Biases in gaps between zeros of Dirichlet L-functions
Abstract
We describe a new family of Dirichlet L-functions that provably have an unusual value distribution. Experimentally, we find that the L-functions in this family have a significant and previously undetected bias in the distribution of gaps between their zeros. These phenomena appear to have an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for which this sum is nonzero and count them. This sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (a set of density zero among all primitive characters). The nonvanishing of this sum implies that there is a main term when one integrates the Z-function associated to a Dirichlet L-function over a long interval. We prove that the L-functions in this family are the only GL(1)/Q or GL(2)/Q cuspidal automorphic L-functions whose Z-functions have a "large" mean value when integrated. We discuss analogous results for the Riemann zeta-function.
Related
- Small and large gaps between zeros of Dirichlet L-functions
- Zeros of quadratic Dirichlet L-functions
- Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions
- Half-sized gaps between zeros of the Riemann zeta function
- Gevrey localization for simple and distinct zeros in a prime-modulus Dirichlet family
BibTeX
@misc{bober2026biases,
title = {{Biases in gaps between zeros of Dirichlet L-functions}},
author = {Jonathan W. Bober and Zhenchao Ge and Micah B. Milinovich},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31744},
eprint = {2610.31744},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31744},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Biases in gaps between zeros of Dirichlet L-functions
AU - Jonathan W. Bober
AU - Zhenchao Ge
AU - Micah B. Milinovich
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31744
UR - https://arxiv.org/abs/2610.31744
AB - We describe a new family of Dirichlet L-functions that provably have an unusual value distribution. Experimentally, we find that the L-functions in this family have a significant and previously undetected bias in the distribution of gaps between their zeros. These phenomena appear to have an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for which this sum is nonzero and count them. This sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (a set of density zero among all primitive characters). The nonvanishing of this sum implies that there is a main term when one integrates the Z-function associated to a Dirichlet L-function over a long interval. We prove that the L-functions in this family are the only GL(1)/Q or GL(2)/Q cuspidal automorphic L-functions whose Z-functions have a "large" mean value when integrated. We discuss analogous results for the Riemann zeta-function.
ER -