THE MEAN SQUARE OF THELOGARITHM OF THE ZETA-FUNCTION

Abstract

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}.