# Second moment estimates for $\zeta(s)$: different methods?

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 $$0\leq \sigma <1$$.

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 $$1/2<\sigma\leq 1$$, $$\zeta(s)=\sum_{n The result is $$\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.$$