TITLE: Moments of the Riemann zeta function
QUESTION [8 upvotes]: Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$

REPLY [13 votes]: This answer is based on Lucia's remark, and is included for completeness.
By (8.111) in Ivić's book "The theory of the Riemann zeta function with applications", we have
$$\int_T^{2T}|\zeta(\sigma+it)|\,dt\asymp_\sigma T,\qquad T\geq 1,\quad 1/2<\sigma<1.$$
Hence, by the functional equation for $\zeta(s)$ and Stirling's approximation, we also have
$$\int_T^{2T}|\zeta(\sigma+it)|\,dt\asymp_\sigma T^{3/2-\sigma},\qquad T\geq 1,\quad 0<\sigma<1/2.$$
In particular, the answer to the original question is negative.