Let
$$I(\sigma,T)=\int\_0^T |\zeta(\sigma+ i t)|^2 dt.$$
Unconditional bounds and asymptotics for $I(\sigma,T)$, $1/2\leq \sigma <1$, have been known since Hardy and Littlewood (see Chapter 7 of Titchmarsh). What about $0<\sigma<1/2$?


One can of course use the functional equation to try to obtain something useful. I am interested in whether there are results in the literature I can reference (there have to be!). In particular, I would like to see an unconditional result with the right constant in front of the main term.


(The main term should be of the form $c\_\sigma T^{2-2\sigma}$, $c\_\sigma$ a constant.)



Bounding the second moment of $|\zeta(\sigma+i t)|^2$ for $0<\sigma<1$
Ingham has a nice discussion of this in the introduction of his 1926 [paper.](https://mathscinet.ams.org/mathscinet-getitem?mr=1575391)


