Strong Chinburg's conjecture on Mahler measures and Dirichlet L-values

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Let m(P)m(P) denote the Mahler measure of a non-zero polynomial PP, let L′( ⋅ ,−n)L'(\,\cdot\,,-n) denote the derivative of a Dirichlet LL-function, and let χ−f≔(−f.)\chi_{-f}\coloneqq\left(\frac{-f}{.}\right) be an odd quadratic character. Strong Chinburg's conjecture. For every odd quadratic character χ−f\chi_{-f} and every n∈Nn\in\mathbb{N}, there exists a non-zero polynomial Qf∈Z[z1,…,zn+1]Q_f\in\mathbb{Z}[z_1,\ldots,z_{n+1}] and a rational number rfr_f such that

rfm(Qf)=L′(χ−f,−n).r_fm(Q_f)=L'(\chi_{-f},-n).

This conjecture concerns the proposed relationship between Mahler measures and special values of Dirichlet LL-functions. The source presents it as one of Chinburg's conjectures and does not state a resolution.

References

Primary source

Marie-José Bertin and Mahya Mehrabdollahei, “An exact family of bivariate polynomials and Variants of Chinburg's Conjectures”, arXiv:2407.20634 (2025).

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