lehmer’s problem
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Author(s):  
JAN-WILLEM M. VAN ITTERSUM ◽  
BEREND RINGELING ◽  
WADIM ZUDILIN

Abstract Motivated by a famous question of Lehmer about the Mahler measure, we study and solve its analytic analogue.


2020 ◽  
pp. 1-32
Author(s):  
W. Lück

We consider the problem whether for a group [Formula: see text] there exists a constant [Formula: see text] such that for any [Formula: see text]-matrix [Formula: see text] over the integral group ring [Formula: see text] the Fuglede–Kadison determinant of the [Formula: see text]-equivariant bounded operator [Formula: see text] given by right multiplication with [Formula: see text] is either one or greater or equal to [Formula: see text]. If [Formula: see text] is the infinite cyclic group and we consider only [Formula: see text], this is precisely Lehmer’s problem.


2016 ◽  
Vol 11 (1) ◽  
pp. 79-139 ◽  
Author(s):  
Jean-Louis Verger-Gaugry

AbstractLet n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers $(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for $\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures ${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for $\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, 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 of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.


10.37236/2834 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Graeme Taylor ◽  
Gary Greaves

We solve Lehmer's problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer's number $\tau_0 = 1.17628\dots$.


2011 ◽  
Vol 54 (1) ◽  
pp. 155-162 ◽  
Author(s):  
ZHANG WENPENG

AbstractLet q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.


2011 ◽  
Vol 61 (4) ◽  
pp. 909-916
Author(s):  
Xiaowei Pan ◽  
Wenpeng Zhang

2008 ◽  
Vol 04 (04) ◽  
pp. 587-596 ◽  
Author(s):  
NICOLAE CIPRIAN BONCIOCAT

We obtain explicit lower bounds for the Mahler measure for nonreciprocal polynomials with integer coefficients satisfying certain congruences.


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