On summatory functions for automorphic L-functions

Abstract

Let λ f (n) denote the nth normalized Fourier coefficient of a primitive holomorphic cusp form f for the full modular group. Let Δ(x, f ⨂ f) be the error term in the asymptotic formula of Rankin and Selberg for $$ \sum\limits_{n \leqslant x} {{\lambda_f}{{(n)}^2}.} $$ It is proved that Δ(x, f ⨂ f) = Ω (x 3/8) and $$ \sum\limits_{n \leqslant x} {{\lambda_f}\left( {{n^2}} \right) = \Omega \left( {{x^{{{1} \left/ {3} \right.}}}} \right).} $$ Other summatory functions associated with automorphic L-functions also are studied. Bibliography: 22 titles.