In this article, we prove non-vanishing results for L-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen [4] to forms of half-integral weight.

In this paper, we present a quantitative result for the number of sign changes for the sequences {a(n j )} n≥1 , j = 2, 3, 4 of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group SL 2 (Z). We also prove a similar kind of quantitative result for the number of sign changes of the q-exponents c(p) (p vary over primes) of certain generalized modular functions for the congruence subgroup Γ 0 (N ), where N is square-free.

In this article we give a description of the kernel of the restriction map for Jacobi forms of index 2 and obtain the injectivity of D 0 ⊕ D 2 on the space of Jacobi forms of weight 2 and index 2. We also obtain certain generalization of these results on certain subspace of Jacobi forms of square-free index m.

Abstract. We study the signs of the Fourier coefficients of a newform. Let f be a normalized newform of weight k for Γ 0 (N ). Let a f (n) be the nth Fourier coefficient of f . For any fixed positive integer m, we study the distribution of the signs of {a f (p m )} p , where p runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power Lfunctions.