Counting Bundles

TL;DR

This paper introduces new types of zeta functions associated with semi-stable bundles and reductive groups, exploring their fundamental properties, conjectures on zeros, and potential for uniformity in number theory.

Contribution

It defines genuine zetas of two types, establishes their basic properties, and proposes conjectures on their zeros and uniformity, advancing the understanding of zeta functions in algebraic geometry and number theory.

Findings

Established rationality and functional equations for the new zetas

Proposed conjectures on zeros and uniformity of the zetas

Introduced group zetas for reductive groups and semi-stable bundles

Abstract

We introduce new genuine zetas. There are two types, i.e., the pure non- abelian zetas defined using semi-stable bundles, and the group zetas defined for reductive groups. Basic properties such as rationality and functional equation are obtained. Moreover, conjectures on their zeros and uniformity are given.