Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic

Abstract

By analogy with the Riemann zeta function at positive integers, for each finite field ${\mathbb{F}}_{p^r}$ with fixed characteristic $p$, we consider Carlitz zeta values ${\zeta }_r\left(n\right)$ at positive integers $n$. Our theorem asserts that among the zeta values in the set ${\bigcup }_{r=1}^{\infty }\left\{{\zeta }_r\left(1\right),{\zeta }_r\left(2\right),{\zeta }_r\left(3\right),\dots \right\}$, all the algebraic relations are those relations within each individual family $\left\{{\zeta }_r\left(1\right),{\zeta }_r\left(2\right),{\zeta }_r\left(3\right),\dots \right\}$. These are the algebraic relations coming from the Euler–Carlitz and Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.