  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 $k \to
\infty.$
