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Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials


Published:20070801
Printed: Aug 2007
T. Erdélyi
D. S. Lubinsky
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Abstract
We investigate large sieve inequalities such as
\[
\frac{1}{m}\sum_{j=1}^{m}\psi ( \log  P( e^{i\tau
_{j}})  ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left(
\log [ e P( e^{i\tau })  ] \right) \,d\tau
,
\]
where $\psi $ is convex and increasing, $P$ is a polynomial or an
exponential of a potential, and the constant $C$ depends on the degree of $P$,
and the distribution of the points $0\leq \tau _{1}<\tau _{2}<\dots<\tau
_{m}\leq 2\pi $. The method allows greater generality and is in some ways
simpler than earlier ones. We apply our results to estimate the Mahler
measure of Fekete polynomials.