arXiv subjects

Simon Rutard

Publications and source records attributed to Simon Rutard.

3 recordsLinked to original sources

arXiv2026

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

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.

math.NT↗

arXiv2025

Values and derivative values at nonpositive integers of generalized multiple Hurwitz zeta functions

We establish the meromorphic continuation of certain multiple zeta functions of generalized Hurwitz type. From this meromorphic continuation, we obtain explicit formulas for their (derivative) values at nonpositive integers along a given direction. As an application, we provide explicit formulas for some values and derivative values of the Witten zeta functions $ζ_{\mathfrak{g}_2}$ and $ζ_{\mathfrak{so}(5)}$. Furthermore, by employing a Meinardus-type theorem, we investigate the asymptotic behavior of the number of $n$-dimensional representations of the exceptional Lie algebra $\mathfrak{g}_2$.

math.NT↗

arXiv2025

Values at non-positive integers of partially twisted multiple zeta-functions II

We study the values at non-positive integer points of multi-variable twisted multiple zeta-functions, whose each factor of the denominator is given by polynomials. The fully twisted case was already answered by de Crisenoy. On the partially twisted case, in one of our former article we studied the case when each factor of the denominator is given by linear forms or power-sum forms. In the present paper we treat the case of general polynomial denominators, and obtain explicit forms of the values at non-positive integer points. Our strategy is to reduce to the theorem of de Crisenoy for the fully twisted case, via the multiple Mellin-Barnes integral formula. We observe that in some cases the obtained values are transcendental.

math.NT↗