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LOCAL SUPREMA OF DIRICHLET POLYNOMIALS AND ZEROFREE REGIONS OF THE RIEMANN ZETA-FUNCTION
Published online by Cambridge University Press: 22 August 2014
Abstract
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A new family of zerofree region of the Riemann Zeta-function ζ is identified by using Turán's (P. Turán, Eine neue Methode inter Analysis und deren Anwendungen (Akadémiai Kiadó, Budapest, Hungary, 1953); Analytic number theory, Proc. Symp. Pure Math., vol. XXIV (Amer. Math. Soc. Providence, RI, 1972)) localization criterion linking zeros of ζ with uniform local suprema of sets of Dirichlet polynomials expanded over the primes. The proof is based on a randomization argument. An estimate for local extrema for some finite families of shifted Dirichlet polynomials is established by preliminary considering their local increment properties by means of Montgomery-Vaughan's variant of Hilbert's inequality. A covering argument combined with Turán's localization criterion allows to conclude.
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- Copyright © Glasgow Mathematical Journal Trust 2014
References
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