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 = T4/3P3(log T) + O\varepsilon(T7/6 + \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.