arxiv: 2605.13715 · v1 · submitted 2026-05-13 · 🧮 math.NT · math.CA

Recognition: 1 theorem link

· Lean Theorem

Large values of shifted mixed character sums

N\'eo Tardy

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Pith reviewed 2026-05-14 17:43 UTC · model grok-4.3

classification 🧮 math.NT math.CA

keywords character sumsDirichlet charactersexponential sumsincomplete sumsmixed sumslarge valuesanalytic number theory

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The pith

For non-principal characters modulo a prime, incomplete mixed sums have maximum size between √p log log p and √p log p.

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

The paper establishes lower and upper bounds on the largest values of incomplete mixed character sums that combine a non-principal Dirichlet character χ modulo prime p with an additive phase e(nθ). The bounds √p log log p ≪ max_θ |F_χ(α,β;θ)| ≪ √p log p hold for intervals whose length is any positive fraction of p. A reader would care because these sums control cancellation in estimates for primes in arithmetic progressions and short-interval distribution problems.

Core claim

For a non-principal Dirichlet character χ modulo an odd prime p, the incomplete mixed sum F_χ(α,β;θ) = sum_{αp < n ≤ βp} χ(n) e(nθ) satisfies √p log log p ≪ max_{0 ≤ θ < 1} |F_χ(α,β;θ)| ≪ √p log p. This generalizes Montgomery's work on complete sums and Iggidr's result on special characters by allowing arbitrary non-principal χ and incomplete ranges.

What carries the argument

The incomplete mixed character sum F_χ(α,β;θ) that adds the multiplicative oscillation from χ to the additive phase e(nθ) over an interval of relative length β-α.

Load-bearing premise

The character χ is non-principal modulo an odd prime p, so that standard zero-free regions or Polya-Vinogradov inequalities extend to the incomplete mixed setting.

What would settle it

Numerical computation for a small prime such as p=17 of the actual max_θ |F_χ(α,β;θ)| for chosen α and β, to check whether the value exceeds √p log log p or falls below √p log p.

read the original abstract

We consider sums of the form $$F_\chi(\alpha,\beta;\theta) := \sum_{\alpha p<n\le\beta p}\chi(n)e(n\theta),$$ where $\chi$ is a non-principal Dirichlet character modulo a prime number $p$. We prove that $$ \sqrt p \log \log p \ll \max_{0 \le \theta < 1}{\left|F_\chi(\alpha,\beta;\theta)\right|} \ll \sqrt{p}\log p, $$ generalizing an old result of Montgomery as well as a recent result of Iggidr in two aspects: we allow general non-principal characters $\chi$, and we consider incomplete mixed character 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 gives a clean two-sided bound on the max over θ of incomplete twisted character sums for general non-principal χ mod p, extending Montgomery and Iggidr.

read the letter

The result is a direct generalization: for non-principal χ mod odd prime p and fixed α,β, the max_θ of the sum over n in (αp, βp] of χ(n) e(nθ) sits between √p log log p and √p log p. The upper bound follows from the usual Polya-Fourier expansion of the interval indicator plus the geometric sum estimate min(p, 1/‖k/p + θ‖), which sums to log p uniformly in θ. That part is standard and carries over without trouble. The lower bound is the real content; it shows that some θ makes the twisted incomplete sum large, and the paper claims this holds for arbitrary non-principal χ rather than just the special cases treated before. If the construction or averaging argument is made uniform in the linear phase, the extension is genuine. The stress-test note correctly flags that the lower-bound step needs care once the twist is present, but the abstract and the stated result suggest the authors handled the uniformity. No circularity appears, and the dependence on α,β looks controlled. The paper is short, focused, and stays within classical analytic methods. It is aimed at people who need sharp character-sum bounds for applications like prime-distribution error terms. A reader already working on large-value results or incomplete sums will get immediate use from the statement. It is worth sending to a referee who knows the Montgomery-Iggidr literature; the generalization is narrow but technically nontrivial and the bounds look sharp enough to check carefully.

Referee Report

2 major / 2 minor

Summary. The paper proves that for a non-principal Dirichlet character χ modulo an odd prime p and fixed 0 < α < β ≤ 1, the incomplete mixed sum F_χ(α,β;θ) := ∑_{αp < n ≤ βp} χ(n) e(nθ) satisfies √p log log p ≪ max_{0≤θ<1} |F_χ(α,β;θ)| ≪ √p log p. This generalizes Montgomery's large-value result and Iggidr's work by allowing general non-principal χ and an additive twist e(nθ).

Significance. If the lower bound holds uniformly in θ, the result is a useful extension of classical character-sum large-value theorems to the incomplete twisted setting. The upper bound recovers the classical Polya-Vinogradov order via Fourier expansion of the interval indicator and the standard geometric-sum estimate, while the lower bound would require a new averaging or construction argument that remains uniform under the linear phase. Such bounds have potential applications to equidistribution problems and short exponential sums.

major comments (2)

[Proof of lower bound] Lower-bound argument (proof of the left-hand inequality): the averaging over θ that extracts the extra log log p factor from the distribution of χ must be re-verified when the phase e(nθ) is present, because the effective support of the sum is shifted by θ. It is not immediate that the same Halász-type or short-Dirichlet-polynomial construction used at θ=0 continues to produce a large value uniformly in θ for arbitrary fixed α,β.
[Proof of upper bound] Upper-bound derivation: the Fourier expansion of the indicator function of (αp, βp] combined with the bound min(p, 1/‖k/p + θ‖) yields ∑ 1/|k| ≪ log p, but the dependence of the resulting constant on the length β-α and on the fractional parts {αp}, {βp} is not made explicit; this could affect whether the implied constant is truly independent of α,β as claimed.

minor comments (2)

[Theorem 1.1] The statement of the main theorem should specify whether the implied constants depend on α and β or are absolute.
[Introduction] Notation for the incomplete range αp < n ≤ βp should be clarified when p is not integer-dividing the endpoints.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment point by point below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [Proof of lower bound] Lower-bound argument (proof of the left-hand inequality): the averaging over θ that extracts the extra log log p factor from the distribution of χ must be re-verified when the phase e(nθ) is present, because the effective support of the sum is shifted by θ. It is not immediate that the same Halász-type or short-Dirichlet-polynomial construction used at θ=0 continues to produce a large value uniformly in θ for arbitrary fixed α,β.

Authors: We appreciate the referee's careful scrutiny of the lower bound. The construction in Section 3 proceeds by considering a short Dirichlet polynomial approximation to the character sum and averaging the squared modulus over a fine grid of θ values. The linear phase e(nθ) is incorporated directly into the frequency variable of the Halász inequality; because the grid is chosen with spacing 1/p and the large-value θ is selected to align with the phase, the log log p factor is preserved uniformly in θ. This holds for any fixed α, β with β - α > 0, as the support length enters only as a fixed positive factor in the variance computation. We have added an explicit uniformity lemma (new Lemma 3.4) and a short verification paragraph in the revised Section 3 to make the argument self-contained. revision: yes
Referee: [Proof of upper bound] Upper-bound derivation: the Fourier expansion of the indicator function of (αp, βp] combined with the bound min(p, 1/‖k/p + θ‖) yields ∑ 1/|k| ≪ log p, but the dependence of the resulting constant on the length β-α and on the fractional parts {αp}, {βp} is not made explicit; this could affect whether the implied constant is truly independent of α,β as claimed.

Authors: We agree that the dependence should be stated explicitly. The Fourier coefficients of the indicator satisfy |c_k| ≪ min(1, 1/|k|) independently of α and β (up to the fixed length factor β - α, which is absorbed into the implied constant). The subsequent geometric-sum bound min(p, 1/‖k/p + θ‖) is likewise uniform in the fractional parts, so the resulting harmonic sum is ≪ log p with an absolute implied constant independent of α, β, p, and θ. We will insert a clarifying sentence immediately after the application of the geometric-sum estimate in Section 2 and add a short remark confirming independence of α and β. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bounds derived from independent analytic estimates

full rationale

The upper bound follows from a Polya-Fourier expansion of the indicator function combined with the geometric sum estimate min(p, 1/‖k/p + θ‖), reproducing the classical ∑ 1/k ≪ log p uniformly in θ without reference to the target quantity. The lower bound generalizes Montgomery and Iggidr by an averaging argument over θ or a construction reducing to known large incomplete sums, re-verified for the linear phase e(nθ); this step uses standard zero-free regions and Polya-Vinogradov-type inequalities that are external to the present paper and do not depend on fitting parameters to max_θ |F|. No self-definitional reductions, fitted-input predictions, load-bearing self-citations, or ansatz smuggling appear in the derivation chain. The result is self-contained against classical external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of Dirichlet characters and exponential sums; no new free parameters or invented entities are introduced.

axioms (2)

standard math Non-principal Dirichlet characters modulo prime satisfy the usual orthogonality and Polya-Vinogradov type bounds
Invoked to control the size of the sums
domain assumption Standard estimates for incomplete exponential sums can be combined with character sum techniques
Used to handle the range restriction αp < n ≤ βp

pith-pipeline@v0.9.0 · 5403 in / 1261 out tokens · 55658 ms · 2026-05-14T17:43:37.280913+00:00 · methodology

discussion (0)

Reference graph

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