Rankin–Selberg method

The Rankin–Selberg method represents Dirichlet series built from automorphic forms as integrals against Eisenstein series and studies them by unfolding those integrals.

For two weight-$k$ level-one cusp forms, use the primitive real-analytic Eisenstein series and $d\mu =dxdy/y^2$. The Rankin–Selberg unfolding identity isInitially this follows by absolute convergence, unfolding to $0\le x<1$, and integrating the Fourier series in $x$. Exponential cusp decay and polynomial growth of the primitive real-analytic Eisenstein series make the integral holomorphic for $\mathrm{Re}w>1$, continuing the Dirichlet series to $\mathrm{Re}s>k$.

For real $w>1$, the finite Rankin–Selberg integral $I(f,f,w)$ has a nonnegative integrand. Tonelli theorem unfolds it without an a priori bound on the Dirichlet series. Integration over $x$ gives ${\int }_0^1|f(x+iy){|}^{2}dx={\sum }_{n\ge 1}|a_n{|}^2{e}^{-4\pi ny}$. A second application of Tonelli theorem gives the finite sum in the displayed identity with $s=w+k-1$. Positivity, rather than analytic continuation alone, proves absolute convergence.

By positive unfolding of a cusp-form square, $\sum |a_n{|}^2{n}^{-\beta }<\infty$ for $\beta >k$. Choose $k<\beta <2\sigma -1$. The Cauchy-Schwarz inequality bounds the absolute sum for the L-function of a cusp form by

For cusp forms $f=\sum a_n{q}^n$ and $g=\sum b_n{q}^n$, their Rankin–Selberg convolution in the elementary normalization is $L(f,g,s)={\sum }_{n\ge 1}{a}_n\overline{b_n}{n}^{-s}$.

For normalized level-one Hecke eigenforms, let ${\alpha }_p+{\beta }_p=a_p$, ${\alpha }_p{\beta }_p=p^{k-1}$, and similarly ${\gamma }_p+{\delta }_p=b_p$, ${\gamma }_p{\delta }_p=p^{k-1}$. The Hecke multiplication relations give $a_{p^r}=({\alpha }_p^{r+1}-{\beta }_p^{r+1})/({\alpha }_p-{\beta }_p)$, with the repeated-root case obtained by continuity. Multiplying the two expressions and summing four geometric series proves the displayed local factor. Multiplicativity gives an Euler product; multiplying by $\zeta (2s-2k+2)$ cancels its local numerator and produces the degree-four convolution factors.

For two level-one weight-$k$ cusp forms, $F(s)=\sum a_n{\overline{b}}_n{n}^{-s}$. Unfolding against the nonholomorphic lattice sum normalized by ${\pi }^{-w}\Gamma (w)$ gives the displayed completion as the integral of $f\overline{g}{y}^k$ against that Eisenstein series. Its functional equation is $\mathcal{R}(s)=\mathcal{R}(2k-1-s)$, and its only possible simple poles are at $k-1,k$. The raw series is obtained by dividing out the gamma and zeta factors, so it does not inherit the same unqualified pole statement; zeros of the denominator zeta factor can yield additional possible poles. The raw series is regular at $k-1$ and can have a simple pole at $k$.

If $f$ and $g$ have respective weights $k$ and $l$, and $r=k-l>2$, unfolding the weight-$r$ Eisenstein series gives