260 research outputs found
Log-sine evaluations of Mahler measures, II
We continue the analysis of higher and multiple Mahler measures using
log-sine integrals as started in "Log-sine evaluations of Mahler measures" and
"Special values of generalized log-sine integrals" by two of the authors. This
motivates a detailed study of various multiple polylogarithms and worked
examples are given. Our techniques enable the reduction of several multiple
Mahler measures, and supply an easy proof of two conjectures by Boyd.Comment: 35 page
Log-sine evaluations of Mahler measures
We provide evaluations of several recently studied higher and multiple Mahler
measures using log-sine integrals. This is complemented with an analysis of
generating functions and identities for log-sine integrals which allows the
evaluations to be expressed in terms of zeta values or more general
polylogarithmic terms. The machinery developed is then applied to evaluation of
further families of multiple Mahler measures.Comment: 25 page
A generalization of a theorem of Boyd and Lawton
The Mahler measure of a nonzero -variable polynomial is the integral
of on the unit -torus. A result of Boyd and Lawton says that the
Mahler measure of a multivariate polynomial is the limit of Mahler measures of
univariate polynomials. We prove the analogous result for different extensions
of Mahler measure such as generalized Mahler measure (integrating the maximum
of for possibly different 's), multiple Mahler measure (involving
products of for possibly different 's), and higher Mahler measure
(involving ).Comment: 9 page
Densities of short uniform random walks
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic to
expect similar results for more than five steps. New results are also presented
concerning the moments of uniform random walks and, in particular, their
derivatives. Relations with Mahler measures are discussed.Comment: 32 pages, 9 figure
Hyper-Mahler measures via Goncharov-Deligne cyclotomy
The hyper-Mahler measures and are evaluated in closed form via
Goncharov-Deligne periods, namely -linear combinations of multiple
polylogarithms at cyclotomic points (complex-valued coordinates that are roots
of unity). Some infinite series related to these hyper-Mahler measures are also
explicitly represented as Goncharov-Deligne periods of levels , , ,
, , , and .Comment: (v1) i+30 pages, 5 tables. (v2) i+37 pages, 7 tables. Results
improved and enriched. Maple and Mathematica worksheets available as
ancillary files. (v3) 47 pages, 8 tables. Reformatted and corrected. (v4) 51
pages, 8 tables. Accepted versio
Mahler measures and computations with regulators
In this work we apply the techniques that were developed in [Lalin: An
algebraic integration for Mahler measure] in order to study several examples of
multivariable polynomials whose Mahler measure is expressed in terms of special
values of the Riemann zeta function or Dirichlet L-series. The examples may be
understood in terms of evaluations of regulators. Moreover, we apply the same
techniques to the computation of generalized Mahler measures, in the sense of
Gon and Oyanagi [Gon, Oyanagi: Generalized Mahler measures and multiple sine
functions]Comment: 5 figure
On two conjectures of Sun concerning Apéry-like series
We prove two conjectural identities of Z.-W. Sun concerning Apéry-like series. One of the series is alternating, whereas the other one is not. Our main strategy is to convert the series and the alternating series to log-sine-cosine and log-sinh-cosh integrals, respectively. Then we express all these integrals using single-valued Bloch–Wigner–Ramakrishnan–Wojtkowiak–Zagier polylogarithms. The conjectures then follow from a few rather non-trivial functional equations of those polylogarithms in weights 3 and 4
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