Fredholm Theory and Optimal Test Functions for Detecting Central Point Vanishing Over Families of L-functions. (arXiv:1708.01588v1 [math.NT])

The Riemann Zeta-Function is the most studied L-function; it's zeroes give information about the prime numbers. We can associate L-functions to a wide array of objects, and in general, the zeroes of these L-functions give information about those objects. For arbitrary L-functions, the order of vanishing at the central point is of particular important. For example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing at the central point of an elliptic curve L-function is the rank of the Mordell-Weil group of that elliptic curve. The Katz-Sarnak Density Conjecture states that this order vanishing and other behavior are well-modeled by random matrices drawn from the classical compact groups. In particular, the conjecture states that an average order vanishing over a family of L-functions can be bounded using only a given weight function and a chosen test function, phi. The conjecture is known for many families when the test functions are suitably restricted. It is 查看全文>>