Application of Rankin-Selberg theory Let $f$ be a Hecke-Maass form for $SL(2)$, and its L-function defined by the Fourier coefficients $\lambda_f(n)$, i.e.
$$L(s,f) = \sum_{n=1}^\infty \frac{\lambda_f(n)}{n^s}$$
I would like to understand the proof of the bound on average
$$\sum_{n < X} |\lambda_f(n)|^2 \ll X$$
I read that it is an easy consequence of "Rankin-Selberg theory". I know Rankin-Selberg integral representations, but what is the relation with this bound and how to deduce it?
 A: As @reun suggests (and I do not claim originality in that essay of mine: in some interview that I've lost the reference to, R. Rankin said that his advisor Ingham already knew of this way give an integral presentation of the Eisenstein series...), what you are asking is indeed a relatively weak conclusion from the Rankin (1939)-Selberg (1940) integral representation of $\sum |\lambda_f(n)|^2/n^s$ as an integral of $|f|^2$ against an Eisenstein series.
For both of them, the goal was to use some subtler info (an old lemma of Landau about generalized Dirichlet series with functional equations of a certain form) to obtain not just the main term in an asymptotic, but also a power-reduced error term... from which they could make incremental progress on Ramanujan's conjecture about the size of the Fourier coefficients $\tau(n)$ of the holomorphic level-on cuspform of weight $12$... State of the art at that time, prior to P. Deligne's 1970s work using Weil Conjectures, etc.
An asymptotic without error term should follow (as @reuns remarks) just from relatively crude properties of the Eisenstein series, obtainable, for example, from that Mellin transform trick, and then a Tauberian theorem. Also, possibly, from a suitable adaptation of the argument of D.J. Newman from 1980 for the Prime Number Theorem (without a Tauberian theorem, etc.), one could obtain an asymptotic (without error term) for the indicated sum. I've not ever tried to write out such a thing, so I cannot really confirm this possibility.
As for many things, the Iwaniec-Kowalski book (AMS) on analytic number theory is one of the best starting points for references, and if I were wanting to give a real answer to this question, I'd start by looking there. (A sign of my degree of endorsement of that source is that I own two copies, one for office, one for home-office...)
Since the $n-1,1$ degenerate-data Eisenstein series (also called "Epstein zeta function") for $GLn$ admits an analogous, elementary, integral presentation, and the Rankin-Selberg integral presentation of the "product" $L$-function for two cuspforms for $GLn$ is entirely analogous, one should be able to obtain at least an asymptotic for the analogous sum in greater generality.
