## Regulator of modular units and Mahler measures.(English)Zbl 1386.11129

Summary: We present a proof of the formula, due to Mellit and Brunault, which evaluates an integral of the regulator of two modular units to the value of the $$L$$-series of a modular form of weight 2 at $$s=2$$. Applications of the formula to computing Mahler measures are discussed.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11F11 Holomorphic modular forms of integral weight 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) 11R27 Units and factorization

### Keywords:

Mahler measure; modular unit; $$L$$-series; modular form

### Software:

PARI/GP; SageMath
Full Text:

### References:

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