## 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.
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