06021898

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06021898 Lal{\'\i}n, Matilde Sinha, Kaneenika Higher Mahler measure for cyclotomic polynomials and Lehmer's question. Ramanujan J. 26, No. 2, 257-294 (2011). 2011 Springer, Norwell, MA EN higher Mahler measures Lehmer's question cyclotomic polynomials zeta values

doi:10.1007/s11139-010-9278-6

Let $P \in \mathbb{C}[x]$ be a nonzero polynomial and let $k \in \mathbb{N}$. The $k$-higher (logarithmic) Mahler measure of $P$ is defined by $$m_k(P):= \frac{1}{2\pi i} \int_{|x|=1} \log^k|P(x)| \frac{dx}{x}.$$ In their main result (Theorem 4) the authors show that for each integer $s \geq 1$ and each $P \in \mathbb{Z}[x]$ which is not a monomial we have $m_{2s}(P) \geq (\pi^2/12)^s$ if $P$ is reciprocal and $m_{2s}(P) \geq (\pi^2/48)^s$ if $P$ is non-reciprocal. Another result (Theorem 6) is devoted to the evaluation of $m_3(P)$ for a polynomial $P \in \mathbb{C}[x]$ with all roots on the unit circle. The paper also contains a number of other interesting results on this $k$-higher generalization of Mahler's measure. Art\=uras Dubickas (Vilnius)