Irrationality and transcendence questions in the "poor man's adèle ring"

Abstract

We discuss arithmetic questions related to the "poor man's adèle ring" $\mathcal{A}$ whose elements are encoded by sequences $(t_p{)}_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb{Z}/p\mathbb{Z}$. Our main theorem is about the $\mathcal{A}$-transcendence of the element $(F_p(q){)}_p$, where $F_n(q)$ (Schur's $q$-Fibonacci numbers) are the $(1,1)$-entries of $2\times 2$-matrices $$\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}1& 1\\ q& 0\end{array}\right)\left(\begin{array}{cc}1& 1\\ q^2& 0\end{array}\right)\cdots \left(\begin{array}{cc}1& 1\\ q^{n-2}& 0\end{array}\right)$$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH.

7 pages

References in corpus (2)

• On the distribution of the order and index of g(mod p) over residue classes II
• On finite analogues of Euler's constant