Rankin-Selberg Method
Notations and Definitions
\(H\) : Upper half plane of complex plane \(\mathbb{C}\).
\(f,g\) : holomorphic cuspforms on the upper half plane \(H\) of weight \(2k\).
\(\Gamma\) : \(\Gamma = SL(2,\mathbb{Z})\) subgroup.
\(P\) : Upper triangular matrices in \(\Gamma\).
Let the Fourier expansions of \(f,g\) be:
\begin{equation} \begin{aligned} & f(z) = \sum_{n>0} a_n e^{2\pi i n z}
& g(z) = \sum_{n>0} b_n e^{2\pi i n z}
\end{aligned} \end{equation}
The Peterson inner product of modular forms is:
\[\langle f,g \rangle = \int_{\Gamma \ H} f(z) \overline{g(z)} y^{2k} \frac{dx\,dy}{y^2}.\]The Eisenstein Series is:
\[\begin{aligned} & E_s(z) = \frac{1}{2} \cdot \sum_{c,d \in \mathbb{Z} \textrm{coprime} } \frac{y^s}{\mid cz+d \mid^{2s}} = \sum_{\gamma \in P\backslash \Gamma }\textrm{Im}(\gamma z)^s \\ & \tilde{E_s}(z) = 2 \zeta(2s) E_s(z) = \sum_{(c,d)\neq (0,0) } \frac{y^s}{\mid cz+d \mid^{2s}} \\ \end{aligned}\]Results
Theorem1: The inner product is an L-function:
\[\langle f\cdot E_s , g \rangle = (4\pi)^{-(s+2k-1)} \Gamma(s+2k-1) \sum_{n\geq 1} \frac{a_n\overline{b_n}}{n^{s+2k-1}}.\]And the function
\[\xi(2s) \langle f\cdot E_s , g \rangle = (4\pi)^{-(s+2k-1)} \xi(2s) \Gamma(s+2k-1) \sum_{n\geq 1} \frac{a_n\overline{b_n}}{n^{s+2k-1}}.\]has an analytic continuation to \(\mathbb{C}\) with poles at most at \(s = 0, 1\). In the equation \(\xi(s) = \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s)\), and \(\Gamma,\zeta\) are famous Gamma and Zeta functions.
Theorem2: The function
\[\pi^{-s} \Gamma(s) \tilde{E_s}(z) = 2\pi^{-s} \Gamma(s) \zeta(2s) E_s(z).\]has a meromorphic continuation to \(s\in \mathbb{C}\) with poles only at \(s = 0, 1\), which are simple, and is invariant under \(s \to 1-s\). The residue of \(E_s(z)\) at \(s = 1\) is the constant function \(3/\pi\). The function \(E_s(z)\) has no pole in \(\mathrm{Re}(s) > \frac{1}{2}\) other than at \(s = 1\).
The reason behind it is as follows: Consider the integral representation (a Mellin transformation)
\[\int_{0}^{\infty} t^{2s} ( \Theta(tg) -1 ) \frac{dt}{t},\]where
\[\Theta(g) = \sum_{v\in \mathbb{Z}^2} \varphi(v\cdot g),\quad \varphi(v) = e^{-\pi |v|^2},\quad g\in GL(2,\mathbb{R})\, \textrm{or}\, SL(2,\mathbb{R}).\]Actually the continuation follow from anothor form of left hand right of the above equation:
\[\int_{1}^{\infty} t^{2s} (\Theta(tg) -1) \frac{dt}{t} + \int_{1}^{\infty} t^{2-2s} (\Theta(tg) -1) \frac{dt}{t} + \frac{1}{2s-2} - \frac{1}{s}.\]Functional Equations
For zeta funtion \(\zeta(z)\):
\[\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s).\]For Gamma function we have Euler’s reflection formula:
\[\Gamma(s)\Gamma(1-s) = \frac{\pi}{\sin(\pi s)}.\]For Riemann’s Xi function
\[\xi(s) = \pi^{-s/2} \Gamma(\frac{s}{2}) \zeta(s),\]we get its functional equation in simpler form
\[\xi(s) = \xi(1-s).\]And for Eisenstein Series,
\[\xi(2s) E_s = \xi(2-2s) E_{1-s}.\]