Absolute convergence of cusp-form L-series from square coefficients
= Absolute convergence of cusp-form L-series from square coefficients
{title2=$\sum_{n\geq1}|a_n|n^{-\sigma}<\infty\quad(\sigma>(k+1)/2)$}
By <positive unfolding of a cusp-form square>, $\sum|a_n|^2n^{-\beta}<\infty$ 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
$$
\left(\sum_{n\geq1}|a_n|^2n^{-\beta}\right)^{1/2}\left(\sum_{n\geq1}n^{-(2\sigma-\beta)}\right)^{1/2}<\infty.
$$