Mahler/Zeta Correspondence

TL;DR

This paper establishes a novel connection between Mahler measures and zeta functions through quantum walks, extending the relationship from one-dimensional to higher-dimensional cases and comparing with classical random walks.

Contribution

It introduces a new relation linking Mahler measures and zeta functions specifically for quantum walks, including higher-dimensional cases, which was not previously known.

Findings

Established a relation between Mahler measure and zeta function for quantum walks.

Extended the relation from one-dimensional to higher-dimensional quantum walks.

Compared quantum walk results with higher-dimensional random walks.

Abstract

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks. Afterwards we deal with higher-dimensional quantum walks. For comparison with the case of the quantum walk, we also treat the case of higher-dimensional random walks. Our results bridge between the Mahler measure and the zeta function via quantum walks for the first time.