Sharp Upper Bounds for Moments of Dedekind Zeta Functions

Benjamin Durkan; Kamalakshya Mahatab; Nilmoni Karak

REVIEW 3 major objections 4 minor 24 references

Under GRH, Dedekind zeta moments of finite Galois extensions provably match the conjectured sharp order, with no epsilon loss.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 10:00 UTC pith:BK7BF3TF

load-bearing objection Sharp unshifted moment bound is probably right; the shifted-moment theorem has a real uniformity gap at large shifts. the 3 major comments →

arxiv 2606.27516 v3 pith:BK7BF3TF submitted 2026-06-25 math.NT

Sharp Upper Bounds for Moments of Dedekind Zeta Functions

Benjamin Durkan , Nilmoni Karak , Kamalakshya Mahatab This is my paper

classification math.NT MSC 11M5011R4211S40

keywords Dedekind zeta functionmomentsGeneralized Riemann Hypothesisshifted momentsprime splittingeffective prime-splitting density theoremlarge deviationsideal-counting function in short intervals

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that, assuming the Generalized Riemann Hypothesis, the 2k-th moment of the Dedekind zeta function of any finite Galois extension K/Q is at most a constant times T (log T)^{[K:Q]k^2}. This is the exponent predicted by conjectures, and it removes the extra (log T)^ε that appeared in the best previous conditional upper bound. The proof also handles shifted moments, tracking how values at points separated by b decorrelate through an explicit function g(|b|). Because of this, the paper obtains matching large-deviation estimates and short-interval bounds for the ideal-counting function r_K(n). A sympathetic reader would care because the upper side of the moment conjecture for Dedekind zeta functions is now closed under GRH for all Galois extensions, going beyond the solvable cases where it was previously known.

Core claim

The central claim is Theorem 1.1: for a finite Galois extension K/Q, under GRH for ζ_K, and for any positive exponents a_j and shifts b_j with |b_j|≤T/2, the shifted moment ∫_0^T ∏_{j=1}^{2k} |ζ_K(1/2+i(t+b_j))|^{a_j} dt is bounded by T (log T)^{[K:Q](a_1^2+...+a_{2k}^2)/4} times a product over pairs of the correlation function g(|b_i-b_j|)^{[K:Q] a_i a_j/2}. Setting all a_j=1 and b_j=0 gives the unshifted bound ∫_0^T |ζ_K(1/2+it)|^{2k} dt ≪ T (log T)^{[K:Q]k^2}, the conjecturally sharp order. The theorem thus closes the upper side of the moment problem for Dedekind zeta functions of finite Galois extensions, conditionally on GRH.

What carries the argument

The machinery is a dyadic 'good set/bad set' decomposition of the t-integral, in which the moment integral is expressed through truncated Taylor expansions of a Dirichlet polynomial, together with the effective prime-splitting density theorem. The key input is an identity (Lemma 3.8): for a Galois extension, the sum of cos(α log p)/p over primes that split completely equals (1/[K:Q]) times the same sum over all primes, up to O(1). Proposition 3.9 feeds this into the variance of the Dirichlet polynomial, compressing the naive exponent [K:Q]^2 Σa_j^2/4 down to [K:Q]Σa_j^2/4. The correlation function g(x), defined piecewise (log T, 1/x, or log log x depending on scale), records how strongly val

Load-bearing premise

The whole result sits on the effective prime-splitting density estimate (3.4)–(3.5), which under GRH says the count of completely split rational primes up to x is (1/[K:Q])π(x) plus an error O(x e^{-c√log x}); a weaker error term would remove the saving of one factor [K:Q] and the bound would revert to the old [K:Q]^2k^2 shape.

What would settle it

For a fixed Galois extension such as the splitting field of x^3−2, numerically compare Σ_{p≤x, p splits completely} cos(α log p)/p with (1/3)Σ_{p≤x} cos(α log p)/p for α=0 and α=1 up to x=10^9; if the difference grows beyond O(1) as x grows, the key Lemma 3.8 fails and the exponent compression in Proposition 3.9 collapses.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

For every finite Galois K/Q, under GRH the 2k-th moment of ζ_K has the expected size T(log T)^{[K:Q]k^2}; no (log T)^ε remains.
The shifted-moment theorem gives a precise correlation law: values of ζ_K at points separated by ≤1/log T are nearly fully correlated, and as the separation grows the correlation decays like 1/|b| (or log log |b|) exactly as the g-function prescribes.
The large-deviation estimate Corollary 1.3 says that, at the scale of log log T, the probability that log|ζ_K(1/2+it)| is V standard deviations above its mean is at most e^{-V^2/2}, matching Gaussian heuristics.
For the ideal-counting function r_K(n), the variance in short intervals of length δ is O(δ T (log T)^{[K:Q]}), and pointwise, away from a small exceptional set, the count is the main term plus O(√δ (log t)^{[K:Q]/2}ψ(t)).
Since the result requires only that K/Q be Galois, the previous obstruction for non-solvable Galois groups is removed; the upper side of the moment conjecture for Dedekind zeta functions is conditionally closed for all finite Galois extensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same split-prime compression may carry over to moments of Artin L-functions attached to irreducible characters of Gal(K/Q), since the factorization of ζ_K into such L-functions suggests each character inherits a similar prime-splitting-weighted variance; the paper does not pursue this.
For non-Galois fields, the uniform factor 1/[K:Q] is unavailable, so the true moment exponent may be strictly larger than [K:Q]k^2; a numerical study of a non-Galois cubic field's fourth moment would test whether the Galois obstruction is real.
The large-deviation bound plausibly extends to the full Selberg central-limit-theorem regime V ≍ √(log log T) with the correct constant, since the variance of log|ζ_K| appears to be ([K:Q]/2) log log T; proving the CLT would be a natural next step.
Because the shifted moment bound holds for shifts up to T/2, it might be combined with the explicit formula to constrain the pair correlation of low-lying zeros of ζ_K, though the authors do not discuss this.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Sharp unshifted moment bound is probably right; the shifted-moment theorem has a real uniformity gap at large shifts.

read the letter

Two things to know. First, the unshifted result (Corollary 1.2) is a genuine advance: assuming GRH it gets the conjectured [K:Q]k^2 exponent for all finite Galois extensions, removing the epsilon and the solvability restriction. The Chebotarev split-prime reduction in Proposition 3.9 is the right idea, and it checks out for the unshifted case. Second, the shifted-moment theorem as written does not go through for large shifts. The stress-test is correct. Lemma 3.8 states O_alpha(1), but the proof's error term is (1+alpha) times an integral; that is O(alpha), not O(1) uniformly. In Theorem 1.1, alpha = |b_i-b_j| can be as large as T/2, so the error is O(T), which cannot be absorbed into the g factors. The shifted-moment bound for large shifts is therefore unsupported. The missing contribution from nontrivial Artin factors is real, around log log log alpha, not O(1). I suspect the theorem is salvageable — one should use the character expansion and bound the Artin factors by the same g function — but the present proof does not do that.

What is good: the paper is clearly structured, the reduction to completely split primes is conceptually clean, and the Harper-style moment accounting in Lemma 4.1 is handled carefully with the corrected exponent. The applications to short intervals and large deviations rely on Corollary 1.2, so those appear safe. The references and attribution to prior work are accurate; there is no fitted-parameter or circularity problem.

Soft spots are modest aside from the shifted gap: the arXiv abstract claims non-Galois coverage and lists three authors, while the body restricts to Galois extensions and lists two. The second half of Lemma 4.3 (the (4.4) part) has its proof omitted. There are also small slips like the spurious factor 20 in (4.7) and the equality at (5.34) is sloppy. These are cosmetic and fixable.

This is a paper for analytic number theorists working on moments of L-functions. The unshifted moment bound is important and probably correct; the shifted-moment theorem needs revision. A serious referee should see it, but the referee report must require either a uniform version of Lemma 3.8 or an explicit bound on the Artin-character contribution for large shifts. If the authors fix that, the paper is strong.

Referee Report

3 major / 4 minor

Summary. The paper assumes GRH for ζ_K and uses Harper's moment method together with the Chebotarev density theorem to prove sharp upper bounds for shifted moments of Dedekind zeta functions of finite Galois extensions K/Q (Theorem 1.1). The advertised exponent is [K:Q](a_1^2+...+a_{2k}^2)/4, with pairwise correlation factors g(|b_i-b_j|)^{[K:Q]a_i a_j/2}. From this the authors derive Corollary 1.2: ∫_0^T |ζ_K(1/2+it)|^{2k} dt ≪_{K,k} T(log T)^{[K:Q]k^2}, removing the (log T)^ε loss from Milinovich–Turnage-Butterbaugh and extending Hagen from solvable to all finite Galois extensions. Applications to large deviations and short-interval sums of r_K(n) are also given. The paper is careful and expository, but the shifted-moment theorem contains a uniformity gap in the Chebotarev step.

Significance. If the proof is repaired, the result is significant: it would settle, under GRH, the conjecturally sharp upper-bound side of the moment conjecture for Dedekind zeta functions of finite Galois extensions, with a clean conceptual reason (complete splitting lowers the exponent from [K:Q]^2 k^2 to [K:Q]k^2). The paper gives no fitted parameters and uses only external inputs (GRH, effective Chebotarev, Chandee, Harper–Heap), which is a methodological strength. The key innovation—using the Chebotarev density theorem to replace the split-prime variance by a rational-prime variance—is attractive and, for the unshifted moment, essentially sound. The main theorem, however, is not established as written for shifts of size comparable to T, because the Chebotarev approximation is not uniform in the shift parameter. The unshifted corollary and the applications that use it are largely unaffected, but the shifted-moment claim is load-bearing and needs a substantial, though plausibly local, repair.

major comments (3)

[§3, Lemma 3.8 / Prop. 3.9, and (5.18)] The O(1) in Proposition 3.9 is not uniform in α. In the partial summation of Lemma 3.8, the remainder contributes E(x) ≪ e^{-c√log x} + (1+α)∫_2^x e^{-c√log t}/t dt, which grows with α; the notation O_α(1) hides this. Since Theorem 1.1 allows |b_i-b_j| as large as T/2, using Proposition 3.9 with α=|b_i-b_j| is not justified. Equivalently, the difference between the split-prime sum and (1/[K:Q])Σ_p cos(α log p)/p is (1/[K:Q])Re log(ζ_K/ζ)(1+1/log x+iα)+O(1), which under GRH can be of size log log log α. In the range 10≤α≤e^T this has the same size as log g(α), so the passage to line (5.18) is unsupported. Corollary 1.2 (α=0) is unaffected, but Theorem 1.1's shifted-moment claim is not proved as written.
[Corollary 1.2 proof] The proof says 'putting a_j=1 and b_j=0 for all j', which only gives an integer number 2k of factors. The corollary is stated for all real k>0. This is easily repaired by applying Theorem 1.1 to a single factor with exponent 2k (or by a separate convexity argument), but as written the implication is incomplete.
[Metadata abstract vs. body] The abstract in the paper's metadata claims results for 'arbitrary number fields' and says the results apply to both Galois and non-Galois extensions. The body proves results only for finite Galois extensions and explicitly states in §1 that the non-Galois case 'appears to require new ideas and remains open'. This contradiction must be resolved before publication.

minor comments (4)

[Definition of g, line after Theorem 1.1] The piecewise definition of g is inconsistent at x=10: the second branch gives 1/10 while the third gives log log 10. The intended value (presumably the larger one, since an upper bound is needed) should be specified.
[Theorem 1.1 / Corollary 1.2 notation] The phrase 'where 2k is a fixed positive integer' in Theorem 1.1 conflicts with the later use of k as an arbitrary real exponent in Corollary 1.2. This contributes to the ambiguity in the proof of Corollary 1.2.
[Lemma 3.8] The statement 'for every fixed α≥0' should be replaced by an explicit uniformity statement. As written, the O_α(1) notation invites the reader to believe the constant is independent of α, which is not the case for α up to T/2.
[Throughout] Minor typographical issues include 'extenstion' in Lemma 3.7 and the repeated use of 'Harper' for the arXiv preprint [11] without a journal reference; please update the references if available.

Circularity Check

0 steps flagged

No significant circularity found; external GRH/Chebotarev inputs drive the derivation.

full rationale

No load-bearing step reduces to its own input. Theorem 1.1 is derived from GRH via Chandee's log-bound, Harper's dyadic large-deviation framework, and an effective Chebotarev approximation for split primes. Each input is external and is not equivalent to the moment bound: GRH for ζ_K constrains zeros, Chandee's lemma is a general L-function bound, Harper's lemmas are generic Dirichlet polynomial estimates, and Lemma 3.3 is a known RH prime-sum estimate. Proposition 3.9 uses Lemma 3.8 to replace split-prime sums by rational-prime sums with O(1) error, which produces the improved [K:Q] factor; this is a theorem under GRH, not a parametrization of the desired conclusion. The correlation function g is stated in the theorem and then bounded through Lemma 3.3; it is not fitted after the fact to force the exponent. Corollary 1.2 is a specialization of Theorem 1.1, not a fitted input. There are no self-citations and no imported uniqueness assertion. The paper explicitly notes the non-Galois case is open, so it is not claiming a conclusion it has baked into its assumptions. The possible failure of uniformity in α in Lemma 3.8 raised by the skeptic is a technical correctness concern — the O(1) could conceal nontrivial Artin factors — not a circularity; even if true, it would invalidate the proof for large shifts rather than show the theorem was assumed.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 0 invented entities

The derivation rests on GRH (assumed), effective Chebotarev, and three cited analytic lemmas; all inputs are external and none contains the target bound. The truncation constants are hand-chosen proof parameters that do not enter the final exponent, so no number is fitted to the target result. No new entities are introduced; the Artin-factorization discussion in §1 is motivation only and is not used in the proof.

free parameters (1)

Harper-method truncation constants (20, 1000, 1/10, 3/4, and the '+50' in a)
Hand-chosen proof parameters in the dyadic decomposition α_i = 20^{i-1}/(log log T)^2, the truncation exponent θ_i ≈ √(a' α_i^{-3/4}), the cutoff α_J ≈ e^{-1000a'}, and M ≈ 1/(10[K:Q]α_{j+1}). The final bounds are insensitive to their exact values; no data is fitted and the target exponent is not among them.

axioms (6)

domain assumption Generalized Riemann Hypothesis for ζ_K(s)
Headline assumption stated in Theorem 1.1 and Corollaries 1.2–1.4. Also supplies RH for ζ(s) (Lemma 3.3) and the effective Chebotarev quality used in Lemma 3.8.
standard math Effective Chebotarev density theorem (identity class of Gal(K/Q))
Eq. (3.3)–(3.4): π_split(x) = [K:Q]^{-1} Li(x) + O(x^{1/2} log x), converted to R(x) ≪ x e^{-c√log x}. Classical theorem (Lagarias–Odlyzko); carries the whole exponent-saving from [K:Q]^2 to [K:Q].
standard math Chandee's Theorem 2.1 (Lemma 3.4 in the paper)
Quoted from [4]; gives the smoothed GRH-conditional bound for log|ζ_K(1/2+it)| that Proposition 3.6 is built on. Requires ζ_K to satisfy standard L-function axioms; the paper invokes |Λ_K(n)| ≤ [K:Q]Λ(n).
standard math Harper's cosine-product lemma (Lemma 3.1 = [11, Prop 2.5]) and Heap's truncated-exponential identity (Lemma 3.2 = [13, Eq. 37])
Quoted without proof; they convert Dirichlet-polynomial moments into the exponential of a prime sum in §5.1–5.3.
standard math Prime-correlation estimate Σ_{p≤x} cos(α log p)/p (Lemma 3.3, from [24, Lem. 2])
Determines the piecewise definition of the correlation function g and hence the cross-terms of Theorem 1.1.
standard math K/Q finite Galois; |Λ_K(n)| ≤ [K:Q]Λ(n)
Galois uniformity makes Λ_K(p) supported exactly on completely split primes with value [K:Q] log p (Proposition 3.9); the degree bound on Λ_K is used in Lemmas 3.5–3.7. The first fact is the reason non-Galois fields are left open.

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read the original abstract

Assuming the Generalised Riemann Hypothesis, we establish conjecturally sharp upper bounds for shifted moments of products of Dedekind zeta functions of arbitrary number fields. This improves results of Milinovich and Turnage-Butterbaugh and extends a recent result of Hagen. As applications, we obtain mean-square bounds for short-interval sums of the coefficients of Dedekind zeta functions and upper bounds for the large deviations of Dedekind zeta functions. Our results apply to both Galois and non-Galois extensions.

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 3 linked inside Pith

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Akbary and B

A. Akbary and B. Fodden, Lower bounds for power moments ofL-functions, Acta Arith.151(2012), no. 1, 11–38

2012

[2] [2]

Arguin and E

L.-P. Arguin and E. C. Bailey, Large deviation estimates of Selberg’s Central Limit Theorem and applications, Int. Math. Res. Not. IMRN23(2023), 20574–20612

2023

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Arguin, E

LP. Arguin, E. Bailey, A. Roberts, Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function. arXiv preprint arXiv:2604.25579 (2026)

Pith/arXiv arXiv 2026

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Chandee, Explicit upper bounds forL-functions on the critical line, Proc

V. Chandee, Explicit upper bounds forL-functions on the critical line, Proc. Amer. Math. Soc.137(2009), no. 12, 4049–4063

2009

[5] [5]

Chandee, On the correlation of shifted values of the Riemann zeta function, Q

V. Chandee, On the correlation of shifted values of the Riemann zeta function, Q. J. Math.62(2011), no. 3, 545–572

2011

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N. G. Chebotar¨ ev, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitution- sklasse geh¨ oren, Math. Ann.95(1926), no. 1, 191–228

1926

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J. B. Conrey et al., Autocorrelation of random matrix polynomials, Comm. Math. Phys.237(2003), no. 3, 365–395

2003

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J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments ofL-functions, Proc. London Math. Soc. (3)91(2005), no. 1, 33–104

2005

[9] [9]

M. J. Curran, Correlations of the Riemann zeta function, Mathematika70(2024), no. 4, Paper No. e12268, 14 pp

2024

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M. V. Hagen, Sharp conditional moment bounds for products ofL-functions, Q. J. Math.76(2025), no. 2, 683–713

2025

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A. J. Harper, Sharp conditional bounds for moments of the Riemann zeta function, arXiv preprint arXiv.1305.4618 (2013)

Pith/arXiv arXiv 2013

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W. P. Heap, Moments of the Dedekind zeta function and other non-primitiveL-functions, Math. Proc. Cambridge Philos. Soc.170(2021), no. 1, 191–219

2021

[13] [13]

W. P. Heap, On the splitting conjecture in the hybrid model for the Riemann zeta function, Forum Math.35(2023), no. 2, 329–362

2023

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Landau, Einf¨ uhrung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea, New York, 1949

E. Landau, Einf¨ uhrung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, Chelsea, New York, 1949

1949

[15] [15]

Motohashi, A note on the mean value of the Dedekind zeta-function of the quadratic field, Math

Y. Motohashi, A note on the mean value of the Dedekind zeta-function of the quadratic field, Math. Ann.188(1970), 123–127

1970

[16] [16]

M. B. Milinovich and C. L. R. Turnage-Butterbaugh, Moments of products of automorphicL-functions, J. Number Theory 139(2014), 175–204

2014

[17] [17]

Munsch, Shifted moments ofL-functions and moments of theta functions, Mathematika63(2017), no

M. Munsch, Shifted moments ofL-functions and moments of theta functions, Mathematika63(2017), no. 1, 196–212

2017

[18] [18]

N. C. Ng, Q. Shen and P.-J. Wong, Shifted moments of the Riemann zeta function, Canad. J. Math.76(2024), no. 5, 1556–1586

2024

[19] [19]

Lang,Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970

S. Lang,Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970

1970

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M. Radziwi l l and K. Soundararajan, Continuous lower bounds for moments of zeta andL-functions, Mathematika59(2013), no. 1, 119–128

2013

[21] [21]

Radziwill, Large deviations in Selberg’s central limit theorem

M. Radziwill, Large deviations in Selberg’s central limit theorem. arXiv preprint arXiv:1108.5092 (2011)

Pith/arXiv arXiv 2011

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Sono, Continuous lower bounds for the moments of Dedekind zeta-functions, J

K. Sono, Continuous lower bounds for the moments of Dedekind zeta-functions, J. Number Theory188(2018), 335–356

2018

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Soundararajan, Moments of the Riemann zeta function, Ann

K. Soundararajan, Moments of the Riemann zeta function, Ann. of Math. (2)170(2009), no. 2, 981–993

2009

[24] [24]

Szab´ o, High moments of theta functions and character sums, Mathematika70(2024), no

B. Szab´ o, High moments of theta functions and character sums, Mathematika70(2024), no. 2, Paper No. e12242, 37 pp. Nilmoni Karak, Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India. Email address:nilmonikarak@gmail.com, nilmonimath@kgpian.iitkgp.ac.in Kamalakshya Mahatab, Department of Mathematics, Indian Instit...

2024