Non-vanishing Of L-functions At The Central Point

The non-vanishing of L-functions at the central point is an important sub-ject in contemporary number theory,which has important implications in Birch-Swinnerton-Dyer conjecture,spectral deformation theory and classical analytic num-ber theory.Currently,a typical approach in the study of non-vanishing problems is to consider non-vanishing in a family of L-functions at the central point.An effec-tive way is to establish asymptotic formulas for mean value estimates.In this thesis,we consider the non-vanishing of the first derivative of Rankin-Selberg L-functions on GL(3)x GL(2)at the central point and the non-vanishing of the product of two twisted L-functions at the central point.By establishing asymptotic formulas for their mean value estimates,we obtain non-vanishing results accordingly.We first study the mean value estimate of the first derivative of Rankin-Selberg L-functions on GL(3)x GL(2)at the central point.Let f be a fixed self-dual Hecke-Maass cusp form for SL(3,Z)and {μj} be an orthogonal basis of odd Hecke-Maass cusp forms for SL(2,Z).We establish an asymptotic formula for the average of the first-derivative of the Rankin-Selberg L-functions of f and μj at the central point.This implies the non-vanishing result for the first derivative of these L-functions at the central point.Our techniques include Kuznetsov formula for an orthogonal basis of odd Hecke-Maass cusp forms,Voronoi summation formula for GL(3)and stationary phase method.We next study the mean value estimate of the product of two twisted L-functions at the central point.Suppose that q and r are two distinct large primes.Let g be a cuspidal Hecke eigenform of level 1 and even weight k1 and Hk2(q)be the set of cus-pidal Hecke eigenforms of level q and even weight k2.We establish an asymptotic formula for the averages of the product of the central values of two L-functions of modular forms f∈Hk2(q)and g twisted by primitive Dirichlet character of modular r under certain assumptions.This implies the non-vanishing result for these prod-ucts of twisted L-functions at the central point.To prove this result,we will use the same trick as Kiral-Young[26]on Bessel functions,balanced and unbalanced ap-proximate functional equations,Petersson formula and Voronoi summation formula for GL(2).