Explicit Prime Densities for the Rank of Appearance in Lucas Sequences

TL;DR

This paper derives explicit formulas for the density of primes with a given divisibility property related to the rank of appearance in Lucas sequences, extending previous work to all sequences and divisors.

Contribution

It provides comprehensive closed-form formulas for prime densities in Lucas sequences, completing prior research by covering all sequences and divisors.

Findings

Derived closed-form formulas for prime densities

Extended results to all Lucas sequences and divisors

Complete the characterization of prime distributions in this context

Abstract

Let $U$ be a Lucas sequence, $p$ be prime, and $\rho_U(p)$ be the rank of appearance of $p$ in $U$. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\mid \rho_U(p)$, where $d\geq 1$ is a fixed integer. Our results complete the work of Sanna ($2022$) by covering all $U$ and all $d\geq 1$.