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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

Software:

PARI/GP; SageMath
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References:

[1] DOI: 10.1090/S0894-0347-97-00228-2 · Zbl 0913.11027
[2] DOI: 10.1142/S1793042110003174 · Zbl 1201.11098
[3] DOI: 10.2307/1970966 · Zbl 0354.14007
[4] DOI: 10.1007/978-1-4614-6642-0_20 · Zbl 1316.11038
[5] DOI: 10.4064/aa134-3-7 · Zbl 1195.11083
[6] DOI: 10.1112/S0024609304003510 · Zbl 1064.11037
[7] DOI: 10.2140/ant.2007.1.87 · Zbl 1172.11037
[8] DOI: 10.1080/10586458.1998.10504357 · Zbl 0932.11069
[9] Bertin, J. Reine Angew. Math. 569 pp 175– (2004)
[10] DOI: 10.1112/S0010437X11007342 · Zbl 1260.11062
[11] DOI: 10.1353/ajm.2005.0037 · Zbl 1127.11041
[12] Rodriguez Villegas, Topics in Number Theory pp 17– (1997)
[13] Mellit, Explicit Methods in Number Theory pp 1990– (2011)
[14] DOI: 10.1007/978-3-642-65663-7
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