= Rankin–Selberg method
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= Rankin–Selberg unfolding
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The <Rankin–Selberg method> relates arithmetic <Dirichlet series> to integrals of <automorphic forms> against <Eisenstein series>. Expanding the <Eisenstein series> as a sum over a <coset> space lets one replace an integral over a <fundamental domain> by an integral over a simpler region. This step is called unfolding.
For a weight-$k$ level-one <cusp form> $f(\tau)=\sum_{n\ge1}a_ne^{2\pi in\tau}$, the <primitive real-analytic Eisenstein series> $E(\tau,w)$ gives, initially for large $\operatorname{Re}w$,
$$
\int_{\Gamma(1)\backslash\mathbb H}|f(\tau)|^2y^k E(\tau,w)\,\frac{dx\,dy}{y^2}
=\frac{\Gamma(w+k-1)}{(4\pi)^{w+k-1}}\sum_{n\ge1}\frac{|a_n|^2}{n^{w+k-1}}.
$$
Unfolding produces the strip $0\le x<1$, integration in $x$ removes unequal <Fourier modes>, and integration in $y$ supplies the <gamma function>. The <Rankin–Selberg integral for holomorphic cusp forms> makes this link precise for two forms. Analytic properties of the <Eisenstein series> then control the coefficient <Dirichlet series>. For real $w>1$, positivity also proves convergence through <positive unfolding of a cusp-form square>; analytic continuation alone would not justify convergence of the original sum.
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