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arxiv: 1908.10346 · v3 · submitted 2019-08-27 · 🧮 math.NT

The fourth moment of Dirichlet L-functions along a coset and the Weyl bound

Ian Petrow , Matthew P. Young This is my paper

Pith reviewed 2026-05-24 16:04 UTC · model grok-4.3

classification 🧮 math.NT
keywords Dirichlet L-functionsfourth momentsubconvexityWeyl boundcharacter cosetsconductorLindelof hypothesis
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The pith

A fourth moment bound along cosets of characters yields the Weyl subconvex bound for Dirichlet L-functions of every conductor.

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

The paper proves a Lindelof-on-average upper bound for the fourth moment of Dirichlet L-functions of fixed conductor q, averaged over characters lying in a coset of the subgroup modulo d, but only under the arithmetic condition that q* divides d. This restricted average is then used to finish an earlier argument and obtain a subconvex bound of Weyl strength for each individual L-function, with no remaining conditions on the size of q. A sympathetic reader would care because subconvex bounds control the size of central values and have direct implications for the distribution of primes and zeros. The argument proceeds by establishing the moment bound in the special case where the coset condition holds and then invoking prior techniques to pass from the average to the individual bound.

Core claim

We prove a Lindelof-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q* divides d. As a consequence we establish a Weyl-strength subconvex bound for all Dirichlet L-functions with no restrictions on the conductor.

What carries the argument

The fourth moment of L-functions along a coset of characters modulo d, under the divisibility condition q* divides d, which supplies the average input needed for amplification.

If this is right

  • The Weyl subconvex bound holds for L(1/2 + it, chi) uniformly in the conductor q.
  • All previous conductor restrictions on the Weyl bound for Dirichlet L-functions are removed.
  • The same average input can be fed into amplification to reach the individual bound at any height t.

Where Pith is reading between the lines

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

  • If the coset condition q* divides d can be removed, the fourth-moment result would apply to a larger set of averages.
  • The method might extend to produce similar average bounds for other moments or for L-functions in different families.
  • The resulting subconvexity could be inserted into existing zero-density estimates to improve error terms in prime-number theorems.

Load-bearing premise

The fourth-moment bound holds only when q* divides d, and the passage from this average to the individual Weyl bound relies on amplification techniques whose applicability without further restrictions is taken from prior work.

What would settle it

An explicit numerical check, for a small prime q where q* does not divide d, showing that the fourth moment over the coset exceeds the Lindelof average size, or an explicit Dirichlet L-function whose central value exceeds the Weyl bound.

read the original abstract

We prove a Lindel\"of-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that $q^2|(q^*)^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.

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

1 major / 0 minor

Summary. The paper proves a Lindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q^* divides d (with q^* the least positive integer such that q^2 divides (q^*)^3). As a consequence, the authors claim to complete their prior work and obtain a Weyl-strength subconvex bound for every Dirichlet L-function with no restrictions on the conductor.

Significance. If the deduction from the restricted coset moment to the unrestricted individual bound holds, the result would be a substantial advance: it would remove all conductor restrictions from the Weyl bound for Dirichlet L-functions, a longstanding goal with implications for many applications in analytic number theory. The coset-average approach itself appears technically novel.

major comments (1)
  1. [Abstract / consequence section] Abstract and the section deriving the consequence: the fourth-moment bound is established only under the condition q^* | d, yet the central claim is that this yields the Weyl bound for arbitrary q with no restrictions. The manuscript must explicitly verify that the specific form of the coset (modulo d with the divisibility constraint) is compatible with the amplification or related techniques from the authors' earlier work without introducing fresh conditions on q; this compatibility is load-bearing for the unrestricted conclusion but is presented as immediate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this point about the deduction from the coset moment to the unrestricted Weyl bound. We address the concern directly below.

read point-by-point responses

  1. Referee: [Abstract / consequence section] Abstract and the section deriving the consequence: the fourth-moment bound is established only under the condition q^* | d, yet the central claim is that this yields the Weyl bound for arbitrary q with no restrictions. The manuscript must explicitly verify that the specific form of the coset (modulo d with the divisibility constraint) is compatible with the amplification or related techniques from the authors' earlier work without introducing fresh conditions on q; this compatibility is load-bearing for the unrestricted conclusion but is presented as immediate.

    Authors: We agree that the compatibility of the coset (under the condition q^* | d) with the amplification from our prior work should be verified explicitly rather than left implicit. In the earlier paper the amplification applies to any coset of characters modulo d and requires no further conditions on q once such a d is fixed. One may always choose d to be a multiple of q^* (for instance d = q^*), which satisfies the divisibility hypothesis without restricting the conductor q in any way. The resulting coset is admissible for the amplification method, and no new constraints on q arise. We will add a short paragraph in the consequence section spelling out this choice of d and confirming that the prior amplification applies verbatim. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior work completes the subconvexity claim; new fourth-moment estimate under q^*|d supplies independent content

full rationale

The abstract presents a new Lindelöf-on-average fourth-moment bound that holds precisely when q^* divides d. The passage to an unrestricted Weyl bound is described as finishing the authors' previous work via amplification or related techniques. This constitutes one minor self-citation that is not load-bearing for the novel moment estimate itself. No equation or derivation in the provided text reduces a claimed prediction to a fitted input or prior result by construction, and the central moment bound retains independent mathematical content against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard properties of L-functions rather than new postulates; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math Dirichlet L-functions admit analytic continuation and satisfy a functional equation
    Fundamental property used implicitly for all moment calculations of L-functions.
  • standard math Approximate functional equations express central values via Dirichlet polynomials
    Standard tool invoked for bounding moments along cosets.

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