The Riemann hypothesis for Weng's zeta function of $Sp(4)$ over $\mathbb{Q}$

TL;DR

This paper proves the Riemann hypothesis for Weng's zeta function associated with the symplectic group of degree four, extending previous results on high rank zeta functions and their properties.

Contribution

It establishes the Riemann hypothesis for a new class of zeta functions linked to the symplectic group Sp(4), generalizing earlier work on high rank and parabolic subgroup zeta functions.

Findings

Riemann hypothesis proven for Weng's zeta function of Sp(4)

Confirms standard properties of the zeta function in this context

Includes a general construction for zeta functions of Sp(2n)

Abstract

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function. For the rank two case, the Riemann hypothesis is proved for a general number field. Recently, he defined more general new zeta function associated to a pair of reductive group and its maximal parabolic subgroup. As well as high rank zeta functions, the new zeta function satisfies standard properties of zeta functions.In this paper, we prove that the Riemann hypothesis of Weng's zeta function attached to the sympletic group of degree four.This paper includes an appendix written by L. Weng, in which he explains a general construction for zeta functions associated to $Sp(2n)$.