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Mean-values of the Riemann zeta-function

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 February 2010

K. Soundararajan
K. Soundararajan
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, U.S.A.
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Extract

Let

Asymptotic formulae for Ik(T) have been established for the cases k=1 (Hardy-Littlewood, see [13]) and k = 2 (Ingham, see [13]). However, the asymptotic behaviour of Ik(T) remains unknown for any other value of k (except the trivial k = 0, of course). Heath-Brown, [6], and Ramachandra, [10], [11], independently established that, assuming the Riemann Hypothesis, when 0≤K≤2, Ik(T) is of the order T(log T)k2 One believes that this is the right order of magnitude for Ik(T) even when k = 2 and indeed expects an asymptotic formula of the form

where Ck is a suitable positive constant. It is not clear what the value of Ck should be.

MSC classification

Secondary: 11M06: $zeta (s)$ and $L(s, chi)$
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 158 - 174
Copyright
Copyright © University College London 1995

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References

1.Balasubramanian, R.Conrey, J. B. and Heath-Brown, D. R.. Asymptotic mean square of the Riemann Zeta-function and a Dirichlet polynomial. J. Reine Angew. Math., 357 (1985), 161–181.Google Scholar
2.Balasubramanian, R. and Ramachandra, K.. Proof of some conjectures on the mean value of Titchmarsh series, I. Hardy Ramanujan J., 13 (1990), 1–20.Google Scholar
3.Conrey, J. B. and Ghosh, A.. Mean Values of the Riemann Zeta-function. Mathematika, 31 (1984), 159–164.CrossRefGoogle Scholar
4.Conrey, J. B. and Ghosh, A.. Mean Values of the Riemann Zeta-function, III. Proc. of the Amalfi Conference on Analytic Number Theory (1992), 35–59.Google Scholar
5.Gonek, S. M.. A formula of Landau and Mean Values of ζ(s). In Topics in Analytic Number Theory (edited by Graham, S. W. and Vaaler, J. D.) 92–97.Google Scholar
6.Heath-Brown, D. R.. Fractional Moments of the Riemann Zeta-function. J. Lond. Math. Soc. (2), 24 (1981), 65–78.Google Scholar
7.Heath-Brown, D. R.. Fractional Moments of the Riemann Zeta-function, II. Quart. J. of Math. Oxford (2), 44 (1993), 185–197.CrossRefGoogle Scholar
8.Ivic, A.. Mean Values of the Riemann Zeta function. T.I.F.R. Lectures in Mathematics and Physics, 82 (1991).Google Scholar
9.Ramachandra, K.. Application of a Theorem of Montgomery and Vaughan to the Zetafunction. J. Lond. Math. Soc. (2), 10 (1975), 482–486.Google Scholar
10.Ramachandra, K.. Some Remarks on the mean value of the Riemann Zeta-function and other Dirichlet series, II. Hardy Ramanujan J. 3 (1980), 1–24.Google Scholar
11.Ramachandra, K.. Some Remarks on the mean value of the Riemann Zeta-function and other Dirichlet series, III. Ann. Acad. Sci. Fenn. Ser. A. I. 5 (1980), 145–158.Google Scholar
12.Selberg, A.. Note on a paper of L. G. Sathe. J. of the Indian Math. Soc. B. 18 (1954), 83–87.Google Scholar
13.Titchmarsh, E. C.. The theory of the Riemann Zeta-function. Clarendon Press (second edition), Oxford (1986).Google Scholar