Generalized Mahler Measures and Multiple Sine Functions

Abstract

We introduce a generalized Mahler measure. It has relations to multiple sine functions and Dirichlet L-functions. In particular, we are able to express special values of Dirichlet L-functions by sum of logarithmic generalized Mahler measures.

Mahler measures and computations with regulators

“…The first case was known to Gon and Oyanagi [GO04]. To our knowledge, the other two cases are new results.…”

“…Lalín [Lal03] m 1 + x + 2y + (1 − x)z = 7 2π 2 ζ(3) + log 2 2 Lalín [Lal03] m (1 − y)(1 + x) + (1 − x)z = 28 5π 2 ζ(3) Condon [Con03] m (1 + x 1 )(1 + x) + (1 − x 1 )(1 + y)z = 24 π 3 L(χ −4 , 4) Lalín [Lal03,Lal06] Generalized Mahler measures were introduced by Gon and Oyanagi [GO04] who studied their basic properties, computed some examples, and related them to multiple sine functions and special values of Dirichlet L-functions. Given f 1 , .…”

A Generalization of a Theorem of Boyd and Lawton

“…. , x n ], (not necessarily distinct) nonzero polynomials, the generalized Mahler measure is defined in [GO04] by…”

“…These objects have been related to special values of the Riemann zeta function and L-functions ( [GO04,La08] for generalized Mahler measure, [KLO08, Sa10, BS, BBSW] for multiple Mahler measure), but the nature of this relationship is not as well understood than in the classical case.…”

Multiple q-Mahler measures and zeta functions

“…This paper belongs to a series of papers making attempts to extend Mahler measures as in [GO,K1,K2,KO,O1,O2] mainly related to multiple sine functions and their generalizations. Our multiple q-deformation examples are mainly suggested by the usual q-deformation examples treated in [K1,K2,KO,O2].…”