The mean square of Dirichlet series associated with automorphic forms
The author considers real analytic automorphic forms of real weight \(k\), multiplier system \(\chi\) and eigenvalue \(\lambda={1\over 4}-\rho^ 2\) \((\text{Re} \rho\geq 0)\). Theorem 4.1 is an extension of the well-known Hecke-Maaß correspondence between automorphic forms and pairs of Dirichlet series \(L^ \pm(f,s)\) with functional equation. The main results of the work under review are mean square estimates for the Dirichlet series \(L^ \pm(f,s)\). If \(f\) is not a cusp form, the author proves that \[ \int^ T_ 0| L^ \pm(f,\text{Re} \rho+it)|^ 2\;dt=T(\log T)^ \alpha(B^ \pm+O((\log T)^{-1})) \] where \(B^ \pm\) are constants and \(\alpha\) is either 1, 2 or 4, provided that the underlying group \(\Gamma\) is a congruence subgroup of \(SL_ 2(\mathbb{Z})\). If \(\Gamma\) is a finitely generated Fuchsian group of the first kind with cusps \(\infty,0\), the same result holds with \(o(1)\) instead of \(O((\log T)^{-1})\), and there are similar results for cusps forms. These results are applied to the Dirichlet series associated with theta functions corresponding to positive definite or indefinite quadratic forms, that is, to Epstein zeta-functions and Siegel zeta-functions. In the case of the Riemann zeta-function, the author deduces Ingham's fourth power moment estimate from his result. (The reviewer is angry at the fact that in this work on real analytic automorphic forms the names of H. Maaß and W. Roelcke appear only in wrong orthography -- and even A. Selberg lost his ``g'' at least once).
scientific article; zbMATH DE number 164940
scientific article