A rank two zeta and its zeros

Abstract

Theory of zeta functions plays a central role in arithmetic. In this paper, we use a new approach to study them. More precisely, we first reveal an intrinsic relation between higher rank zeta functions and Epstein zeta functions, and expose a fundamental relation between stability of lattices and distances of the corresponding modular points to cusps. Applying to rank two, we then explicitly express the associated zeta functions in terms of Dedekind zetas. Based on such an expression, finally, we show that all zeros of rank two zetas are entirely sitting on the critical line whose real part equals to 1 2 . As such, this work is built up on classics of number theory. Many fine pieces of algebraic and analytic number theory are beautifully unified under our zetas: