Fourier coefficients of cusp forms and automorphic f-functions

Abstract

In the beginning of the paper, we review the summation formulas for the Fourier coefficients of holomorphic cusp forms for γ, γ=SL(2,ℤ), associated with L-functions of three and four Hecke eigenforms. Continuing the known works on the L-functions Lf,ϕ,ψ(s) of three Hecke eigenforms, we prove their new properties in the special case of Lf,ϕ,ψ(s). These results are applied to proving an analogue of the Siegel theorem for the L-function Lf(s) of the Hecke eigenform f(z) for γ (with respect to weight) and to deriving a new summation formula. Let f(z) be a Hecke eigenform for γ of even weight 2k with Fourier expansion\(f(z) = \Sigma _{n = 1}^\infty a(n)e^{2\pi inz} \). We study a weight-uniform analogue of the Hardy problem on the behavior of the sum ⌆p≤xa(p) log p and prove new estimates from above for the sum ⌆n≤xa(F(n))2, where F(x) is a polynomial with integral coefficients of special form (in particular, F(x) is an Abelian polynomial). Finally, we obtain the lower estimate $$L_4 (1) + |L'_4 (1)| \gg \frac{1}{{(\log k)^c }},$$ where L4(s) is the fourth symmetric power of the L-function Lf(s) and c is a constant. Bibliography: 43 titles.