On the Mahler measure of $1+X+1/X+Y+1/Y$

TL;DR

This paper proves a conjecture linking the Mahler measure of a specific Laurent polynomial to the L-series of an elliptic curve with conductor 15, revealing a deep connection between polynomial measures and elliptic curve L-functions.

Contribution

It establishes a proven formula connecting the Mahler measure of a particular Laurent polynomial to the L-series of a specific elliptic curve, confirming a prior conjecture.

Findings

Confirmed the conjectured formula relating Mahler measure and L-series.

Established a new link between polynomial measures and elliptic curve invariants.

Provided evidence supporting the broader connection between Mahler measures and number theory.

Abstract

We prove a conjectured formula relating the Mahler measure of the Laurent polynomial $1+X+X^{-1}+Y+Y^{-1}$, to the $L$-series of a conductor 15 elliptic curve.