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

Mahler measures, short walks and log-sine integrals

The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years — very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of log-sine integrals we provide systematic evaluations of various higher and multiple Mahler measures. The evaluations in terms of log-sine integrals become particularly useful in light of the fact that log-sine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for log-sine integrals. Our initial interest in considering Mahler measures stems from a study of uniform random walks in the plane as first introduced by Pearson. The main results on the moments of the distance traveled by an n-step walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures

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

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