# Mahler measures of a family of non-tempered polynomials and Boyd's conjectures

## TL;DR

This paper establishes a new identity linking Mahler measures of non-tempered and tempered polynomials, evaluates some Mahler measures via special functions, and proves Boyd's conjectures for certain elliptic curves.

## Contribution

It introduces a novel identity connecting Mahler measures of different polynomial families and confirms Boyd's conjectures for conductor 30 elliptic curves.

## Key findings

- Derived an identity relating Mahler measures of non-tempered and tempered polynomials.
- Expressed Mahler measures in terms of special values of L-functions and logarithms.
- Proved Boyd's conjectures for conductor 30 elliptic curves.

## Abstract

We prove an identity relating Mahler measures of a certain family of non-tempered polynomials to those of tempered polynomials. Evaluations of Mahler measures of some polynomials in the first family are also given in terms of special values of $L$-functions and logarithms. Finally, we prove Boyd's conjectures for conductor $30$ elliptic curves using our new identity, Brunault-Mellit-Zudilin's formula and additional functional identities for Mahler measures.