Annales Academić Scientiarum Fennicć
Mathematica
Volumen 32, 2007, 269-277

ON THE MEAN SQUARE OF THE ZETA-FUNCTION AND THE DIVISOR PROBLEM

Aleksandar Ivic

Universite u Beogradu, Katedra Matematike RGF-a
Dusina 7, 11000 Beograd, Serbia; ivic 'at' rgf.bg.ac.yu

Abstract. Let \Delta(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of |\zeta(1/2 + it)|. If E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi) with \Delta^*(x) = -\Delta(x) + 2\Delta(2x) - 1/2 \Delta(4x), then we obtain the asymptotic formula

\int_0^T (E^*(t))² dt = T^4/3P₃(log T) + O_\varepsilon(T^{7/6 + \varepsilon}),

where P₃ is a polynomial of degree three in log T with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.

2000 Mathematics Subject Classification: Primary 11N37, 11M06.

Key words: Dirichlet divisor problem, Riemann zeta-function, mean square of |\zeta(1/2 + it)|, mean square of E^*(t).

Reference to this article: A. Ivic: On the mean square of the zeta-function and the divisor problem. Ann. Acad. Sci. Fenn. Math. 32 (2007), 269-277.

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