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arXiv · 2510.21515

Mahler measure, motivic regulators and Dirichlet $L$-values

Wei He·Jungwon Lee
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Abstract

Inspired by the work of Deninger, we present a formula that relates the Mahler measure of a two-variable variant of cyclotomic polynomial to regulator of class in motivic cohomology associated to cyclotomic fields and linear combination of special values of the derivative of Dirichlet $L$-functions. The formula is derived by studying the Beilinson regulator map applied to systematically constructed elements in the motivic cohomology group. Under linear independence hypothesis on the derivative of partial Dirichlet $L$-values at $s=0$ and $-1$, we study a Galois module structure of the relevant motivic cohomology and obtain the refined identity for a single $L$-value.

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Wei He, Jungwon Lee. 2025-10-24. Mahler measure, motivic regulators and Dirichlet $L$-values. https://arxiv.org/abs/2510.21515

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