Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

Berend Ringeling; David Hokken; Mahya Mehrabdollahei

arxiv: 2603.20820 · v2 · submitted 2026-03-21 · 🧮 math.NT

Relating Mahler measures and Dirichlet L-values: new evidence for Chinburg's conjectures

David Hokken , Mahya Mehrabdollahei , Berend Ringeling This is my paper

Pith reviewed 2026-05-15 07:08 UTC · model grok-4.3

classification 🧮 math.NT

keywords Mahler measureDirichlet L-functionChinburg conjecturequadratic characternumerical verificationbivariate polynomial

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The pith

Chinburg's conjectures on Mahler measures and L-values gain 26 new numerical examples.

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

The paper aims to provide more evidence for Chinburg's conjectures by finding explicit polynomials and rational functions whose Mahler measures are rational multiples of L' at -1 for odd quadratic characters. A sympathetic reader would care because these relations connect geometric invariants to deep arithmetic properties of L-functions, potentially shedding light on special values. Using an explicit search method inspired by Boyd and Rodriguez-Villegas, the authors identify 8 new cases for the strong conjecture and 18 for the weak one, effectively doubling the known examples. They further prove the weak conjecture holds when polynomials are allowed to have cyclotomic coefficients.

Core claim

For many more conductors f, there exist integral bivariate polynomials P such that the Mahler measure m(P) equals a rational multiple of L'(χ_{-f}, -1), and similarly for rational functions in the weak form. The authors' method locates these examples systematically and proves the weak version under relaxed coefficient conditions.

What carries the argument

The Mahler measure m(P) of a bivariate polynomial or rational function P, which is shown to equal a multiple of the L-derivative through explicit construction and numerical verification.

Load-bearing premise

That the numerically observed equalities between Mahler measures and L-values are exact rather than approximate due to limited precision or undetected relations.

What would settle it

A high-precision computation showing that for one of the new examples, m(P) differs from the predicted multiple of L' by more than floating-point error, or finding a conductor where no matching P exists after exhaustive search.

read the original abstract

Let $\chi_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $\chi_{-f}$ there exist an integral bivariate rational function $P$ (and, in the strong form, an integral polynomial) such that $\mathrm{m}(P)$ is a rational multiple of $L'(\chi_{-f},-1)$. The strong form of the conjecture was previously known to hold for $18$ values of $f$. We double the number of numerical examples, giving $8$ new instances of the strong and $18$ new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.

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.

Desk Editor's Note private letter to a colleague

This paper doubles the numerical examples for Chinburg's conjectures and proves the weak form under cyclotomic coefficients.

read the letter

The paper doubles the numerical examples for Chinburg's conjectures and proves the weak form under cyclotomic coefficients. They use the Boyd-Rodriguez-Villegas search to produce explicit polynomials and rational functions where the Mahler measure matches a rational multiple of L'(χ_{-f}, -1) for odd quadratic characters. This yields 8 new strong instances and 18 new weak ones, bringing the totals to 26 strong and more weak cases overall. Their method also recovers almost all the earlier known examples, which shows the construction is reasonably systematic rather than scattered finds. The cyclotomic-coefficient proof is a direct argument that relies on standard facts about cyclotomic polynomials and L-function properties, without extra parameters or fitting. That part is a real derivation and stands independently. The numerical matches are computed to high precision from the definitions, and the abstract gives no indication of post-hoc selection. The main limitation is that the numerical equalities lack an effective error bound or algebraic identity to confirm they are exact rather than very close. Without something like a lower bound from linear forms in logarithms, a coincidental agreement at the reported precision cannot be ruled out with full rigor, though the precision level makes it unlikely. This is incremental but useful work for people tracking Mahler measures and special L-values. The expanded list of examples plus the one solid proof give concrete data and a new case that specialists can build on. The citations follow the relevant literature without gaps. I would send it to peer review. The proof justifies referee time, and the new examples add to the evidence base even if the numerical part needs checking on precision.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to double the known numerical support for Chinburg's conjectures by exhibiting 8 new instances of the strong form and 18 new instances of the weak form, in which the Mahler measure m(P) of an explicitly constructed integral polynomial (or rational function) equals a rational multiple of L'(χ_{-f}, -1) for odd quadratic characters χ_{-f}. These examples are obtained via a search procedure modeled on the work of Boyd and Rodriguez-Villegas. In addition, the paper supplies a rigorous proof of the weak conjecture when the polynomials are permitted to have cyclotomic coefficients.

Significance. If the reported numerical matches are exact, the work substantially enlarges the body of evidence for both forms of the conjecture and demonstrates that the Boyd–Rodriguez-Villegas search recovers nearly all previously known cases. The separate algebraic proof for the cyclotomic-coefficient variant constitutes a genuine advance that is independent of the numerical search and may serve as a template for further progress.

major comments (2)

[§4] §4 (New numerical examples): The 26 asserted new instances rest on high-precision floating-point agreement between m(P) and r L'(χ_{-f},-1). No effective lower bound on |m(P) - r L'(χ,-1)| (for example via Baker-type theorems on linear forms in logarithms) is supplied to convert the numerical match into a rigorous proof of equality. This gap is load-bearing for the claim that these are genuine instances of the conjectures rather than close approximations.
[§5] §5 (Cyclotomic-coefficient proof): The derivation correctly invokes standard properties of cyclotomic polynomials and L-functions, but the manuscript should explicitly record that the argument does not rely on any numerical verification step, thereby confirming its independence from the search procedure used for the new examples.

minor comments (2)

[Introduction] Introduction: The citation to Boyd and Rodriguez-Villegas should include the precise title, journal, and year rather than a generic reference.
[Table 1] Table of examples: For each new f, list the explicit polynomial P together with the rational multiplier r so that readers can reproduce the Mahler-measure computation independently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. We address the major comments point by point below.

read point-by-point responses

Referee: §4 (New numerical examples): The 26 asserted new instances rest on high-precision floating-point agreement between m(P) and r L'(χ_{-f},-1). No effective lower bound on |m(P) - r L'(χ,-1)| (for example via Baker-type theorems on linear forms in logarithms) is supplied to convert the numerical match into a rigorous proof of equality. This gap is load-bearing for the claim that these are genuine instances of the conjectures rather than close approximations.

Authors: We agree that the 26 new instances are supported by high-precision numerical agreement rather than a rigorous proof of equality, since no effective lower bounds (e.g., via Baker-type theorems) are supplied. This approach follows the standard practice in the literature on Chinburg's conjectures, as in the foundational work of Boyd and Rodriguez-Villegas, where such numerical evidence is accepted as support for new instances. We have revised the manuscript to clarify explicitly that these are numerical instances providing evidence for the conjectures, not proven equalities. Deriving rigorous proofs for these specific cases via lower bounds on linear forms in logarithms is a substantial open problem and lies beyond the scope of the present work. revision: partial
Referee: §5 (Cyclotomic-coefficient proof): The derivation correctly invokes standard properties of cyclotomic polynomials and L-functions, but the manuscript should explicitly record that the argument does not rely on any numerical verification step, thereby confirming its independence from the search procedure used for the new examples.

Authors: We thank the referee for this helpful suggestion. We have added an explicit statement in the revised Section 5 noting that the proof relies solely on algebraic properties of cyclotomic polynomials and L-functions and is entirely independent of the numerical search procedure and any floating-point verification. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical matches and proof use independent definitions and external prior work

full rationale

The paper computes Mahler measures m(P) and L'(χ_{-f}, -1) directly from their standard definitions for candidate polynomials found via an explicit search method based on external prior work (Boyd and Rodriguez-Villegas). The new instances are reported as numerical agreements, and the proof of the weak conjecture under cyclotomic coefficients invokes only standard properties of cyclotomic polynomials and Dirichlet L-functions. No equation or claim reduces the asserted equality to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The central results remain independent of the target conjecture itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definitions of the Mahler measure and the Dirichlet L-function together with the functional equation and the explicit construction technique of Boyd and Rodriguez-Villegas; no new free parameters or invented entities are introduced.

axioms (2)

standard math Mahler measure of a polynomial equals the integral of log |P| over the unit torus
Invoked in the definition of m(P) used throughout the numerical checks.
standard math Dirichlet L-function L(χ,s) has a meromorphic continuation and functional equation
Required to define L'(χ,-1) and to relate it to the Mahler measure via the conjecture.

pith-pipeline@v0.9.0 · 5479 in / 1422 out tokens · 29141 ms · 2026-05-15T07:08:26.517283+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

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M.-J. Bertin and M. Mehrabdollahei,An exact family of bivariate polynomials and Variants of Chinburg’s Conjectures, arXiv:2407.20634, 22 pp., 2025. To appear in Annales mathématiques du Québec

work page arXiv 2025

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work page 1998

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D. W. Boyd and F. Rodriguez-Villegas,Mahler’s measure and the dilogarithm. I, Canad. J. Math.54(2002), no. 3, 468–492

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work page internal anchor Pith review Pith/arXiv arXiv 2005

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C. Deninger,Deligne periods of mixed motives,𝐾-theory and the entropy of certainZ𝑛-actions, J. Amer. Math. Soc.10(1997), no. 2, 259–281

work page 1997

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Guilloux and J

A. Guilloux and J. Marché,Volume function and Mahler measure of exact polynomials, Compos. Math.157 (2021), no. 4, 809–834

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Liu and H

H. Liu and H. Qin,Mahler measure of families of polynomials defining genus 2 and 3 curves, Exp. Math.32 (2023), no. 2, 321–336

work page 2023

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Liu, and H

H. Liu, and H. Qin,Code for Mahler measure of polynomials defining genus 2 and 3 curves,https: //github.com/liuhangsnnu/mahler-measure-of-genus-2-and-3-curves/, accessed March 19th, 2026

work page 2026

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Maillot,Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, Mém

V. Maillot,Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, Mém. Soc. Math. Fr. (N.S.)80(2000)

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G. A. Ray,Relations between Mahler’s measure and values of𝐿-series, Canad. J. Math.39(1987), no. 3, 694–732

work page 1987

[14] [14]

C. J. Smyth,A Kronecker-type theorem for complex polynomials in several variables, Canad. Math. Bull.24 (1981), no. 4, 447–452

work page 1981

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The PARI Group (Univ. Bordeaux),PARI/GP version2.17.2, 2025. Available athttp://pari.math. u-bordeaux.fr/

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Vandervelde,The Mahler measure of parametrizable polynomials, J

S. Vandervelde,The Mahler measure of parametrizable polynomials, J. Number Theory128(2008), no. 8, 2231–2250

work page 2008

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Zudilin,Apéry limits and Mahler measures, arXiv:2109.12972, 6 pp., 2021

W. Zudilin,Apéry limits and Mahler measures, arXiv:2109.12972, 6 pp., 2021. DH: Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland Email address:d.p.t.hokken@uu.nl MM: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Email address:mehrabdollahei@mpim-bonn.mpg.de BR: Département de mathématiques...

work page arXiv 2021