Special zeta Mahler functions

TL;DR

This paper studies a family of special zeta Mahler functions linked to certain Laurent polynomials, providing explicit formulas, examples, properties, and exploring their relation to Mahler measures, revealing phenomena similar to the Riemann Hypothesis.

Contribution

It introduces explicit formulas and properties for a new family of zeta Mahler functions associated with specific Laurent polynomials, expanding understanding of their behavior and connections.

Findings

Established explicit formulas for the ZMFs.

Identified an RH-type phenomenon in these functions.

Explored the relationship between ZMFs and Mahler measures.

Abstract

In 1969, I. Bernstein and S. Gelfand introduced an object, which is now called the zeta Mahler function (ZMF, also zeta Mahler measure) and related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials $k + (x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})$, where $k$ is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore relations with the Mahler measure.