PreprintarXiv licencearXiv
Zeros of quadratic Dirichlet L-functions
Abstract
We consider Dirichlet L-functions $L(s, χ)$ where χ is a quadratic character to modulus q. We show that for χ even (resp. odd) L(σ, χ)≠ 0 for all σ>0 provided that q≤ 109 (resp. q≤ 1010). In particular, this result rules out Siegel--Landau zeros for such L-functions. Combined, these results improve on previous work by nearly four orders of magnitude. We also list some applications of this work.
Related
- Small and large gaps between zeros of Dirichlet L-functions
- Biases in gaps between zeros of Dirichlet L-functions
- Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions
- The first moment of quadratic Dirichlet L-functions in the even hyperelliptic ensemble
- Large values of derivatives of quadratic Dirichlet L-functions
BibTeX
@misc{platt2026zeros,
title = {{Zeros of quadratic Dirichlet $L$-functions}},
author = {Dave Platt and Tim Trudgian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29018},
eprint = {2610.29018},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29018},
}
RIS
TY - UNPB
TI - Zeros of quadratic Dirichlet L-functions
AU - Dave Platt
AU - Tim Trudgian
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29018
UR - https://arxiv.org/abs/2610.29018
AB - We consider Dirichlet L-functions $L(s, χ)$ where χ is a quadratic character to modulus q. We show that for χ even (resp. odd) L(σ, χ)≠ 0 for all σ>0 provided that q≤ 10^9 (resp. q≤ 10^10). In particular, this result rules out Siegel--Landau zeros for such L-functions. Combined, these results improve on previous work by nearly four orders of magnitude. We also list some applications of this work.
ER -