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On the Mahler measure of $(1+x)(1+y)+z$

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arxiv 2305.02992 v1 pith:SKAYRTIQ submitted 2023-05-04 math.NT math.AGmath.KT

François Brunault This is my paper · ORCID
classification math.NTmath.AGmath.KT
keywords mahlermeasureauthorboydcertainclassescomputationconductor
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abstract

We prove a conjecture of Boyd and Rodriguez Villegas relating the Mahler measure of the polynomial $(1+x)(1+y)+z$ and the value at $s=3$ of the $L$-function of an elliptic curve of conductor $15$. The proof makes use of the computation by Zudilin and the author of the regulator of certain $K_4$ classes on modular curves.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The asymptotic Mahler measure of Gaussian periods

    math.NT 2025-07 accept novelty 8.0 of 10

    For fixed k, the asymptotic Mahler measure of Gaussian periods of conductor kn+1 is n times the Mahler measure of the cyclovariety x0+F_k(x)=0, which is asymptotically (1/2)log k.

  2. The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces

    math.NT 2024-12 conditional novelty 7.0 of 10

    Under Goncharov's and Beilinson's conjectures, the Mahler measure of (x+1)(y+1)(z+1)+t is shown to be a rational linear combination of L'(f7,-1) and zeta'(-2), with f7 the weight-3 level-7 modular form.

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