A Proof of the Riemann Hypothesis for the Weng Zeta Function of Rank 3 for the Rationals

Abstract

Let F be an algebraic number field. Recently, L. Weng introduced zeta functions of rank n associated to F as a generalization of the Iwasawa-Tate zeta integral from the Arakelov geometric point of view. In the article, we call them the Weng zeta function of rank n. In the case of rank one, the Weng zeta function coincide with the Dedekind zeta function of F . The background and precise definition of Weng zeta functions were published in [4, sec.B.4]. One remarkable fact for Weng zeta functions is that we can prove the Riemann hypothesis in the case of rank 2. It had been done by the author and J.C. Lagarias in [2] for the rational number field and was extended to the case of general number field F by Weng in [4, sec.C.4]. The proof of the Riemann hypothesis for a zeta function of rank 2 depends on the explicit expression for it. In this spring, Weng obtained an explicit expression for the zeta function of rank 3 in the case of F = Q. It is given in [5] this volume. By using his explicit formula, the author proved the Riemann hypothesis for the zeta function of rank 3 over the rational number field. In the article we give the proof of the Riemann hypohesis for it and the idea of the proof in self-contained fashion, as far as possible. The zeta function ζF,3(s) of rank 3 is obtained by an integral of the completed Epstein zeta function of rank 3 over a moduli space of semi-stable lattices of rank 3 over F . In detail, see [5, sec.5.3, chap.9] (in [5] our ζF,3(s) is denoted by ξF,3(s)). In the case F = Q, Theorem 4 in [5] assert that the explicit expression for ζQ,3(s) is given by