Rank Two Non-Abelian Zeta and Its Zeros

TL;DR

This paper establishes a connection between rank two non-abelian zeta functions and classical zeta functions, demonstrating that all their zeros lie on the critical line, advancing understanding in number theory.

Contribution

It explicitly expresses rank two non-abelian zeta functions in terms of Dedekind zeta functions and proves their zeros are on the critical line, linking stability and lattice geometry.

Findings

Zeros of rank two non-abelian zeta functions lie on the critical line.

Explicit relation between non-abelian and Dedekind zeta functions.

Connection between lattice stability and zeta function properties.

Abstract

In this paper, we first reveal an intrinsic relation between non-abelian zeta functions and Epstein zeta functions for algebraic number fields. Then, we expose a fundamental relation between stability of lattices and distance to cusps. Next, using these two relations, we explicitly express rank two zeta functions in terms of the well-known Dedekind zeta functions. Finally, based on such an expression, we show that all zeros of rank two non-abelian zeta functions are entirely sitting on the critical line whose real part equals to 1/2. This is an integrated part of our Geo-Arithmetic Program.