Speaker: Prof. Zhou fan Venue: B1044 Time: 14:00-14:50, 15:00-15:50   Dec. 29 Title: New Types of Voronoi Formulae on GL(n) Abstract: The Voronoi formula is an analog of Poisson summation formula for automorphic forms. Previously, one Voronoi formula was known for Maass forms on GL(n). It has great application in analytic number theory of automorphic forms and their L-functions. We discover new types of Voronoi formulae for automorphic forms on GL(n) for n>=4. There are [n/2] different Voronoi formulae on GL(n), which are summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums of dimension k=1,2,...,[n/2]. Title: Hecke Eigenvalues of Maass Forms and Ramanujan Conjecture Abstract: Let us have a primitive Hecke-Maass cusp form of level N with the Laplacian eigenvalue 1/4+t^2. In this talk, we show that there exists a prime p (does not divide N) such that the Ramanujan conjecture holds at p and p < < (nt)^c for some absolute constant c>0. In fact, c can be taken as 0.27332. This can be viewed as a Linnik's problem for the Ramanujan conjecture. In addition, we prove that the natural density of such primes p at which the Ramanujan conjecture holds is at least 34/35. This is an improvement over an earlier similar result of Kim and Shahidi in Dirichlet density. Both results are joint work with Wenzhi Luo.