Title: Non-vanishing of Dirichlet L-functions
Speaker: Rizwanur Khan - University of Mississippi
Abstract: L-functions are fundamental objects in number theory. At the central point s = 1/2, an L-function L(s) is expected to vanish only if there is some deep arithmetic reason for it to do so (such as in the Birch and Swinnerton-Dyer conjecture), or if its functional equation specialized to s = 1/2 implies that it must. Thus when the central value of an L-function is not a "special value", and when it does not vanish for trivial reasons, it is conjectured to be nonzero. In general it is very difficult to prove such non-vanishing conjectures. For example, nobody knows how to prove that L(1/2, \chi) is nonzero for all primitive Dirichlet characters \chi. In such situations, analytic number theorists would like to prove 100% non-vanishing in the sense of density, but achieving any positive percentage is still valuable and can have important applications. In this talk, I will discuss work on establishing such positive proportions of non-vanishing for Dirichlet L-functions.
