62 research outputs found
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 -variable polynomial is the integral
of on the unit -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 for possibly different 's), multiple Mahler measure (involving
products of for possibly different 's), and higher Mahler measure
(involving ).Comment: 9 page
Conjectures of sums of divisor functions in associated to symplectic and orthogonal regimes
In [arXiv:2107.01437], the authors studied the mean-square of certain sums of
the divisor function over the function field in the
limit as 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
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