Trivial zeros of zeta functions of type $\mathrm{A}_r$

Source 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 $\mathrm{A}_r$. 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 $\mathrm{A}_r$ 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 $\mathrm{A}_r$ at non-positive integer points and a possible connection of such zeros and non-trivial identities among classical Eisenstein series.