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Thursday, October 24, 2019

**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.