## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Tuesday, September 19, 2023 10:30 AM EDT

**Zhenchao Ge, Department of Pure Mathematics, University of Waterloo**

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

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. IviÄ‡Â 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

Event tags

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

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