Twelfth moment of Dirichlet L-functions to prime power moduli

Daniel White; Djordje Mili\'cevi\'c

Twelfth moment of Dirichlet L-functions to prime power moduli

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Pith's one-line read This paper proves that the twelfth power moment of Dirichlet L-functions summed over all characters modulo an odd prime power q=p^n is bounded by p^A q^{2+ε}, the conjecturally sharp size.

desk verdict This paper proves the first sharp q-aspect twelfth moment for Dirichlet L-functions to prime power moduli, and the p-adic stationary-phase arguments at its core deserve careful refereeing. read the letter →

arxiv 1908.04833 v2 pith:WVPH2ZM7 submitted 2019-08-13 math.NT

Djordje Milićević , Daniel White This is my paper · ORCID

classification math.NT MSC 11M0611L0711L4026E30

keywords DirichletL-functionstwelfthmomentprimepowermodulip-adicstationaryphaseexponentialsumslargevaluesshortsecondmoments

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 that the twelfth power moment of Dirichlet L-functions at the central point, summed over all characters modulo an odd prime power q=p^n, is bounded by p^A $q^{{2+ε}}$ for some absolute constant A>0. This is the conjecturally sharp size for this family, up to the usual q^ε loss, and it is the q-aspect analogue of the classical twelfth moment bound for the Riemann zeta function. The proof passes through an equivalent large-value statement: for any V, the number of primitive characters with |L(1/2,χ)|>V is at most p^A $q^{{2+ε}}$ $V^{{-12}}$. The reader should care because a moment bound of this strength is the sharpest uniform control available on how often central values of these L-functions are large, and because the proof introduces a p-adic large-sieve mechanism that may work for other families.

What carries the argument

The load-bearing mechanism is the p-adic method of stationary phase, packaged as Lemmas 3 and 4: a complete exponential sum with a p-adically analytic phase reduces to the contribution of points where the derivative vanishes, and each such contribution is evaluated explicitly through quadratic Gauss sums. The bridge into this framework is Lemma 2, which writes any primitive character modulo p^n as χ(1+kp)=e(A log_p(1+kp)/p^n) for a p-adic unit A. This turns character sums into the p-adically analytic exponential sums Kχ(m;Q1), whose square-root cancellation is shown in Lemma 6 and whose orthogonality over pairs of characters is shown in Proposition 2, a bound of size $Q^{{1/2}}$ for complete sums of products with an additive twist. These two estimates are what make the large-sieve step work.

What would settle it

Compute the complete exponential sums Kχ(m;p^n) explicitly for a small odd prime such as p=3 and conductors n=3,4, for all primitive characters and all m, and compare the results with the evaluation in Lemma 6; any mismatch there would trace back to the p-adic logarithmic representation of characters and would directly refute the main estimate. Equivalently, one can test the identity χ(1+kp)=e(A log_p(1+kp)/p^n) numerically for all k modulo p^n for every primitive character of a small conductor.

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Extended reading notes

Core claim

The central claim is Theorem 1: for every odd prime p and every q=p^n, ∑_{χ mod q} |L(1/2,χ)|^{12} ≪_ε p^A $q^{{2+ε}}$. Since summation by parts makes the moment and large-value estimates equivalent, the paper proves Theorem 2, which bounds the number of primitive characters with central value exceeding V by p^A $q^{{2+ε}}$ $V^{{-12}}$. The new work is in the short second moment: after an approximate functional equation and a trace-function decomposition, the argument reduces the bound to square-root cancellation in complete sums of products of the exponential sums Kχ(m;Q1), established by p-adic stationary phase. A Cauchy-Schwarz and large-sieve step then converts this orthogonality into the displayed moment bound. If correct, the theorem confirms the sharp predicted moment in the prime power aspect for every odd prime.

Load-bearing premise

The proof's load-bearing premise is the quoted representation of primitive characters modulo p^n as exponentials of p-adic logarithms with a p-adic unit multiplier (Lemma 2); if that representation fails at some prime or precision, the complete exponential sums and the square-root cancellation estimates no longer follow, and with them the moment bound.

Editorial extensions

If this is right

The moment bound is sharp up to q^ε, so the twelfth moment of these L-functions is as small as the conjectural optimal bound predicts for every odd prime power modulus.
The equivalent large-value bound shows that very few characters can have very large central values: the count above V is at most p^A q^{2+ε} V^{-12}.
A single character with |L(1/2,χ)| > q^{1/6+ε/2} would force the total twelfth moment above the theorem's bound, so the theorem recovers the standard subconvexity bound L(1/2,χ) ≪ q^{1/6+ε} for primitive characters modulo prime powers.
Together with the smooth square-free modulus case, the result covers both families where the sharp twelfth moment is now known, pointing to the remaining difficulty in moduli with several prime factors.
Because the moment and large-value statements are equivalent via integration by parts, the theorem also constrains the distribution of the large set R(V;q) at every scale of V.

Reading between the lines

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

The orthogonality mechanism in Proposition 2 is the seed of a genuinely p-adic large sieve; it should extend to products of more than two Kχ factors and therefore to moments of order higher than 12, with new stationary-phase bookkeeping at degenerate critical points.
A natural next target is a hybrid or multi-prime-power modulus with finitely many well-separated prime power factors, where the same machinery could in principle unify the prime-power and square-free results.
A concrete check of the method would be to compute the matrices (Kχ(m;Q1)) for small p and n and test approximate row orthogonality numerically, providing independent evidence for the large-sieve step before any theoretical extension is attempted.
Because the paper's overview isolates the weighted trace sum as the only obstacle, a further application of stationary phase to more complicated phases should yield the sharper large-value statement in the intermediate range without changing the moment bound itself.

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Editorial analysis

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

Desk Editor's Note

read the letter

Punchline: this is a genuine q-aspect twelfth moment for prime-power moduli, a family where none existed before. It is not a trivial rewrite of Nunes or Heath-Brown; the p-adic stationary-phase machinery is doing real work, and the paper deserves a serious referee.

The new content is concentrated in Lemma 6 and Proposition 2. Lemma 6 gives a complete evaluation of the relevant character-exponential sum K_χ and splits it into two analytic pieces; Proposition 2 gives square-root cancellation in complete sums of products with additive twists. I checked the main dependency chain: Theorem 2 -> summation by parts -> Theorem 1, with Proposition 3 as the aggregate engine. The stress-test note's verification of the derivative identity and the stationary-phase congruence in Proposition 2 is consistent with the text. I did not find a circular step. Lemma 2 is quoted from the author's earlier paper, but that is a published lemma with a parameter-free proof, and all uses here have tilde{q} > p^2, so the edge cases never enter.

Soft spots are real but not disabling. The proof of Proposition 2 is compressed; the verification of the hypotheses of Lemma 3 is summarized rather than fully displayed, and a referee should ask for the expanded version. The same goes for a couple of "easily" steps in section 6. The introduction's "large sieve" language is informal; the actual argument is Cauchy-Schwarz, which is fine, but readers should not expect a formal sieve. The bound has an unspecified p^A and the paper does not try to optimize it; that is consistent with the claim. The paper also explicitly notes that the sharper range result analogous to Nunes's Theorem 1.2 is not obtained; that is an honest limitation, not a defect.

The citation pattern looks appropriate: Heath-Brown, Nunes, Milićević's sub-Weyl paper, Postnikov, Blomer–Milićević. No data fitting, no invented entities, no hidden assumption of the target bound.

Who is this for: analytic number theorists working on moments and subconvexity, especially anyone interested in p-adic analogues of archimedean methods. I would bring it to a reading group. It is a within-subfield advance that fills a clear gap. Recommendation: send it to peer review; it deserves careful referee time. My own confidence in the main theorem is moderate rather than high only because the p-adic arguments are dense and not machine-checked, but I saw no red flag.

Referee Report

0 major / 5 minor

Summary. The paper proves a q-aspect analogue of Heath-Brown's twelfth moment estimate for Dirichlet L-functions, with the modulus restricted to odd prime powers. The main theorem states that for every odd prime p and every q=p^n, the sum over characters modulo q of |L(1/2,chi)|^{12} is bounded by O_epsilon(p^A q^{2+epsilon}). The proof follows the Heath-Brown/Nunes architecture: it establishes a short second moment S_2(chi), evaluates the relevant complete character sums K_chi by p-adic stationary phase (Lemma 6), proves square-root cancellation for sums of products of these sums with additive twists (Proposition 2), and then aggregates the short second moments over sets of characters (Proposition 3). This yields the large-value estimate Theorem 2, from which Theorem 1 follows by summation by parts. The paper is largely self-contained apart from Lemma 2, which is quoted from Milićević's earlier work and expresses primitive characters on principal units through the p-adic logarithm.

Significance. If the proof is correct, the paper establishes the Lindelöf-consistent upper bound for the twelfth moment in the depth aspect, complementing Nunes' result for smooth square-free moduli. The main innovations are the explicit evaluation of the complete character sum K_chi via p-adic stationary phase and the square-root cancellation estimate for sums of products in Proposition 2; these are substantial technical contributions that should be of independent use. The paper also organizes the Heath-Brown large-sieve argument into a clean general proposition, and the polynomial dependence on p is explicit. The result is exactly the expected q-aspect analogue of the Heath-Brown twelfth moment bound, with the same large-value consequence.

minor comments (5)

[4.2, Proposition 2] The proof of the key estimate (30) compresses the verification that the phase sigma satisfies the hypotheses of Lemma 3 into the phrases "Expanding the difference of roots..." and "a moment's reflection." I recommend expanding this verification, including explicit p-adic valuations of sigma^{(k)}/k! and a clear derivation that the stationary-phase congruence has only O(1) solutions modulo rt*(Q) via the four congruences and Hensel's lemma. The steps appear correct, but this is the central arithmetic input and the present level of detail makes independent verification unnecessarily hard.
[1 (Theorem 1) and 6 (proof of Theorem 2)] Theorem 1 is stated for all characters chi modulo q, while the proof in Section 6 counts only primitive characters. Add an explicit reduction showing that imprimitive characters contribute no more than a p^A q^epsilon factor; for example, use the local factor at p and sum the primitive contribution over conductors d|q. Without this comment the reader must reconstruct the reduction.
[2.3, Lemma 4] The displayed evaluation uses the notation p^{n/2} when n may be odd, which is a real power rather than an integer. Clarify how this is interpreted together with the factors Delta_f, or rewrite the exponent in terms of floor(n/2) and ceil(n/2). This would remove an avoidable source of confusion in a foundational lemma.
[4.2, statement (2) of Proposition 2] Add a remark that when chi = chi' one has delta_q(chi,chi') = q/p and hence Q = 1 for every proper divisor tilde q of q. The "vanishes unless |v|_p = 1" assertion therefore implicitly concerns distinct characters; without this remark the diagonal case appears to contradict the statement.
[3.1, equations (23)-(24)] The passage from the double sums over h' and j' to the single sum over m = h'j' with coefficients A(m;p^eta) should state explicitly that each m arises from divisor pairs in the stated ranges, and that the divisor bound gives A(m;p^eta) << m^epsilon. This bookkeeping is currently too terse.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified.

full rationale

The paper proves the prime-power q-aspect twelfth moment bound by a genuine chain of estimates. Theorem 1 is explicitly derived from Theorem 2 by summation by parts, which is a standard equivalence and not a disguised input. Theorem 2 is then obtained from the large-values set R(V;q), the fourth-moment bound, the Weyl-type bound, and the aggregate short-second-moment estimate Proposition 3. The central arithmetic content is Proposition 2 and Lemma 6, both proved in the present paper using the p-adic stationary phase Lemmas 3 and 4, which are fully proved in Section 2. The only imported structural result is Lemma 2, quoted from the first author's earlier paper [11, Lemma 13] and attributed ultimately to Postnikov; it is a standard explicit representation of a primitive character modulo p^n and does not contain the twelfth-moment bound or any fitted parameter. All applications in the paper have tilde q > p^2, so the relevant cases are covered. The self-citations to [11] provide sub-Weyl bounds and the p-adic analytic setup, but the present paper's new exponential-sum estimates are independent of the target bound. No parameter is fitted to a subset of data and then renamed as a prediction, and no uniqueness theorem is invoked to force the argument. Consequently, the derivation is self-contained relative to its stated external lemmas, and there is no circularity in the sense defined by the analysis rules.

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

No free parameters appear: the proof chooses q1≈V^2 and q2=q1^3, but these are structural proof choices, not fitted constants. No new entities are postulated. The theorem depends on standard analytic number theory inputs and on the first author's previously published p-adic stationary phase results; those are independent published results, not assumptions containing the target bound.

assumptions (6)

standard math Approximate functional equation (Lemma 1, from Iwaniec-Kowalski Theorem 5.3) represents |L(1/2,χ)|^2 as a short smooth sum of length about q^{1/2+ε}.
Invoked in Section 3.1 to begin the short second moment computation; it is a standard theorem, not proved in the paper.
standard math Postnikov-Milićević character representation (Lemma 2, quoted from [11, Lemma 13]): for a primitive character χ modulo p^n there is a p-adic unit A with χ(1+kp)=e(A log_p(1+kp)/p^n).
Quoted in Section 2.2; it is the bridge turning character values into p-adically analytic phases in Lemmas 3, 4, 6, and Proposition 2.
domain assumption Fourth moment bound: ∑_{χ mod q} |L(1/2,χ)|^4 ≪ q^{1+ε} (Heath-Brown [6], Soundararajan [16]).
Used in Section 6 to handle the small-V range and to bound |R(V;q)| outside the critical interval.
domain assumption Weyl subconvexity bound for prime power moduli: L(1/2,χ) ≪ q^{1/6+ε} (Postnikov [15], Milićević [11]).
Used in Section 6 to assert that R2(V;χ) is empty for V>q^{1/6+ε}, so only the middle range needs Proposition 3.
standard math Completion and truncation tools: Poisson summation and Lemma 5, which bound incomplete sums via Fourier transforms.
Used in Sections 3.2 and 5 to pass from sums of Kχ products to complete exponential sums.
standard math Hensel's lemma and p-adic analytic continuation properties of square roots (equations (9), (10), (11)).
Used in Lemma 6 and Proposition 2 to locate stationary points of phases and to justify the relevant polynomial congruences modulo p.

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Pith. "Pith review of Twelfth moment of Dirichlet L-functions to prime power moduli." pith.science (2026). https://pith.science/paper/WVPH2ZM7

@misc{pith2026190804833,
  author       = {Pith},
  title        = {Pith review of: Twelfth moment of Dirichlet L-functions to prime power moduli},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WVPH2ZM7}},
  note         = {Machine review of arXiv:1908.04833}
}

read the original abstract

We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic method of stationary phase for sums of products and complement Nunes' bound for smooth square-free moduli.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 8 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

p-adic analytic twists and strong subconvexity

Valentin Blomer and Djordje Mili´ cevi´ c. “p-adic analytic twists and strong subconvexity”. In: Ann. Sci. ´Ec. Norm. Sup´ er. (4)48.3 (2015), pp. 561–605. doi: 10.24033/asens.2252

work page doi:10.24033/asens.2252 2015

[2] [2]

Decoupling, exponential sums and the Riemann zeta function

J. Bourgain. “Decoupling, exponential sums and the Riemann zeta function”. In: J. Amer. Math. Soc. 30.1 (2017), pp. 205–224. doi: 10.1090/jams/860

doi:10.1090/jams/860 2017

[3] [3]

The cubic moment of central values of automorphic L- functions

J. B. Conrey and H. Iwaniec. “The cubic moment of central values of automorphic L- functions”. In: Ann. of Math. (2) 151.3 (2000), pp. 1175–1216. doi: 10.2307/121132

doi:10.2307/121132 2000

[4] [4]

S. W. Graham and G. Kolesnik. van der Corput’s method of exponential sums . Vol. 126. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1991, pp. vi+120. doi: 10.1017/CBO9780511661976

doi:10.1017/cbo9780511661976 1991

[5] [5]

Hybrid bounds for Dirichlet L-functions

D. R. Heath-Brown. “Hybrid bounds for Dirichlet L-functions”. In: Invent. Math. 47.2 (1978), pp. 149–170. doi: 10.1007/BF01578069

doi:10.1007/bf01578069 1978

[6] [6]

The fourth power mean of Dirichlet’s L-functions

D. R. Heath-Brown. “The fourth power mean of Dirichlet’s L-functions”. In: Analysis 1.1 (1981), pp. 25–32. doi: 10.1524/anly.1981.1.1.25

doi:10.1524/anly.1981.1.1.25 1981

[7] [7]

The twelfth power moment of the Riemann-function

D. R. Heath-Brown. “The twelfth power moment of the Riemann-function”. In: Quart. J. Math. Oxford Ser. (2) 29.116 (1978), pp. 443–462. doi: 10.1093/qmath/29.4.443

doi:10.1093/qmath/29.4.443 1978

[8] [8]

Mean-Value Theorems in the Theory of the Riemann Zeta-Function

A. E. Ingham. “Mean-Value Theorems in the Theory of the Riemann Zeta-Function”. In: Proceedings of the London Mathematical Society s2-27.1 (1928), pp. 273–300. doi: 10.1112/ plms/s2-27.1.273

work page 1928