On the index of appearance of a Lucas sequence

TL;DR

This paper investigates the distribution of primes related to Lucas sequences, establishing an asymptotic formula for the count of primes with a given index of appearance, under the Generalized Riemann Hypothesis.

Contribution

It provides a new asymptotic estimate for the distribution of primes in Lucas sequences based on their index of appearance, under GRH and mild assumptions.

Findings

Asymptotic formula for prime counts with fixed index of appearance

Explicit examples and numerical data included

Functions involved are effectively computable

Abstract

Let $\mathbf{u} = (u_n)_{n \geq 0}$ be a Lucas sequence, that is, a sequence of integers satisfying $u_0 = 0$, $u_1 = 1$, and $u_n = a_1 u_{n - 1} + a_2 u_{n - 2}$ for every integer $n \geq 2$, where $a_1$ and $a_2$ are fixed nonzero integers. For each prime number $p$ with $p \nmid 2a_2D_{\mathbf{u}}$, where $D_{\mathbf{u}} := a_1^2 + 4a_2$, let $\rho_{\mathbf{u}}(p)$ be the rank of appearance of $p$ in $\mathbf{u}$, that is, the smallest positive integer $k$ such that $p \mid u_k$. It is well known that $\rho_{\mathbf{u}}(p)$ exists and that $p \equiv \big(D_{\mathbf{u}} \mid p \big) \pmod {\rho_{\mathbf{u}}(p)}$, where $\big(D_{\mathbf{u}} \mid p \big)$ is the Legendre symbol. Define the index of appearance of $p$ in $\mathbf{u}$ as $\iota_{\mathbf{u}}(p) := \left(p - \big(D_{\mathbf{u}} \mid p \big)\right) / \rho_{\mathbf{u}}(p)$. For each positive integer $t$ and for every $x > 0$, let $\mathcal{P}_{\mathbf{u}}(t, x)$ be the set of prime numbers $p$ such that $p \leq x$, $p \nmid 2a_2 D_{\mathbf{u}}$, and $\iota_{\mathbf{u}}(p) = t$. Under the Generalized Riemann Hypothesis, and under some mild assumptions on $\mathbf{u}$, we prove that \begin{equation*} \#\mathcal{P}_{\mathbf{u}}(t, x) = A\, F_{\mathbf{u}}(t) \, G_{\mathbf{u}}(t) \, \frac{x}{\log x} + O_{\mathbf{u}}\!\left(\frac{x}{(\log x)^2} + \frac{x \log (2\log x)}{\varphi(t) (\log x)^2}\right) , \end{equation*} for all positive integers $t$ and for all $x > t^3$, where $A$ is the Artin constant, $F_{\mathbf{u}}(\cdot)$ is a multiplicative function, and $G_{\mathbf{u}}(\cdot)$ is a periodic function (both these functions are effectively computable in terms of $\mathbf{u}$). Furthermore, we provide some explicit examples and numerical data.