AUTOMORPHIC L-FUNCTIONS IN THE WEIGHT ASPECT


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Abstract

Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). LetIt is proved that for large Kwhere ∈ > 0 is arbitrary. For f ∈ Sk(Γ)+, let L(s, sym2 f) denote the symmetric square L-function. It is proved that as k → ∞ the frequenceconverges to a distribution function G(x) at every point of continuity of the latter, and for the corresponding characteristic function an explicit expression is obtained. Bibliography: 17 titles.

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