Moments of the Riemann Zeta Function – Analytic Number Theory

Question

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

Answer 1

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.