Zeta functions for $G_2$ and their zeros

TL;DR

This paper introduces two new zeta functions linked to the exceptional group G_2 and proves they satisfy the Riemann Hypothesis, advancing understanding of automorphic forms and L-functions.

Contribution

It defines two novel zeta functions for G_2 associated with its maximal parabolic subgroups and proves they satisfy the Riemann Hypothesis, a significant theoretical breakthrough.

Findings

Two new zeta functions for G_2 are introduced.

Both zeta functions are shown to satisfy the Riemann Hypothesis.

The results connect automorphic forms with deep number theory conjectures.

Abstract

The exceptional group $G_2$ has two maximal parabolic subgroups $P_{long}$, $P_{short}$ corresponding to the so-called long root and short root. In this paper, the second author introduces two zeta functions associated to $(G_2,P_{long})$ and $(G_2,P_{short})$ respectively, and the first author proves that these zetas satisfy the Riemann Hypothesis.