sqrt(log X log2 X) and r(p) are taken verbatim from Lemma 2.1, not fitted to the desired lower bound; the final constant (1+o(1)) emerges from the counting lemma's diagonal contribution, not from tuning lambda to match the theorem statement. There is no self-citation chain: the cited references [1], [2], [3], [5], [6], [7], [8] are all external works, and the paper does not invoke a uniqueness theorem from its own authors. The only questionable step is mathematical rather than circular: after setting r_f(n)=f(n)r(n), the text asserts 'Since |f(n)|=1, we can obtain M2 = phi(q) sum_{an=bm} r(a)r(b) + ...'. For a merely multiplicative, unit-valued f, this replacement requires f(n)f(a)=f(na) and f(m)f(b)=f(mb) on diagonal pairs, i.e. complete multiplicativity or an extra averaging argument for the phase defects f(na)/(f(n)f(a)). The text does not supply that argument. This is a potential gap in the proof as written and a correctness risk, but it is not circularity: the claimed result is not equivalent to an input by construction, and no fitted quantity is renamed as a prediction. Accordingly, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The ledger is small: the result has no invented entities and no fitted constants beyond the resonator's saddle-point parameters (λ and the prime support range), both inherited from Hough's method and absorbed into the o(1). The three genuine inputs are the standard orthogonality of characters with the an,bm ≤ q range reduction, the borrowed resonance counting lemma (Lemma 2.1), and the paper's own assumption on f. The hidden fourth input is complete multiplicativity of f on diagonal products, which the proof uses but the theorem does not state.

free parameters (2)
  • resonator scale λ = λ = sqrt(log X log_2 X), X = q/N
    Chosen to optimize the second-moment lower bound in Section 3. Suboptimal choices affect only the (1+o(1)) term, so it is a standard saddle-point parameter rather than a target-fitted constant, but it is a hand-chosen parameter on which the condition log N > 3λ log_2 λ depends.
  • resonator prime weight r(p) = λ√(q log p) for λ ≤ p ≤ exp((log λ)^2), 0 otherwise (exact normalization ambiguous in the garbled text)
    The shape and support of the resonator are imported from Lemma 2.1 (Hough's method); the support range is chosen ad hoc to make the counting lemma apply.
assumptions (6)
  • standard math Orthogonality of Dirichlet characters: (1/φ(q)) Σ_χ χ(a)χ̄(b) = 1_{a≡b mod q}
    Used in Section 3 to expand M2 into the diagonal sum over an ≡ bm (mod q).
  • standard math an, bm ≤ q implies an ≡ bm (mod q) if and only if an = bm
    Range condition N X = q in Section 3 with a,b ≤ X and m,n ≤ N; the boundary case an = bm = q is consistent.
  • standard math Lemma 2.1 (resonance counting lemma): Σ_{an=bm} r(a)r(b) ≥ N exp((2+o(1))√(log X/log_2 X)) Σ r(n)² for X = q/N in the stated range
    Stated without proof as 'the main technique in operating the resonance method' and applied in Section 3; it is presumably carried over from Hough [5] or Hilberdink [4], but no proof or theorem reference appears at the statement.
  • domain assumption f is multiplicative and |f(n)| = 1 for every n
    Theorem 1.1's defining assumption on the coefficients; used to bound |D(χ₀)|² ≤ N² and to attempt to remove f from the diagonal.
  • domain assumption q is prime and exp((log q)^{1/2+δ}) ≤ N ≤ √q with δ ∈ (0,1/100)
    Theorem 1.1; the lower bound on N ensures log N > 3λ log_2 λ, which the text verifies in Section 3.
  • ad hoc to paper Complete multiplicativity of f on the diagonal pairs (n,a) and (m,b)
    Implicit in the step 'Since |f(n)| = 1, M2 = φ(q) Σ r(a)r(b) + ...' in Section 3; not stated in Theorem 1.1, and false for general multiplicative f. This is the hidden premise the proof actually needs.

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Pith. "Pith review of Large values of character sums with multiplicative coefficients." pith.science (2026). https://pith.science/paper/4JJ3AVUM

@misc{pith2026250809750,
  author       = {Pith},
  title        = {Pith review of: Large values of character sums with multiplicative coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JJ3AVUM}},
  note         = {Machine review of arXiv:2508.09750}
}
abstract

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

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Large character sums with multiplicative coefficients

    math.NT 2025-09 conditional novelty 5.0 of 10

    An Omega lower bound near sqrt(N) exp((sqrt(2)+o(1)) sqrt(log(q/N) log_3(q/N)/log_2(q/N))) is claimed for character sums weighted by completely multiplicative f, but the extra positivity condition in the statement for...

Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages · cited by 1 Pith paper

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