Trivial zeros of zeta functions of type Ar
Abstract
In this paper, we investigate the values of zeta functions of root systems at non-positive integer points. In particular, we focus on the zeta functions of type Ar. Since non-positive integer points are included in the set of singularities of zeta functions of root systems, we introduce the \emph{ordered limit values} to define those values. We study four particular ordered limit values and give explicit formulas for them, or recurrence formulas that allow us to calculate them inductively. We then prove that three ordered limit values of zeta functions of type Ar vanish under some conditions that can be regarded as a multivariable analogue of the trivial zeros of the Riemann zeta function. Finally, we discuss the other zeros of zeta functions of type Ar at non-positive integer points and a possible connection of such zeros and non-trivial identities among classical Eisenstein series.
Related
- Proportions of the non-trivial zeros of the Riemann zeta function
- A Direct Proof of the Simplicity of Non-Trivial Zeros of the Riemann Zeta Function
- Low Lying Zeros of Dedekind Zeta Functions
- Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions
- Value-distribution of Euler--Zagier type multiple zeta-functions with identical arguments on the critical line
BibTeX
@misc{rutard2026trivial,
title = {{Trivial zeros of zeta functions of type $\mathrmA_r$}},
author = {Simon Rutard and Takeshi Shinohara},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28354},
eprint = {2609.28354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28354},
}
RIS
TY - UNPB
TI - Trivial zeros of zeta functions of type A_r
AU - Simon Rutard
AU - Takeshi Shinohara
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28354
UR - https://arxiv.org/abs/2609.28354
AB - In this paper, we investigate the values of zeta functions of root systems at non-positive integer points. In particular, we focus on the zeta functions of type A_r. Since non-positive integer points are included in the set of singularities of zeta functions of root systems, we introduce the \emph{ordered limit values} to define those values. We study four particular ordered limit values and give explicit formulas for them, or recurrence formulas that allow us to calculate them inductively. We then prove that three ordered limit values of zeta functions of type A_r vanish under some conditions that can be regarded as a multivariable analogue of the trivial zeros of the Riemann zeta function. Finally, we discuss the other zeros of zeta functions of type A_r at non-positive integer points and a possible connection of such zeros and non-trivial identities among classical Eisenstein series.
ER -