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Irrationality of values of L-functions for real Dirichlet characters
Abstract
Let χ be a non-trivial real Dirichlet character and s be a sufficiently large integer. In this paper, we prove that among the first s non-critical special values of the Dirichlet L-function associated with χ at least c√(s/(log s)) are irrational, where c is a constant independent of s. This is a substantial improvement on work of Fischler when $4$ divides the conductor of χ. Otherwise, our constant improves on work of Mistiaen. The proof adapts some techniques of Lai and Yu for odd zeta values to the case of L-functions. The lack of positivity arguments is overcome by applying an irrationality criterion by Lai and Sprang originating from the p-adic setting.
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BibTeX
@misc{kumar2026irrationality,
title = {{Irrationality of values of $L$-functions for real Dirichlet characters}},
author = {Veekesh Kumar and Riccardo Tosi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41458},
eprint = {2610.41458},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41458},
}
RIS
TY - UNPB
TI - Irrationality of values of L-functions for real Dirichlet characters
AU - Veekesh Kumar
AU - Riccardo Tosi
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41458
UR - https://arxiv.org/abs/2610.41458
AB - Let χ be a non-trivial real Dirichlet character and s be a sufficiently large integer. In this paper, we prove that among the first s non-critical special values of the Dirichlet L-function associated with χ at least c√(s/(log s)) are irrational, where c is a constant independent of s. This is a substantial improvement on work of Fischler when $4$ divides the conductor of χ. Otherwise, our constant improves on work of Mistiaen. The proof adapts some techniques of Lai and Yu for odd zeta values to the case of L-functions. The lack of positivity arguments is overcome by applying an irrationality criterion by Lai and Sprang originating from the p-adic setting.
ER -