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

In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.Comment: 14 page

Subword complexity and Laurent series with coefficients in a finite field

Decimal expansions of classical constants such as $\sqrt2$, $\pi$ and $\zeta(3)$ have long been a source of difficult questions. In the case of Laurent series with coefficients in a finite field, where no carry-over difficulties appear, the situation seems to be simplified and drastically different. On the other hand, Carlitz introduced analogs of real numbers such as $\pi$, $e$ or $\zeta(3)$. Hence, it became reasonable to enquire how "complex" the Laurent representation of these "numbers" is. In this paper we prove that the inverse of Carlitz's analog of $\pi$, $\Pi_q$, has in general a linear complexity, except in the case $q=2$, when the complexity is quadratic. In particular, this implies the transcendence of $\Pi_2$ over \F_2(T). In the second part, we consider the classes of Laurent series of at most polynomial complexity and of zero entropy. We show that these satisfy some nice closure properties

Algebraic independence of arithmetic gamma values and Carlitz zeta values

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for F_q[theta] and provide complete algebraic independence results for them.Comment: 15 page

Prolongations of t-motives and algebraic independence of periods

In this article we show that the coordinates of a period lattice generator of the $n$-th tensor power of the Carlitz module are algebraically independent, if $n$ is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for $t$-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another ingredient is a theorem which shows hypertranscendence for the Anderson-Thakur function $\omega(t)$, i.e. that $\omega(t)$ and all its hyperderivatives with respect to $t$ are algebraically independent.Comment: 21 pages; v1->v2: extended the basic notation for better readability, corrected typos; final version to appear in Documenta Mathematic