Geometric Gamma values and zeta values in positive characteristic

TL;DR

This paper explores the algebraic relations among special values of Thakur's geometric Gamma-function and Carlitz zeta-values in positive characteristic, showing they originate from fundamental functional equations and relations.

Contribution

It demonstrates that all algebraic relations among these special values are derived from known functional equations and Frobenius relations, clarifying their algebraic structure.

Findings

All algebraic relations arise from standard functional equations.

Relations are governed by Euler-Carlitz and Frobenius p-th power relations.

The work parallels classical results in positive characteristic setting.

Abstract

In analogy with values of the classical Euler Gamma-function at rational numbers and the Riemann zeta-function at positive integers, we consider Thakur's geometric Gamma-function evaluated at rational arguments and Carlitz zeta-values at positive integers. We prove that, when considered together, all of the algebraic relations among these special values arise from the standard functional equations of the Gamma-function and from the Euler-Carlitz relations and Frobenius p-th power relations of the zeta-function.