Irregularities of Dirichlet L-functions and a parity bias in gaps of zeros

Zhenchao Ge, Department of Pure Mathematics, University of Waterloo

The integral of Hardy's Z-function from $0$ to $T$ measures the occurrence of its sign changes. Hardy proved that this integral is $o(T)$ from which he deduced that the Riemann zeta-function has infinitely many zeros on the critical line. A. Ivić conjectured this integral is $O(T^{1/4})$ and $\Omega_{\pm}(T^{1/4})$ as $T\to\infty$. These estimates were proved, independently, by M.A. Korolev and M. Jutila.

In this talk, we will show that the analogous conjecture is false for the Z-functions of certain "special" Dirichlet L-functions. In particular, we show that the integral of the Z-function of a Dirichlet L-functions from $0$ to $T$ is asymptotic to $c_\chi T^{3/4}$ and we classify precisely when the constant $c_\chi$ is nonzero. Somewhat surprisingly, numerical evidence seems to suggest that the unexpectedly large mean value is caused by a currently unexplained parity bias in the gaps between the zeros of these "special" Dirichlet L-functions.

This is joint work with Jonathan Bober and Micah Milinovich.

MC 5501