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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

MICHEL J. G. WEBER
MICHEL J. G. WEBER*
Affiliation:
IRMA, 10 rue du Général Zimmer, 67084 Strasbourg Cedex, France e-mail: michel.weber@math.unistra.fr
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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.

Keywords

Primary: 11M26Secondary: 26D0560G17
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 3 , September 2014 , pp. 643 - 655
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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