## PIMS- U of Lethbridge Number Theory and Combinatorics Seminar: Micah Milinovich

- Date: 01/28/2021
- Time: 12:00

Online

Irregularities of Dirichlet L-functions and a Chebyshev-type bias for zeros

We describe a thin family of Dirichlet L-functions which have an irregular and perhaps unexpected behavior in their value distribution. This behavior has an arithmetic explanation and corresponds to the nonvanishing of a certain Gauss type sum. We give a complete classification of the characters for which these sums are nonzero and count the number of corresponding characters. It turns out that this Gauss type sum vanishes for 100% of primitive Dirichlet characters but there is an infinite (but zero density) subfamily of characters where the sum is nonzero.

Experimentally, this thin family of L-functions seems to have a significant and previously undetected bias in distribution of gaps between their zeros. After uncovering this bias, we re-examined the gaps between the zeros of the Riemann zeta-function and discovered an even more surprising phenomenon. If we list the gaps in increasing order and sum over arithmetic progressions of gaps, there seems to be a "Chebyshev-type" bias in the corresponding measures; the sum over certain arithmetic progressions of gaps are much larger than others! These observations seem to go well beyond the Random Matrix Theory model of L-functions.

This is joint work with Jonathan Bober and Zhenchao Ge.

Talks are usually at noon on Monday. All times are Mountain Time. For meeting details, contactÂ Nathan Ng