arXiv subjects

Mahya Mehrabdollahei

Publications and source records attributed to Mahya Mehrabdollahei.

4 recordsLinked to original sources

arXiv2026

Relating Mahler measures and Dirichlet $L$-values: new evidence for Chinburg's conjectures

Let $\chi_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $\chi_{-f}$ there exist an integral bivariate rational function $P$ (and, in the strong form, an integral polynomial) such that $\mathrm{m}(P)$ is a rational multiple of $L'(\chi_{-f},-1)$. The strong form of the conjecture was previously known to hold for $18$ values of $f$. We double the number of numerical examples, giving $8$ new instances of the strong and $18$ new instances of the weak conjecture. Our examples arise from an explicit approach, which also captures almost all of the previously known results, and is based on work of Boyd and Rodriguez-Villegas. Moreover, we prove Chinburg's weak conjecture if we allow cyclotomic coefficients.

math.NT↗

arXiv2025

An exact family of bivariate polynomials and Variants of Chinburg's Conjectures

This article provides some solutions to Chinburg's conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet Character of conductor $f$, $χ_{-f}=\left(\frac{-f}{.}\right)$, there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of $L'(χ_{-f},-1)$. To obtain such solutions for the conjectures we investigate a polynomial family denoted by $P_d(x,y)$, whose Mahler measure has been recently studied. We demonstrate that the Mahler measure of $P_d$ can be expressed as a linear combination of Dirichlet $L$-functions, which has the potential to generate solutions to Chinburg's conjectures. Specifically, we prove that this family provides solutions for conductors $f=3,4,8,15,20$, and $24$. Notably, $P_d$ polynomials also provide intriguing examples where the Mahler measures are linked to $L'(χ,-1)$ with $χ$ being an odd non-real primitive Dirichlet character. These examples inspired us to generalize Chinburg's conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version of Chinburg's conjecture, $P_d$ polynomials provide solutions for conductors $5,7$, and $9$.

math.NT↗

arXiv2022

Limits of Mahler measures in multiple variables

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

math.NT↗

arXiv2021

Mahler measure of $P_d$ polynomials

This article investigates the Mahler measure of a family of 2-variate polynomials, denoted by $P_d, d\geq 1$, unbounded in both degree and genus. By using a closed formula for the Mahler measure introduced in "Volume function and Mahler measure of exact polynomials" (by Guilloux and Marché), we are able to compute $m(P_d)$, for arbitrary $d$, as a sum of the values of dilogarithm at special roots of unity. We prove that $m(P_d)$ converges and the limit is proportional to $ζ(3)$, where $ζ$ is the Riemann zeta function.

math.NT↗