Amoroso's criterion

Amoroso [ MR 98f:11113] has proven the following interesting equivalent to the Riemann Hypothesis. Let $\Phi_n(z)$ be the $n$th cyclotomic polynomial and let $F_N(z) = \prod_{n \le N}\Phi_n(z)$. Let

\begin{displaymath}\tilde h(F_N) = (2\pi)^{-1}\int_{-\pi}^\pi \log^+\vert F(e^{i\theta})\vert~d\theta.\end{displaymath}

Then, $\tilde h(F_n) \ll N^{\lambda + \epsilon}$ is equivalent to the assertion that the Riemann zeta function does not vanish for ${\rm Re}{z} \ge \lambda + \epsilon$.

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