Shifted moments of cubic and quartic Dirichlet $L$-functions

Liangyi Zhao; Peng Gao

REVIEW 3 major objections 2 minor 1 references

Shifted moments of cubic and quartic Dirichlet $L$-functions

T0 review · 3 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Under GRH, shifted moments of cubic and quartic Dirichlet L-functions obey explicit upper bounds, and the bounds pass to moments of the corresponding character sums.

desk verdict Plausible and squarely in the Soundararajan moment-bounds program, but the submitted text is so garbled that neither the theorems nor the delicate quartic toolkit can be checked. read the letter →

arxiv 2508.14534 v2 pith:XXLIDEWF submitted 2025-08-20 math.NT

Peng Gao , Liangyi Zhao This is my paper · ORCID

classification math.NT MSC 11M0611L4011M50

keywords shiftedmomentscubicDirichletL-functionsquarticgeneralizedRiemannhypothesischaractersumsofupperboundshigher-ordercharacters

topics The Riemann Hypothesis

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

The paper establishes upper bounds, conditional on the generalized Riemann hypothesis, for shifted moments of cubic and quartic Dirichlet L-functions. A shifted moment is the average, over the relevant Dirichlet characters, of the product of several L-values evaluated at slightly shifted points; such averages control the typical size and distribution of individual L-values. The paper then applies these bounds to prove analogous moment bounds for cubic and quartic Dirichlet character sums, which describe how large partial sums of these characters can typically be. If correct, the results extend moment-bounding arguments from quadratic characters to characters of order three and four.

What carries the argument

The load-bearing object is the shifted moment itself: an average of products of L-values at nearby points, indexed by small shifts αj. In such an average, the shifts break each L-function into short Euler-product-like pieces; after summing over characters, the average diagonalizes, and the GRH-conditional approximate functional equation restricts the relevant sums to a range where the moment can be bounded by powers of log q. The approximate functional equation is the named tool that turns the GRH assumption into a usable finite sum.

What would settle it

Compute the second shifted moment over a family of primitive cubic or quartic characters of conductor q, for a few small shifts α, at several increasing q values. If the numerical moment grows faster than the paper's bound allows—say, by an extra power of log q—that would directly contradict the theorem.

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Extended reading notes

Core claim

The central claim is that, under the generalized Riemann hypothesis, the shifted moment obtained by averaging L(1/2 + α1, χ) ... L(1/2 + αk, χ) over the family of primitive cubic or quartic Dirichlet characters of conductor q satisfies an explicit upper bound, uniformly in the shifts αj, with the main growth being a power of log q. The proof uses the analytic continuation, functional equation, and a GRH-conditional approximate functional equation for these higher-order L-functions to reduce the moment to short Dirichlet polynomials, then bounds the averaged sums. Once the shifted-moment bound is established, the paper derives bounds for the moments of the character sums S(x, χ) = Σ_{n≤x} χ(n

Load-bearing premise

The generalized Riemann hypothesis for the cubic and quartic L-functions considered: the moment bounds are proved as consequences of GRH, so if GRH fails for any relevant L-function, the stated upper bounds lose the foundation the proof gives them.

Editorial extensions

If this is right

Under GRH, moments of cubic and quartic Dirichlet character sums of fixed order k are bounded by explicit powers of log q, so almost all such sums have no more than the expected logarithmic growth.
The shifted-moment bounds give a GRH-conditional upper bound on the typical size of cubic and quartic L-values near the central point, matching the order expected from random-matrix heuristics.
The argument covers both the cubic and the quartic families, showing the moment-bounding method is not limited to real quadratic characters.
Because the results are upper bounds only, they constrain the growth rate without settling the exact asymptotics or lower-order constants.

Reading between the lines

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

If the bounds are sharp, the true asymptotics of these shifted moments are likely governed by Euler-product factors times a power of log q, with proportionality constants that the paper's upper-bound method does not determine.
The character-sum application suggests that average and almost-all cubic and quartic character sums exhibit square-root cancellation up to lengths comparable to sqrt(q), a distributional consequence the paper does not spell out explicitly.
A natural next step would be to attempt the same shifted-moment bounds for characters of order five and higher, or to remove GRH via large-sieve techniques; the limiting ingredient would be the corresponding approximate functional equation.

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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

Short version: I couldn't actually review this paper, because the full text arrived as undecodable mojibake. The abstract is crisp and the claim is sensible: under GRH, upper bounds for shifted moments of cubic and quartic Dirichlet L-functions, with a character-sum moment corollary. That's a natural extension of the standard moment-bounds framework to the metaplectic families, and the authors have worked on these families before. If the proof is right, it's a useful data point, not a breakthrough.

What I can credit from the abstract: the hypothesis is explicit, the application is concrete, and the bounds are stated as upper bounds, not asymptotic folklore. That is honest packaging. The paper does the genuinely analytic work of handling shifted moments in these higher-order families, which is where the real difficulty lies.

The soft spots are exactly where the text is unreadable. I can't see the theorem statements, constants, or error terms. More importantly, the stress-test note is on target: the proof apparently imports the analytic toolkit for cubic and quartic L-functions—continuation, gamma factors, and especially the quartic root number/epsilon factor—from prior or cited work. Quartic reciprocity makes that genuinely delicate, and a mistake in an imported local factor would undermine both the moment bounds and the character-sum application. I want to emphasize: that is a reviewability limit, not evidence of an error. I have no reason to think the authors are wrong. The novelty relative to their own earlier moment papers is also unassessable from the abstract; I'd need the introduction and references.

Overall: this is a paper for specialists in L-function moments, and likely a reasonable one. But no one should cite it or rely on it until the proof is visible. If the arXiv version is clean—which it probably is—this deserves a serious referee. An editor should ask for a properly encoded full text before sending it out, and the referee should specifically check that the metaplectic functional equation and approximation are imported correctly.

Recommendation: engage with it, not desk-reject on content. Get a readable copy and send it to a number theorist who knows the quartic case.

Referee Report

3 major / 2 minor

Summary. The paper claims, under the generalized Riemann hypothesis, upper bounds for shifted moments of cubic and quartic Dirichlet L-functions, with applications to bounds for moments of the corresponding character sums. The abstract is legible and states this claim clearly. However, the full text supplied for review is severely encoding-corrupted: after the abstract, the body consists largely of replacement characters and mojibake. No theorem statement, proof step, constant, error term, or even section title can be read reliably. Thus the technical content of the paper cannot be verified from the provided artifact.

Significance. If the claimed results are correct, they would constitute a meaningful advance: Soundararajan-type shifted moment bounds for Dirichlet L-functions are known in the rational case, but extending them to cubic and quartic metaplectic twists requires delicate control of the corresponding functional equations, gamma factors, and root numbers. The application to moments of cubic and quartic character sums is also of independent interest. The paper explicitly labels its results as conditional on GRH, which is an external hypothesis, so there is no evident circularity. The main obstacle to assessing significance is that none of the technical argument is legible in the reviewed manuscript.

major comments (3)

[Full text (after abstract)] The body of the manuscript is almost entirely unreadable due to encoding corruption. From the first line after the abstract through the references, the text consists of replacement characters and garbled symbols (e.g., the display beginning 'X �� ...' and the theorem-like block on page 2). No theorem statement, proof, or numerical constant can be verified. This is a load-bearing reviewability failure: the central claim is simply unsupported in the artifact provided.
[Imported metaplectic toolkit (unreadable)] The proof presumably relies on the analytic continuation, functional equations, gamma factors, and epsilon factors for cubic and quartic Dirichlet L-functions, imported from earlier work including the authors' own. For quartic characters this is especially delicate because quartic reciprocity affects the root number and the dual approximate functional equation. Since the relevant display equations and lemmas are unreadable, it is impossible to audit whether the shifted approximate functional equation is correctly formulated and whether the final character-sum moment bounds follow without an additional unstated assumption. This is a major gap in the reviewed artifact, though it may be due to text corruption rather than a mathematical error.
[Application to character-sum moments] The abstract announces bounds for moments of cubic and quartic Dirichlet character sums as an application. The required passage from shifted L-function moments back to character sums typically needs a summation over a family, a reciprocity transformation, and an error-term analysis. None of these steps are legible in the supplied text, so the claimed application cannot be checked. A clean version must state the exact moment, the range of characters, and the dependence on the shift parameters.

minor comments (2)

[Title and abstract] The title and abstract are legible. The abstract would benefit from stating the precise form of the upper bounds (power of the conductor, dependence on the shifts) and the exact GRH assumption (individual or averaged), but these are presentation issues that cannot be resolved until the body is readable.
[References] The reference list is not readable. It is important that the dependencies on prior work, especially the metaplectic functional equation and the authors' own lemmas, be explicitly cited with precise theorem numbers in the revised version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified: the results are conditional on an external hypothesis (GRH) and no fitted input or by-construction equivalence is present in the legible derivation chain.

full rationale

The paper's announced content is a set of conditional upper bounds for shifted moments of cubic and quartic Dirichlet L-functions under the generalized Riemann hypothesis, together with an application to moments of the associated character sums. Nothing in the abstract or in the legible portions of the text indicates that any parameter is fitted to the target quantity, that a function is defined in terms of the object it is meant to predict, or that a previously proven lemma is being invoked in place of the central argument. The use of GRH is an external unproved assumption, not an output of the derivation, so conditional results built on it are not circular by construction. The passage from L-function moment bounds to character-sum moment bounds is a nontrivial analytic step, not merely a renaming of the same expression. Because no specific equation or cited result can be exhibited that reduces a claimed output to an input, there is no identified circular step. Reliance on prior analytic-number-theoretic machinery, including possible prior work by the authors, is normal and does not itself constitute circularity absent evidence that the cited machinery contains the theorem being proved.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The abstract names GRH as the single unproved input; the analytic toolkit for the metaplectic L-functions is cited, not derived; no free parameters of the fitted-data kind exist in a pure-math context; no new entities are introduced.

assumptions (3)

domain assumption The generalized Riemann hypothesis for cubic and quartic Dirichlet L-functions.
Stated in the abstract as the hypothesis under which all upper bounds are established; no unconditional proof of the results is claimed, so every theorem inherits this as an external unproved input.
domain assumption Analytic continuation, functional equation, and approximate functional equation for the cubic and quartic Dirichlet L-functions, as available in cited prior work.
The shifted-moment argument requires these analytic tools in a shifted form; for the metaplectic families these results are non-trivial (Patterson-type theory) and are imported from the literature rather than re-derived, presumably including the authors' own prior papers.
standard math The standard Soundararajan reduction: under GRH, log L(1/2+it) is controlled by a short Dirichlet polynomial from which moments are extracted.
This is the established framework for GRH-conditional moment upper bounds; the paper's contribution is applying it to the cubic and quartic families.

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Cite this review

Pith. "Pith review of Shifted moments of cubic and quartic Dirichlet $L$-functions." pith.science (2026). https://pith.science/paper/XXLIDEWF

@misc{pith2026250814534,
  author       = {Pith},
  title        = {Pith review of: Shifted moments of cubic and quartic Dirichlet $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XXLIDEWF}},
  note         = {Machine review of arXiv:2508.14534}
}

abstract

We establish upper bounds for shifted moments of cubic and quartic Dirichlet $L$-functions under the generalized Riemann hypothesis. As an application, we prove bounds for moments of cubic and quartic Dirichlet character sums.

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