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Power mean values of the Riemann zeta-function

Published online by Cambridge University Press:  26 February 2010

J.-M. Deshouillers  and
H. Iwaniec
J.-M. Deshouillers
Affiliation:
Laboratoire associé au C.N.R.S.no. 226, Université de Bordeaux I, U.E.R. de Mathématiques et d'Informatique, 351, Cours de la Libération, 33405 Talence, Cedex, France.
H. Iwaniec
Affiliation:
Mathematics Institute, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warszawa, Poland.
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Extract

The classical mean value theorem for Dirichlet's polynomials states that

see H. L. Montgomery [7]. This formula is very useful in the theory of the Riemann zeta-function ζ(s). From the approximate functional equation

where | χ(½ + it)| = 1, u, v ≥ 1, 2πuv = t (see E. C. Titchmarsh [8]) it follows that χ(½ + it) can be well approximated by Dirichlet's polynomials of length N< t½.

Type
Research Article
Information
Mathematika , Volume 29 , Issue 2 , December 1982 , pp. 202 - 212
Copyright
Copyright © University College London 1982

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References

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8.Titchmarsh, E. C.. The Theory of the Riemann Zeta-function (Clarendon Press, Oxford, 1951).Google Scholar