Results 1  10
of
65
Some examples of Mahler measures as multiple polylogarithms
 J. Number Theory
"... The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial. Key words: Mahler measure, Lfunctions, polylogarithms, hyperlogarithms ..."
Abstract

Cited by 22 (18 self)
 Add to MetaCart
The Mahler measures of certain polynomials of up to five variables are given in terms of multiple polylogarithms. Each formula is homogeneous and its weight coincides with the number of variables of the corresponding polynomial. Key words: Mahler measure, Lfunctions, polylogarithms, hyperlogarithms, polynomials, Jensen’s formula
FugledeKadison determinants and entropy for actions of discrete amenable groups
 J. Amer. Math. Soc
"... Consider a discrete group Γ and an element f in the integral group ring ZΓ. Then Γ acts from the left on the discrete additive group ZΓ/ZΓf by automorphisms of groups. Dualizing, we obtain a left Γaction on the compact Pontrjagin dual group Xf = ZΓ/ZΓf by continuous automorphisms of groups. By de ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
Consider a discrete group Γ and an element f in the integral group ring ZΓ. Then Γ acts from the left on the discrete additive group ZΓ/ZΓf by automorphisms of groups. Dualizing, we obtain a left Γaction on the compact Pontrjagin dual group Xf = ZΓ/ZΓf by continuous automorphisms of groups. By definition, Xf is a
An explicit formula for the Mahler measure of a family of 3variable polynomials
"... An explicit formula for the Mahler measure of the 3variable Laurent polynomial a + bx 1 + cy +(a + bx + cy)z is given, in terms of dilogarithms and trilogarithms. 2000 Mathematics Subject Classification. 11R06. ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
An explicit formula for the Mahler measure of the 3variable Laurent polynomial a + bx 1 + cy +(a + bx + cy)z is given, in terms of dilogarithms and trilogarithms. 2000 Mathematics Subject Classification. 11R06.
The Mahler measure of algebraic numbers: a survey.” Conference Proceedings
 University of Bristol
, 2008
"... Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though n ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
Abstract. A survey of results for Mahler measure of algebraic numbers, and onevariable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (‘house’) of an algebraic integer are also discussed. Some generalisations are also mentioned, though not to Mahler measure of polynomials in more than one variable. 1.
New 5F4 hypergeometric transformations, threevariable Mahler measures, and formulas for 1/π
 Ramanujan J
"... New relations are established between families of threevariable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1 ..."
Abstract

Cited by 18 (7 self)
 Add to MetaCart
New relations are established between families of threevariable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1
Mahler measure of some nvariable polynomial families
, 2004
"... The Mahler measures of some nvariable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet Lseries, generalizing the results of [13]. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric funct ..."
Abstract

Cited by 14 (12 self)
 Add to MetaCart
The Mahler measures of some nvariable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet Lseries, generalizing the results of [13]. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions.
The arithmetic and geometry of Salem numbers
 Bull. Amer. Math. Soc
, 1991
"... Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and ar ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
Abstract. A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this paper we survey some of the recent appearances of Salem numbers in parts of geometry and arithmetic, and discuss the possible implications for the ‘minimization problem’. This is an old question in number theory which asks whether the set of Salem numbers is bounded away from 1. Contents
Functional equations for Mahler measures of genusone curves
 ALGEBRA AND NUMBER THEORY
"... In this paper we will establish functional equations for Mahler measures of families of genusone twovariable polynomials. These families were previously studied by Beauville [3], and their Mahler measures were considered by Boyd [11] and RodriguezVillegas [19]. Bertin [8], Zagier [26], and Stiens ..."
Abstract

Cited by 13 (11 self)
 Add to MetaCart
In this paper we will establish functional equations for Mahler measures of families of genusone twovariable polynomials. These families were previously studied by Beauville [3], and their Mahler measures were considered by Boyd [11] and RodriguezVillegas [19]. Bertin [8], Zagier [26], and Stienstra [24]. Our functional equations allow us to prove identities between Mahler measures that were conjectured by Boyd. As a corollary, we also establish some new transformations for hypergeometric functions.
Hypergeometric formulas for lattice sums and Mahler measures
, 2010
"... We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd’s conjectured identities between Mahler measures and special values of Lseries of elliptic curves. 1 ..."
Abstract

Cited by 13 (9 self)
 Add to MetaCart
We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd’s conjectured identities between Mahler measures and special values of Lseries of elliptic curves. 1