The Rankin-Selberg method for non-holomorphic automorphic forms (Q1804189)

Let \(\Gamma\) be a Fuchsian group of the first kind, with at least one cusp. The author considers the Rankin-Selberg convolution for \(\Gamma\)-automorphic forms of arbitrary real weight. In this case, two types of Rankin-Selberg zeta-functions arise, corresponding to the Fourier coefficients at positive and negative indexes, respectively. The author proves the analytic continuation and a function equation for these zeta-functions. As an application he proves asymptotic formulas for the summatory function for the squares of the Fourier coefficients. The author uses methods which can also be found in earlier papers of the reviewer [Math. Z. 211, 155-172 (1992; Zbl 0778.11033), Acta Arith. 65, 1-14 (1993; Zbl 0790.11037)], where the same kind of results are obtained. The reviewer is somewhat surprised that the author should not have had knowledge about this.