Let be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function as x → ∞, where q grows with x in a definite way and j = 2,3,4.
Let be a normalized primitive holomorphic cusp form of even integral weight for the full modular group . Denote by the th normalized Fourier coefficient of . We are interested in the average behaviour of the sum for , where and is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power -functions and Rankin-Selberg -functions.
Nous rappelons que Manin décrit l’homologie singulière relative aux pointes de la courbe modulaire comme un quotient du groupe . En s’appuyant sur des techniques de fractions continues, nous donnons une expression indépendante de d’un relèvement de l’action des opérateurs de Hecke de sur .
This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.
We study so-called real zeros of holomorphic Hecke cusp forms, that is, zeros on three geodesic segments on which the cusp form (or a multiple of it) takes real values. Ghosh and Sarnak, who were the first to study this problem, showed the existence of many such zeros if many short intervals contain numbers whose prime factors all belong to a certain subset of the primes.We prove new results concerning this sieving problem which leads to improved lower bounds for the number of real zeros.