Geometric Gamma Values and Zeta Values in Positive Characteristic

In analogy with values of the classical Euler Γ-function at rational numbers and the Riemann ζ-function at positive integers, we consider Thakur’s geometric Γ-function evaluated at rational arguments and Carlitz ζ-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 Γ-function and from the Euler-Carlitz relations and Frobenius p-th power relations of the ζ-function.