I have been reading about "mean value theorems in number theory" such as $$\int_{-T}^T |\zeta(\frac{1}{2} + it)|^4 \, dt \sim T \log(T)^4 $$ How to prove such a result?
One source says it is related to the equidistrbution of the line $\{ (1 + \frac{i}{T})x: x \in \mathbb{R} \} $, but I can't find a reference. The funny thing about is that this set is not a horocycle, an object which I am more familiar with.
Ingham shoed (Mean value theorems in the theory of the Riemann Zeta function, P.L.M.S., (2), 27, (1926), 273-300) $$\int_{1}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{4}dt=\frac{T\log^{4}\left(T\right)}{2\pi^{2}}+O\left(T\log^{2}\left(T\right)\right) $$ as $T\rightarrow\infty $ and $$\int_{1}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{4}dt\sim\frac{T\log^{4}\left(T\right)}{2\pi^{2}} $$ is a consequence of the following theorem
Theorem. As $\delta\rightarrow0$ we have $$\int_{1}^{\infty}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{4}e^{-\delta t}dt\sim\frac{1}{2\pi^{2}}\frac{1}{\delta}\log^{2}\left(\frac{1}{\delta}\right). $$
You can find a proof of it in Titchmarsh, The theory of the Riemann Zeta function, page 166 theorem 7.16.