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 L_f,ϕ,ψ(s) of three Hecke eigenforms, we prove their new properties in the special case of L_f,ϕ,ψ(s). These results are applied to proving an analogue of the Siegel theorem for the L-function L_f(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 L_f(s) and c is a constant. Bibliography: 43 titles.