Weinvestigate the function R(T,σ), which denotes the error term in theasymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt. It is shown thatR(T,σ) is uniformly bounded for σ \ge 1 and almostperiodic in the sense of Bohr for fixed σ \ge 1; hence R(T,σ)= Ω(1) when T \to \infty. In case {1 \over2}<σ<1 is fixed we can obtain the bound R(T,σ) \ll_ϵT\,^{(9-2σ)/8+ϵ}.