The Riemann hypothesis for Weng's zeta function of [formula omitted] over [formula omitted

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 a semi-simple reductive algebraic group and its maximal parabolic subgroup. As well as the high rank zeta function, the new zeta function satisfies standard properties of zeta functions. In this paper, we prove that the Riemann hypothesis for Weng's zeta function attached to the symplectic 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 ( 2 n ) .