Experimental Mathematics

Mahler's measure and special values of {$L$}-functions

David W. Boyd


If $P(x_1$,\,\dots,\,$x_n)$ is a polynomial with integer coefficients, the Mahler measure $M(P)$ of $P$ is defined to be the geometric mean of $|P|$ over the $n$-torus $\T ^n$. For $n = 1$, $M(P)$ is an algebraic integer, but for $n$\raise.5pt\hbox{\footnotesize\mathversion{bold}${}>{}$}$1$, there is reason to believe that $M(P)$ is usually transcendental. For example, Smyth showed that $\log M(1+x+y)=L'(${\mathversion{normal}$\chi$}$_{-3}$,$\,{-}1)$, where {\mathversion{normal}$\chi$}$_{-3}$ is the odd Dirichlet character of conductor $3$. Here we will describe some examples for which it appears that $\log M(P(x$,$\,y)) = r@@L'(E$,$\,0)$, where $E$ is an elliptic curve and $r$ is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.

Article information

Experiment. Math., Volume 7, Issue 1 (1998), 37-82.

First available in Project Euclid: 14 March 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 11Y35: Analytic computations

Mahler measure polynomials computation $L$-function elliptic curve Beilinson conjectures


Boyd, David W. Mahler's measure and special values of {$L$}-functions. Experiment. Math. 7 (1998), no. 1, 37--82. https://projecteuclid.org/euclid.em/1047674271

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