On Dirichlet Products Evaluated at Fibonacci Numbers

TL;DR

This paper explores properties of Dirichlet products evaluated at Fibonacci numbers, deriving new representations, bounds, and fixed points related to primitive prime divisors and Euler functions.

Contribution

It introduces novel representations of Fibonacci numbers using Euler's totient function and establishes bounds and fixed points involving primitive prime divisors.

Findings

Representation of Fibonacci numbers via Euler's totient function

Upper bounds on the number of primitive prime divisors

Identification of a non-trivial fixed point of a sum over primitive divisors

Abstract

In this work we discuss Dirichlet products evaluated at Fibonacci numbers. As first applications of the results we get a representation of Fibonacci numbers in terms of Euler's totient function, an upper bound on the number of primitive prime divisors and representations of some related Euler products. Moreover, we sum functions over all primitive divisors of a Fibonacci number and obtain a non--trivial fixed point of this operation.