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

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

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 $m_k( 1+x_1+x_2),k\in\mathbb Z_{>1}$ and $m_k( 1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb Q$-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 $1$, $2$, $3$, $4$, $6$, $8$, $10$ and $12$.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

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

On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions

International audienceLet $n ≥ 2$ be an integer and denote by $\theta_n$ the real root in $(0, 1)$ of the trinomial$G_{n}(X) = −1 + X + X^n$ . The sequence of Perron numbers $(\theta_{n}^{−1} )_{n≥2}$ tends to 1. We prove thatthe Conjecture of Lehmer is true for $\{\theta_{n}^{−1} | n ≥ 2\}$ by the direct method of Poincar\'e asymptoticexpansions (divergent formal series of functions) of the roots $\theta_n , z_{j,n}$, of $G_{n}(X)$ lying in $|z| <1$, as a function of $n, j$ only. This method, not yet applied to Lehmer’s problem up to theknowledge of the author, is successfully introduced here. It first gives the asymptotic expansionof the Mahler measures ${\rm M}(G_n) = {\rm M}(\theta_{n}) = {\rm M}(\theta_{n}^{-1})$ of the trinomials $G_n$ as a function of $n$only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisotnumber. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. Bythis method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for$\{\theta_{n}^{−1} | n ≥ 2\}$, with a minoration of the house \house\{\theta_{n}^{−1}\}= \theta_{n}^{−1} , and a minoration of the Mahler measure${\rm M}(G_n)$ better than Dobrowolski’s one for $\{\theta_{n}^{−1} | n ≥ 2\}$ . The angular regularity of the roots of $G_n$ , near the unitcircle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu,Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context ofthe Erd\H{o}s-Tur\'an-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions

A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus

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