Multiple arithmetic functions in function fields (Q2020001)

Let \({\mathbb F}\) be a field of characteristic \(p\ge 0\) and let \({\mathbb K}[T]_+\) be the set of monic polynomials in the polynomial ring \({\mathbb K}[T]\) where \({\mathbb K}\) is a field. The author defines the Dirichlet product of \({\mathbb F}\)-valued arithmetic functions on \({\mathbb K}[T]_{+}\) and considers the ring of \({\mathbb F}\)-valued arithmetic functions. The main purpose of the paper is to give an analogue of the \(ABC\)-conjecture in this ring. This is a function field analogue of a result of \textit{A. Zaharescu} and \textit{M. Zaki} [J. Ramanujan Math. Soc. 25, No. 4, 345--354 (2010; Zbl 1247.11009)]. He proves that there exist nonzero solutions \((x,y,z)\) to Fermat's equation \(x^n+y^n=z^n\) and in fact infinitely many of them for any \(n\) in this ring. He also proves that for each positive integers \(m,~n\) there are infinitely many solutions \((x,y)\) to the Catalan equation \(x^m-y^n=1\) in this ring.