Is it possible to get an upper bound better than $\ll\_\sigma T^{3/2-\sigma}$ for $$\int\_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$



Moments of the Riemann zeta function
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.


