On the mean square of Hecke \(L\)-functions associated to holomorphic cusp forms (Q2707561)

This paper brings forth a simple approach to the spectral decomposition of the weighted integral NEWLINE\[NEWLINE \int_{-\infty}^\infty |H_A(\textstyle{1\over 2} + it)|^2w(t) dt. NEWLINE\]NEWLINE Here \(H_A(s) = \sum_{n\geq 1}a(n)n^{-(s+ {k-1\over 2})}\) is the Hecke \(L\)-function associated to the holomorphic cusp form \(A = A(z) = \sum_{n\geq 1}a(n)e(nz)\) of weight \(k\) (\(k\) even and \(\geq 12\)), and \(w(t)\) is a suitable weight function. The basic idea is to evaluate in two different ways the Mellin transform NEWLINE\[NEWLINE I(s) = \int_0^\infty y^{s-1}J(y) dy,\quad J(y) = \langle|A|^2, \overline{K_\psi(iy,\cdot)}\rangle_k, NEWLINE\]NEWLINE where \(K_\psi\) is a suitable kernel function, precisely defined in the text. The first way is by the unfolding method, and the second by using the spectral decomposition.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].