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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 $f(\tau )={\sum }_{n\ge 1}{a}_n{e}^{2\pi in\tau }$, the primitive real-analytic Eisenstein series $E(\tau ,w)$ gives, initially for large $\mathrm{Re}w$,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.