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Negative discrete second moments of Dirichlet $L$-functions

T0 review · 2 major / 2 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read Assuming GRH and simple zeros, discrete second moments of Dirichlet L-functions at their zeros are bounded below by a positive proportion of the conjectured asymptotics, uniformly in the conductor.

desk verdict This paper gives conditional uniform-in-q lower bounds on two discrete second moments of L-functions that recover a β/(1+β) proportion of the expected size and match known fixed-q results. read the letter →

arxiv 2606.25094 v1 pith:F5OKTT56 submitted 2026-06-23 math.NT

Andrew Pearce-Crump This is my paper · ORCID
classification math.NT MSC 11M06
keywords DirichletL-functionsdiscretemomentsGeneralizedRiemannHypothesissimplezerosconductoruniformitynegative
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 proves lower bounds for two discrete moments summed over the non-trivial zeros of L(s, χ) for primitive Dirichlet characters χ. These are the sum of the reciprocal squared of the derivative at the zero and a related sum involving L at twice the zero. The bounds hold assuming the generalized Riemann hypothesis and that the zeros are simple, and they capture the proportion β/(1+β) of the full conjectured size, with β = log T / log(qT). This proportion is 1/2 when q is fixed or small compared to T, and decreases as q grows with T.

What carries the argument

The discrete sums over the ordinates γ of the zeros, with lower bounds obtained by controlling contributions from each simple zero under GRH.

What would settle it

A computation for a specific character χ mod q and large T showing the sum is smaller than the stated lower bound times the conjectured main term, or discovery of a multiple zero violating the assumption.

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Extended reading notes

Core claim

Assuming the Generalised Riemann Hypothesis for L(s,χ) and that the non-trivial zeros ρ=½+iγ of L(s,χ) are simple, the discrete moments ∑_{0<γ≤T} |L'(ρ,χ)|^{-2} and ∑_{0<γ≤T} |L(2ρ,χ²)/L'(ρ,χ)|² are at least a positive constant times β/(1+β) times their conjectured leading asymptotics, where β=log T/log qT, uniformly in the conductor q.

Load-bearing premise

The non-trivial zeros of each L(s,χ) are all simple.

Editorial extensions

If this is right

  • When log q is o(log T), the bounds recover half the conjectured asymptotic, as in fixed q cases.
  • When q = T^A, the captured proportion is 1/(2+A).
  • The results extend theorems of Milinovich and Ng and of Sinha to the Dirichlet setting uniformly in q.
  • The paper conjectures the true leading order asymptotics for these moments and their averages over characters mod q.

Reading between the lines

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

  • If the simple zero assumption holds, these bounds suggest that multiple zeros, if any, are rare enough not to affect the average size.
  • Similar techniques might apply to other families of L-functions beyond Dirichlet characters.
  • The uniformity in conductor could allow applications to moments in short intervals or other arithmetic statistics.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. Assuming GRH for L(s,χ) and simplicity of all non-trivial zeros ρ=1/2+iγ, the paper establishes lower bounds for the discrete moments ∑_{0<γ≤T} |L'(ρ,χ)|^{-2} and ∑_{0<γ≤T} |L(2ρ,χ²)/L'(ρ,χ)|² that are uniform in the conductor q. These bounds recover the proportion β/(1+β) of the conjectured main term (with β=log T/log(qT)), recovering the fixed-q results of Milinovich-Ng and Sinha when β→1 and degrading to 1/(2+A) when q=T^A. The paper also states a conjecture for the true leading asymptotics and their character averages.

Significance. Conditional on standard hypotheses, the result supplies the first uniform-in-q lower bounds for these discrete moments and quantifies the degradation with growing conductor in a controlled way. It directly extends prior fixed-q work while remaining within the scope of GRH+simplicity, and the explicit proportion β/(1+β) is a clean and falsifiable feature of the argument.

major comments (2)
  1. [Abstract and §2] The lower-bound argument relies on the simplicity hypothesis to exclude multiple zeros from the discrete sums; the manuscript should state explicitly (with a reference to the relevant lemma) whether the same proportion is recovered when a zero of multiplicity m>1 is present, or whether the contribution is simply omitted.
  2. [Theorem 1.1] The transition from the GRH+simplicity assumptions to the explicit factor β/(1+β) appears in the main theorem; the paper should isolate the step where the log(qT) denominator arises (likely from the zero-density or truncated Euler-product estimates) so that the dependence on the conductor is transparent.
minor comments (2)
  1. [§3] The second moment involves L(2ρ,χ²); a brief reminder of the functional equation or the relation between χ and χ² at the beginning of §3 would help readers track the character.
  2. [Introduction] The conjecture for the full asymptotic (including the constant factor) is stated in the abstract and introduction; moving the heuristic derivation or reference to the corresponding random-matrix model into an appendix would separate conjecture from proved result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §2] The lower-bound argument relies on the simplicity hypothesis to exclude multiple zeros from the discrete sums; the manuscript should state explicitly (with a reference to the relevant lemma) whether the same proportion is recovered when a zero of multiplicity m>1 is present, or whether the contribution is simply omitted.

    Authors: The manuscript assumes throughout that all non-trivial zeros are simple. Under this hypothesis, zeros of multiplicity m>1 are excluded by definition and their contribution is omitted from the discrete sums (as L'(ρ,χ)=0 renders the terms undefined). Simplicity is invoked to justify the form of the sums in the key estimate of §2. We will add an explicit clarifying remark in §2 referencing the relevant lemma and noting that the stated proportion β/(1+β) holds under the simplicity assumption while multiple zeros are simply omitted. revision: yes

  2. Referee: [Theorem 1.1] The transition from the GRH+simplicity assumptions to the explicit factor β/(1+β) appears in the main theorem; the paper should isolate the step where the log(qT) denominator arises (likely from the zero-density or truncated Euler-product estimates) so that the dependence on the conductor is transparent.

    Authors: We agree that the origin of the factor β = log T / log(qT) should be isolated for clarity. This factor enters in the proof of Theorem 1.1 when the GRH zero-density bound is combined with the truncation length of the Euler product in the approximate functional equation, producing the ratio that yields the proportion β/(1+β). We will revise the proof to add a dedicated remark or paragraph that isolates this step and makes the conductor dependence explicit. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; derivation conditional on external hypotheses

full rationale

The paper states its lower bounds explicitly under the external assumptions of GRH for L(s,χ) and simplicity of all non-trivial zeros. These are not derived within the paper but taken as given; the claimed proportion β/(1+β) of the conjectured main term follows from direct estimation under those hypotheses, recovering prior fixed-q results as a special case when β→1. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The result is self-contained against the stated benchmarks and does not reduce the target bounds to a re-expression of its own inputs.

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

The proof rests on two domain assumptions that are not derived in the paper.

assumptions (2)
  • domain assumption Generalised Riemann Hypothesis for L(s,χ)
    Used to place all non-trivial zeros on the critical line Re(s)=1/2 so that the discrete sums are well-defined at those points.
  • domain assumption Non-trivial zeros ρ of L(s,χ) are simple
    Invoked to ensure that the derivative L'(ρ,χ) is non-zero and that each zero contributes independently to the summed moments.

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Cite this review

Pith. "Pith review of Negative discrete second moments of Dirichlet $L$-functions." pith.science (2026). https://pith.science/paper/F5OKTT56

@misc{pith2026260625094,
  author       = {Pith},
  title        = {Pith review of: Negative discrete second moments of Dirichlet $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5OKTT56}},
  note         = {Machine review of arXiv:2606.25094}
}
abstract

Let $\chi$ be a primitive Dirichlet character modulo $q>1$. Assuming the Generalised Riemann Hypothesis for $L(s,\chi)$ and that the non-trivial zeros $\rho=\tfrac12+i\gamma$ of $L(s,\chi)$ are simple, we prove lower bounds for the discrete moments $\sum_{0<\gamma\le T}|L'(\rho,\chi)|^{-2}$ and $\sum_{0<\gamma\le T}|L(2\rho,\chi^2)/L'(\rho,\chi)|^2$, uniformly in the conductor. The bounds capture the proportion $\beta/(1+\beta)$ of the conjectured asymptotics, where $\beta=\log T/\log qT$: this is one half whenever $\log q=o(\log T)$, recovering for fixed $q$ the Dirichlet analogues of theorems of Milinovich and Ng and of Sinha, and degrades to $1/(2+A)$ when $q=T^{A}$. We conjecture the true leading order asymptotics and their analogues when we average over the family of primitive characters modulo $q$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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