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On mean values of the zeta-function

Published online by Cambridge University Press:  26 February 2010

J. B. Conrey  and
A. Ghosh
J. B. Conrey
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A.
A. Ghosh
Affiliation:
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, U.S.A.
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Extract

Let

It is conjectured that for any k,

for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, C1 = 1, and c2 = (2π2)-1, but there is not even a conjectural value of ck for any other k. However, it is known that the Riemann hypothesis implies

for all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]).

Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 159 - 161
Copyright
Copyright © University College London 1984

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References

1.Heath-Brown, D. R.. Fractional moments of the Riemann zeta-function. J. London Math. Soc. (2), 24 (1981), 65–78.CrossRefGoogle Scholar
2.Ramachandra, K.. Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series, I. Hardy-Ramanujan J., 1 (1978), 1–15.Google Scholar
3.Titchmarsh, E. C.. The Theory of the Riemann Zeta-Function (Clarendon Press, Oxford, 1951).Google Scholar