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arxiv: 2606.27516 · v1 · pith:BK7BF3TF · submitted 2026-06-25 · math.NT

Sharp Bounds for Moments of the Dedekind Zeta Function

Nilmoni Karak , Kamalakshya Mahatab This is my paper

Reviewed by Pith2026-06-29 00:57 UTCgrok-4.3pith:BK7BF3TFopen to challenge →

classification math.NT
keywords Dedekind zeta functionshifted momentsGeneralized Riemann HypothesisGalois extensionsupper boundsanalytic number theoryL-functions
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0 comments X

The pith

Assuming the generalized Riemann hypothesis, shifted moments of the Dedekind zeta function for finite Galois extensions satisfy upper bounds of conjectural order.

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

The paper proves that, under the generalized Riemann hypothesis, the shifted moments of the Dedekind zeta function attached to any finite Galois extension obey upper bounds that match the size predicted by standard conjectures. A sympathetic reader cares because these moments quantify the typical size of the zeta function near the critical line and therefore control how large partial sums over prime ideals can become. The bounds improve earlier estimates of Milinovich and Turnage-Butterbaugh and extend a recent result of Hagen to the Galois setting. If the claim holds, analytic number theorists obtain a uniform description of moment growth that applies equally to the rational field and to all its finite Galois extensions.

Core claim

Assuming the Generalized Riemann Hypothesis, we establish upper bounds of conjectural order of magnitude for shifted moments of the Dedekind zeta function associated with a finite Galois extension.

What carries the argument

The shifted moment, an integral that records the average magnitude of the Dedekind zeta function after a small vertical shift inside the critical strip.

If this is right

  • The upper bounds achieve the exact order of magnitude conjectured for these moments.
  • The result applies uniformly to every finite Galois extension of the rationals.
  • The same order is obtained for all admissible shift parameters inside the critical strip.
  • The estimates improve the range of moments previously treated for the Dedekind zeta function.

Where Pith is reading between the lines

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

  • The same conditional bounds should extend immediately to Artin L-functions attached to the same Galois extensions.
  • Under GRH the moments would then align with the predictions coming from random-matrix models for families of L-functions over number fields.
  • These bounds supply a concrete test that could be checked numerically for small Galois extensions once sufficiently many zeros are computed under the GRH assumption.

Load-bearing premise

The generalized Riemann hypothesis holds for the Dedekind zeta functions of the finite Galois extensions under consideration.

What would settle it

An explicit finite Galois extension together with a numerical or analytic verification that, even assuming GRH, at least one of its shifted moments exceeds the stated upper bound.

read the original abstract

Assuming the Generalized Riemann Hypothesis, we establish upper bounds of conjectural order of magnitude for shifted moments of the Dedekind zeta function associated with a finite Galois extension. This improves results of Milinovich and Turnage-Butterbaugh and extends a recent result of Hagen.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves, under the assumption of the Generalized Riemann Hypothesis for the relevant Dedekind zeta functions, that the shifted moments of ζ_K(s) for a finite Galois extension K/Q attain upper bounds of the conjectural order of magnitude. The result improves the bounds obtained by Milinovich and Turnage-Butterbaugh and extends recent work of Hagen.

Significance. Conditional on GRH, the paper supplies sharp upper bounds for a family of moments that had previously been known only with weaker exponents or under stronger hypotheses. This strengthens the analytic toolkit for studying value distribution of Dedekind zeta functions over number fields and supplies a concrete improvement that can be cited in future work on moment conjectures.

minor comments (2)
  1. [Theorem 1.1] The statement of the main theorem (presumably Theorem 1.1 or 1.2) should explicitly record the dependence of the implied constant on the degree [K:Q] and the discriminant; the current abstract leaves this dependence implicit.
  2. [Introduction] In the introduction, the comparison with Milinovich–Turnage-Butterbaugh is stated only qualitatively; a short table or sentence giving the precise exponent improvement would help readers assess the gain.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report lists no specific major comments under the MAJOR COMMENTS section.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result is an upper bound for shifted moments of the Dedekind zeta function, explicitly conditional on the external Generalized Riemann Hypothesis (GRH) for the relevant zeta functions. The abstract and description indicate the bounds match conjectural order of magnitude and improve prior work by other authors (Milinovich-Turnage-Butterbaugh, Hagen). No self-citations, fitted parameters renamed as predictions, self-definitional steps, or ansatzes smuggled via prior work by the same authors are present in the provided text. The derivation chain relies on GRH as an independent assumption rather than reducing to the paper's own inputs by construction. This is the standard case of a conditional analytic number theory result with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The sole explicit assumption is the Generalized Riemann Hypothesis; no free parameters, invented entities, or additional axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Generalized Riemann Hypothesis for Dedekind zeta functions of finite Galois extensions
    Invoked to obtain the moment bounds.

pith-pipeline@v0.9.1-grok · 5559 in / 1064 out tokens · 26963 ms · 2026-06-29T00:57:36.101342+00:00 · methodology

discussion (0)

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Reference graph

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