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Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus

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Pith's one-line read For prime-power moduli with exponent at least 50, the twisted fourth moment of Dirichlet $L$-functions is evaluated asymptotically with a power-saving error term.

desk verdict New result for prime-power moduli, but the far-apart off-diagonal bound rests on an unproved variation of a cited large sieve; exponent typo is minor. read the letter →

arxiv 2507.18186 v1 pith:GZUTLVMF submitted 2025-07-24 math.NT

Peng Gao , Liangyi Zhao This is my paper · ORCID
classification math.NT MSC 11M06
keywords twistedfourthmomentDirichletL-functionsprimepowermodulusshiftedmomentsapproximatefunctionalequationKloostermansumsKuznetsovtraceformulapower-savingerrorterm
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 an asymptotic formula for the twisted fourth moment of Dirichlet $L$-functions at the central point, for moduli that are powers of a single odd prime. For $q=q_0^{n_0}$ with $n_0\ge 50$, the sum over primitive even characters of four shifted $L$-values times $\chi(a)\chi(b)$ is shown to equal six explicit main terms plus an error of size $q^{\varepsilon}(1+|\alpha|)^4(1+|\beta|)^4(1+|\gamma|)^4(1+|\delta|)^4(ab)^7(q^{1-1/576}+q^{1-1/n_0})$. The main terms match the shape of the conjectured fourth-moment formula, and the error term is a genuine power saving over the trivial size. The result extends earlier prime-modulus evaluations to prime powers and supplies the kind of uniform shifted-moment asymptotics needed for mollification and upper-bound work.

What carries the argument

The central machinery is the approximate functional equation for a product of four Dirichlet $L$-functions (Lemma 2.5), which converts the moment into two double sums over $m,n$ with smooth weights. Character orthogonality (Lemma 2.2) turns the character sum into congruence conditions $ma\pm nb\equiv 0 \pmod d$. The off-diagonal sums are then split: when the summation lengths $M$ and $N$ are far apart, Voronoi summation and a large sieve inequality for Kloosterman sums (Lemma 2.12) control them; when $M$ and $N$ are close, the $\delta$-method detects the congruence, and a second Voronoi summation followed by the Kuznetsov trace formula, which relates sums of Kloosterman sums to spectral data of automorphic forms, and a spectral large sieve bounds the remainder. The final power saving comes from optimizing two parameters $\eta_0=1/576$ and $\eta_1=1/9$.

What would settle it

Check Lemma 2.12 directly for $q=q_0^{n_0}$ with $n_0$ between 50 and 57: if the claimed bound fails for some $r\mid q$, $s\mid r$, then the error term in Theorem 1.1 is not obtained. Also evaluate the final exponent inequality in Section 10 with $n_0=50$ to see whether $q^{1+1/1152+1/18-1/16+1/(4n_0)} \le q^{1-1/576}$ actually holds; if not, the stated range $n_0\ge 50$ is false as written.

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Extended reading notes

Core claim

The discovery is Theorem 1.1: under the size condition $(1+|\alpha|)^4(1+|\beta|)^4(1+|\gamma|)^4(1+|\delta|)^4(ab)^7 \ll q^{\min(1/576,1/n_0)-\varepsilon_0}$, the twisted fourth moment $S(\alpha,\beta,\gamma,\delta;a,b)$ equals the sum of the six terms $S_1,\dots,S_6$ in (1.10) plus a power-saving error. Each of the six terms is an explicit product of zeta factors, powers of $a$ and $b$, and the multiplicative coefficients $\tau_{\alpha,\beta,\gamma,\delta}$, reflecting a distinct pairing of the four shifts; their total is the natural continuation of the known prime-modulus formula and agrees with the conjectured integral-moment formula. The proof obtains this by writing the four-$L$-function product through an approximate functional equation, detecting character orthogonality by congruences, and splitting the resulting sums into diagonal, far-apart, and close-proximity regimes.

Load-bearing premise

The proof rests on a large-sieve inequality for Kloosterman sums modulo prime powers (Lemma 2.12) that is quoted as a variation of a known result rather than proved, and separately the final exponent balance appears to need $n_0 \ge 58$ rather than the stated $n_0 \ge 50$.

Editorial extensions

If this is right

  • For every odd prime $q_0$ and every exponent $n_0\ge 50$, the twisted fourth moment has an asymptotic formula with error $q^{1-1/576}+q^{1-1/n_0}$, a genuine power saving over the main term.
  • The six explicit main terms reproduce the structure of the conjectured moment formula, so the result gives a concrete check of that conjecture for prime-power families.
  • The uniform polynomial dependence on the shifts and on $ab$ makes the formula usable as an input for mollified moments and for upper bounds below the fourth moment.
  • The proof splits the off-diagonal contribution by the relative size of $m$ and $n$, combining elementary congruences, Voronoi summation, and spectral theory in a way that can serve as a template for other families with fixed prime-power conductor.

Reading between the lines

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

  • In the editor's reading, the unproved large-sieve inequality for Kloosterman sums is the main obstacle to lowering the exponent threshold; a proof for all $r\mid q$, $s\mid r$ would likely push $n_0$ well below 50.
  • The uniform polynomial dependence on $ab$ suggests the formula is ready to be used as an input for mollified fourth moments with short mollifiers; for twists of size comparable to $q$, a different treatment would be needed.
  • One testable extension is to check the six-term main shape numerically for small prime powers and small shifts, which would separate the analytic error-term mechanism from any hidden issue in the quoted large sieve.
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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

2 major / 4 minor

Summary. The paper evaluates, for a fixed prime-power modulus q = q0^{n0} with n0 >= 50, the twisted shifted fourth moment of even primitive Dirichlet L-functions, proving an asymptotic formula with a six-term main term and power-saving error O(q^{1-1/576} + q^{1-1/n0}) under the size condition (1.8). The proof combines the approximate functional equation for products of L-functions, character orthogonality, a dyadic decomposition, Voronoi summation, a Kloosterman large sieve in the far-apart case, and the delta method together with the Kuznetsov trace formula and spectral large sieve in the close-range case, following the frameworks of Blomer-Milicevic, Hough, Zacharias, and Liu. The paper is a substantial technical extension of existing fourth-moment results to fixed prime powers.

Significance. If the result is valid, it is a significant extension of the twisted fourth moment for Dirichlet L-functions from prime and factorable moduli to a fixed prime-power modulus, with an explicit main term matching the CFKRS conjecture and a power-saving error. The paper is technically demanding and carefully organized, and it makes clear which ingredients are imported from prior work. The main term and the broad structure of the argument are credible. However, the proof relies on an unproved large-sieve inequality for Kloosterman sums in a prime-power setting, and the final exponent balance in Section 10 is written incorrectly for the stated range n0 >= 50; both issues are load-bearing for Theorem 1.1 and need to be addressed before the claim can be accepted.

major comments (2)
  1. [Lemma 2.12 / Section 5, (5.8)-(5.10), Proposition 5.1] Lemma 2.12 is the critical input for the far-apart off-diagonal estimate. It is applied in (5.8)-(5.9) with r | d | q and s = (r, q1), and it leads directly to Proposition 5.1 and then to the term q^{1+eta0/2+eta1/2} q1^{-1/4} in the final error R in (10.2). The lemma is stated as a 'slight variation' of [5, Theorem 5] and no proof is given. The application requires r and s to be powers of the same prime q0 (since r | q0^{n0} and s | r), so r and s are not coprime and the modulus is not squarefree; this is precisely a range where a variation of a theorem proved in a different setting needs independent verification. If Lemma 2.12 fails in this range, the bound for S_{+,2} + S_{-,2} collapses and Theorem 1.1 is unsupported. The authors should provide a complete proof of Lemma 2.12 or a precise reference that covers prime-power levels with s | r and (r/s, 2) = 1.
  2. [Section 10, exponent balance after (10.2)] The displayed chain q^{1+eta0/2+eta1/2}/q1^{1/4} <= q^{1+1/1152+1/18-1/16+1/(4n0)} <= q^{1-1/576} is not correct as written for n0 = 50. With eta0 = 1/576 and eta1 = 1/9, the second inequality requires 1/1152 + 1/18 - 1/16 + 1/(4n0) <= -1/576, which holds only for n0 >= 58. The conclusion can be repaired, however, by choosing i0 = floor(n0/4) and using q1^{-1/4} = q^{-i0/(4n0)}; for n0 >= 50 this gives i0/(4n0) >= 67/1152, so the desired exponent q^{1-1/576} follows. The authors should either correct the displayed inequality with the sharper choice of q1 or adjust the statement of Theorem 1.1 to the range n0 >= 58.
minor comments (4)
  1. [Section 1, Section 2.9, Section 3, Section 5, Section 7] There are several typographical errors that should be corrected: 'focuse' in Section 1, 'Eisentein' in Section 2.9, 'simliar' in Section 3, 'supscript' in Section 5, and 'Tthe' at the beginning of Section 7.
  2. [Lemma 2.12] The phrase 'a slight variation of given [5, Theorem 5]' is grammatically incomplete; it should read 'a slight variation of [5, Theorem 5]'.
  3. [Section 3, (3.17)] The notation 'M,N ≪ log q' in (3.17) is shorthand for dyadic ranges with O(log q) choices; this should be stated explicitly to avoid ambiguity.
  4. [Section 9, (9.1)] The notation 'T ±;∗∗∗∗ ±,M,N' is introduced without an explicit definition of the four sign patterns; a sentence explaining that ∗∗∗∗ ranges over the four combinations appearing in (6.6) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is built on external theorems and prior work, with no load-bearing self-citation or input recycled as prediction.

full rationale

The paper's derivation is self-contained relative to the external results it invokes: the approximate functional equation (Lemma 2.5), orthogonality of characters (Lemma 2.2), Voronoi summation (Lemma 2.7), the Kuznetsov trace formula (Lemma 2.10), and spectral and Kloosterman large sieve inequalities (Lemmas 2.12, 2.13). The six main terms S_1,...,S_6 are not assumed; they emerge from residue computations and Mellin-transform evaluations in Sections 4, 8, and 9, following the treatments of Young, Zacharias, and Liu. The only self-citation, reference [9], is an application of Hough's and Liu's results and is not used as an input to the proof, so it is not load-bearing. Two non-circular correctness risks should be flagged rather than counted as circularity: (i) Lemma 2.12 is stated as an unproved 'slight variation' of [5, Theorem 5], and the application to prime-power moduli with s=(r,q1) is not checked; if that variation fails, the S_{±,2} bound in Proposition 5.1 and the power saving would collapse. (ii) The displayed inequality in Section 10, q^{1+1/1152+1/18-1/16+1/(4n0)} ≤ q^{1-1/576}, is false as written for n0=50, since the exponent is approximately 1-0.00107, which is larger than 1-1/576≈1-0.001736; however, choosing q1=q0^{floor(n0/4)} gives q1^{-1/4} ≤ q^{-67/1152} and repairs the balance. These are issues of proof coverage and arithmetic verification, not of circular reasoning. No equation in the paper reduces to its own input by construction, and no fitted parameter is relabeled as a prediction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The proof rests on standard analytic number theory lemmas such as the approximate functional equation, orthogonality, Voronoi summation, the Kuznetsov trace formula, and large sieve inequalities. These are external theorems, not fitted inputs. The only hand-chosen constants are proof parameters η0, η1, and q1. No invented entities appear.

free parameters (3)
  • η0 = 1/576
    Chosen in Section 10 to balance error terms; determines the exponent 1-1/576 in the final error.
  • η1 = 1/9
    Chosen in Section 10 to balance error terms; used with η0 to set the final power saving.
  • q1 = q0^{i0} with i0/n0 in (1/4-1/n0, 1/4]
    Chosen in Section 10 as a divisor of q near q^{1/4} to optimize the error bound; central to the final exponent check.
assumptions (7)
  • standard math Approximate functional equation for the product of four Dirichlet L-functions (Lemma 2.5).
    Quoted from Young [24, Proposition 2.4]; it expresses the fourth moment as a sum of two series and is the starting point of Section 3.
  • standard math Orthogonality relation for even primitive characters (Lemma 2.2).
    Quoted from Soundararajan [21]; it evaluates the character sum and creates the diagonal and off-diagonal structure.
  • standard math Voronoi summation formula for divisor-type sums (Lemma 2.7).
    Quoted from Liu [15, Theorem 6.11]; used to transform sums over m and n in Sections 5 and 6.
  • standard math Kuznetsov trace formula (Lemma 2.10).
    Quoted from Zacharias [25, Proposition 2.3]; used to estimate Kloosterman sums via automorphic spectra.
  • domain assumption Large sieve inequalities for Kloosterman sums and for spectral data (Lemmas 2.12, 2.13).
    Lemma 2.12 is stated as a slight variation of [5, Theorem 5] without proof; it is load-bearing for bounding S_{±,2} in Section 5.
  • standard math Subconvexity bound for the Riemann zeta function on the critical line (equation (4.11)).
    Quoted from Iwaniec and Kowalski [12]; used to bound the diagonal error term in Section 4.
  • standard math Kim-Sarnak bound on Hecke eigenvalues (equation (2.11)).
    Quoted from Kim [13]; used to bound spectral contributions in Section 7.

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Pith. "Pith review of Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus." pith.science (2026). https://pith.science/paper/GZUTLVMF

@misc{pith2026250718186,
  author       = {Pith},
  title        = {Pith review of: Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZUTLVMF}},
  note         = {Machine review of arXiv:2507.18186}
}
abstract

We evaluate the twisted four moment on the critical line of the family of Dirichlet $L$-functions to a fixed prime power modulus, obtaining an asymptotic formula with a power saving error term.

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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. On the $\beta=2$ Partition function for Dirichlet $L$-functions in the $q$-aspect

    math.NT 2026-08 conditional novelty 7.0 of 10

    Upper bounds for the q-aspect beta=2 partition function of Dirichlet L-functions and for the typical maximum, matching FHK predictions to second order.

Reference graph

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