Positive unfolding of a cusp-form square (${\sum }_{n\ge 1}|a_n{|}^2{n}^{-s}<\infty (s>k)$)

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