TITLE: Second moment estimates for $\zeta(s)$: different methods?
QUESTION [11 upvotes]: What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$
with an error term $E(T) = O(T^{\alpha+\epsilon})$ with $\alpha=1/2$ or $\alpha=1/3$?
Note I am not asking for the best $\alpha$; I am asking for different ways - the simpler the better - with the intention of choosing one of them to make explicit.
Methods I know:

Proofs based on Atkinson's 1949 formula. This formula yields $\alpha=1/2$ automatically and $\alpha=1/3$ after some smoothing and some work. The problem is that there is an asymptotically tiny error term that seems hard to make explicit without applying the saddle-point method explicitly (twice) - something that I know to be time-consuming and sometimes painful.
Balasubramanian (1978). Again, this doesn't look simple.
Ingham/Atkinson (1939). Short and relatively simple. Yields $E(T) = O(T^{1/2} \log^2 T)$.

I'm also curious about the same question (in the same way) for
$$\int_0^T \left|\zeta\left(\sigma + i t\right)\right|^2 dt,$$
where $0\leq \sigma <1$.

REPLY [2 votes]: Theorem 7.2(A) of Titchmarsh's 'The Theory of the Riemann zeta function' uses a simplified form of the special case $x=t$ of the approximate functional equation of the Riemann zeta function which holds for $1/2<\sigma\leq 1$, 
$$
\zeta(s)=\sum_{n<t} n^{-s} + O(t^{-\sigma}).
$$
The result is 
$$
\int_1^T|\zeta(\sigma+it)|^2 \ dt< AT \min\left(\log T, \frac1{\sigma-\frac12}\right)
$$
uniformly for $1/2\leq \sigma \leq 2$. 
For $0<\sigma<1/2$, the above result and the functional equation may help. 
More precisely, for $\sigma>1/2$, Theorem 7.2 of the same book gives
$$
\int_1^T |\zeta(\sigma+it)|^2 \ dt \sim \zeta(2\sigma) T.
$$