Relations Between Mahler's Measure and Values of L-Series

Abstract

Mahler's measure is a natural generalization of Jensen's formula to polynomials in several variables. Its definition is as follows:The importance of Mahler's measure for polynomials in several variables lies in its connection to Lehmer's classical question which can be phrased in terms of Mahler's measure for polynomials in one variable:Given , are there any polynomials p with integer coefficients in one variable for which ?Surprisingly, Boyd [1] has shown that to answer this question, it is necessary to investigate the larger question involving polynomials in several variables.

The Mahler measure of parametrizable polynomials

“…It states that These equations imply that (29) is equivalent to 6D(iω) + 6D(iω) + 8D(i) = 0. To complete our computation, we employ the Kubert identity for the dilogarithm, discussed in [8]: We remark that a proof of (2) along similar lines exists, in principle. We instead apply our main result to the polynomial from (3), shown here along with its Newton polygon P (x, y) = y 2 + y x 2 + x + 1 + x 4 + x 3 + x 2 + x + 1 −→ This polygon has one interior point, which implies that it generically defines a curve of genus one.…”

“…In light of the five-term identity (7), the dilogarithm vanishes on B ′′ (K), so we obtain a well-defined homomorphism D : B(K) → R.…”

“…It states that These equations imply that (28) is equivalent to 6D(iω) + 6D(iω) + 8D(i) = 0. To complete our computation, we employ the Kubert identity for the dilogarithm, discussed in [7]:…”

Mahler measures, K-theory and values of L-functions

“…where χ is the quadratic character of conductor 3 and L(χ, s) is the Dirichlet series associated to it. Some similar formulas can be found in [6] and [14]. The proofs of these identities however are analytical and do not shed much light on the deeper reasons for this phenomenon.…”

Apéry limits and Mahler measures

“…for the right-hand side in ( 1) is borrowed from the paper by G. Ray [11,Proposition 2]. The connection with L(χ −8 , 2) comes from the general formula…”