On locally repeated values of arithmetic functions over Fq[T]

Abstract

AbstractThe frequency of occurrence of ‘locally repeated’ values of arithmetic functions is a common theme in analytic number theory, for instance in the Erdős–Mirsky problem on coincidences of the divisor function at consecutive integers, the analogous problem for the Euler totient function and the quantitative conjectures of Erdős, Pomerance and Sarkőzy and of Graham, Holt and Pomerance on the frequency of occurrences. In this paper, we introduce the corresponding problems in the setting of polynomials over a finite field, and completely solve them in the large finite field limit.