Arithmetics, Interrupted

I share some of my adventures in mathematical research and homeschooling in the time of COVID-19

An algebraic integration for Mahler measure

There are many examples of several variable polynomials whose Mahler measure is expressed in terms of special values of polylogarithms. These examples are expected to be related to computations of regulators, as observed by Deninger [D] and, later, by Rodriguez-Villegas [R] and Maillot [M]. While Rodriguez-Villegas made this relationship explicit for the two-variable case, it is our goal to understand the three-variable case and shed some light on the examples with more variables. 1

A generalization of a theorem of Boyd and Lawton

The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$).Comment: 9 page

Conjectures of sums of divisor functions in Fq[T]\mathbb F_q[T] associated to symplectic and orthogonal regimes

In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_k(f)$ over the function field $\mathbb{F}_q[T]$ in the limit as $q \to \infty$ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [arXiv:1504.07804] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields

An invariant property of Mahler measure

We exhibit a change of variables that maintains the Mahler measure of a given polynomial. This method leads to the construction of highly non-trivial polynomials with given Mahler measure and settles some conjectural numerical formulas due to Boyd and Brunault.Comment: 12 page

Mahler measure of some n-variable polynomial families

The Mahler measures of some n-variable polynomial families are given in terms of special values of the Riemann zeta function and a Dirichlet L-series, generalizing the results of \cite{L}. The technique introduced in this work also motivates certain identities among Bernoulli numbers and symmetric functions