On locally repeated values of arithmetic functions over $\mathbb F_q[T]$

TL;DR

This paper investigates the frequency of locally repeated values of arithmetic functions over polynomial rings in finite fields, providing complete solutions in the large finite field limit, thus extending classical number theory problems to a new algebraic setting.

Contribution

It introduces and solves the problem of locally repeated values of arithmetic functions over $\\mathbb{F}_q[T]$ in the large finite field limit, extending classical number theory results.

Findings

Complete solutions for the frequency of locally repeated values in polynomial rings over finite fields.

Extension of classical number theory problems to the setting of polynomials over finite fields.

Results hold in the large finite field limit.

Abstract

The frequency of occurrence of "locally repeated" values of arithmetic functions is a common theme in analytic number theory, for instance in the Erd\H{o}s-Mirsky problem on coincidences of the divisor function at consecutive integers, the analogous problem for the Euler totient function, and the quantitative conjectures of Erd\H{o}s, Pomerance and Sark\H{o}zy 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.