Integrals of Z-functions

Number Theory Seminar

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The Z-function associated to an L-function is a rotated real-valued function defined so that |Z(t)| = |L(1/2 + it)|, so that critical zeros of L(s) are real zeros, and presumably sign changes, of Z(t). For the zeta function, the integral of Z(t) is well-understood. I'll discuss the integral for (mostly) Dirichlet L-functions, and some phenomena in the spacings of zeros of zeta and L-functions discovered along the way. These phenomena seem to be governed by the vanishing/non-vanishing of certain Gauss-type sums.
This is joint work with Zhenchao Ge and Micah Milinovich.

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