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THE MEAN SQUARE OF THELOGARITHM OF THE ZETA-FUNCTION

Published online by Cambridge University Press:  01 May 2000

MICHEL BALAZARD  and
ALEKSANDAR IVIĆ
MICHEL BALAZARD
Affiliation:
Laboratoire d'Algorithmique Arithmétique (CNRS), Mathématiques, Université de Bordeaux 1, 351, cours de la Libération, 33405 Talence, [email protected]
ALEKSANDAR IVIĆ
Affiliation:
Katedra Matematike RGF-a, Universiteta u Beogradu, Dju{\breve s}ina 7, 11000 Beograd, Serbia (Yugoslavia)[email protected], [email protected]
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Abstract

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We investigate the function R(T,σ), which denotes the error term in the asymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt. It is shown thatR(T,σ) is uniformly bounded for σ \ge 1 and almost periodic in the sense of Bohr for fixed σ \ge 1; hence R(T,σ) = Ω(1) when T \to \infty. In case {1 \over 2}<σ<1 is fixed we can obtain the bound R(T,σ) \ll_ϵ T\,^{(9-2σ)/8+ϵ}.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 42 , Issue 2 , May 2000 , pp. 157 - 166
Copyright
2000 Glasgow Mathematical Journal Trust