Algebraic numbers close to 1 and variants of Mahler's measure
The author's interest is in bounding the Mahler measure of certain families of polynomials. For this purpose he studies the measure \(\widetilde{h}(R)= \int^1_0\log^+|R(e^{2\pi it})|dt\) for rational functions \(R\). (Although he attributes the introduction of \(\widetilde{h}(R)\) to \textit{M. Mignotte} [Ann. Fac. Sci. Toulouse, V. Ser., Math. 1, 165-170 (1979; Zbl 0421.10022)], it had been used earlier by \textit{K. Mahler} [Ill. J. Math. 8, 1-4 (1964; Zbl 0128.07101)].) One of the main results in a nontrivial and sharp estimate of \(\widetilde{h}(R)\) for \(R(z)=\prod_{n\leq N}(z^n-1)^{\theta_n}\), where the \(\theta_n\) are positive or negative integers. He uses this to estimate Mahler's measure for the family of polynomials \[ G(z)=1+ (z+1)\prod^N_{n=1} (z^{2n-1}-1), \] obtaining the remarkable \[ \log M(G)\leq (1/\pi^2)(\log N)^2+3\log N+7. \] The interest in \(G\) is that it has a root \(\alpha\) with \(|\alpha+1|\leq (N^2+1)2^{-N}\), i.e. an algebraic unit close to \(-1\) with small Mahler measure. This example shows that a recent estimate of \textit{M. Mignotte} and \textit{M. Waldschmidt} [J. Number Theory 47, 43-62 (1994; Zbl 0801.11033)] is close to best possible. Since \(\alpha^k\) is a unit for all odd \(k\leq N\), this also has an application to a recent result of \textit{J. H. Silvermann} [Exp. Math. 4, No. 1, 69-83 (1995; Zbl 0851.11064)], as observed by C. Pinner. Another interesting result has to do with the polynomial \(\Phi(z)= \prod_{n\leq N}\Phi_n(z)\), where \(\Phi_n(z)\) is the \(n\)th cyclotomic polynomial. He shows that the statement \(\widetilde{h}(\Phi) \ll_\varepsilon N^{1/2+\varepsilon}\) for any \(\varepsilon>0\) is equivalent to the Riemann hypothesis.
scientific article; zbMATH DE number 934969