Asymptotic nature of higher Mahler measure

TL;DR

This paper investigates the asymptotic behavior of higher Mahler measures for linear polynomials, revealing that the normalized measure approaches 1/π as the order increases, and introduces new asymptotic results.

Contribution

It provides new asymptotic results for higher Mahler measures of linear polynomials, expanding understanding of their growth behavior.

Findings

Normalized higher Mahler measure approaches 1/π as k increases

Established asymptotic relation for m_k(P) when P(x)=x+r with |r|=1

Derived new asymptotic formulas for higher Mahler measures

Abstract

We consider Akatsuka's zeta Mahler measure as a generating function of higher Mahler measure $m_k(P)$ of a polynomial $P,$ where $m_k(P)$ is the integral of $\log^{k}\left| P \right|$ over the complex unit circle. Restricting ourselves to $\displaystyle P(x)=x+r$ with $|r|=1$ we show some new asymptotic results regarding $m_k(P)$, especially $\left| m_k(P) \right|/k! \to 1/\pi$ as $k \to \infty.$