= Positive unfolding of a cusp-form square
{title2=$\sum_{n\geq1}|a_n|^2n^{-s}<\infty\quad(s>k)$}
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 $\int_0^1|f(x+iy)|^2dx=\sum_{n\geq1}|a_n|^2e^{-4\pi ny}$. 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.
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