## Experimental Mathematics

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

David W. Boyd

#### Abstract

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

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

Dates
First available in Project Euclid: 14 March 2003

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