On the index of appearance of a Lucas sequence

Let \(\varvec{u} = (u_n)_{n \ge 0}\) $\mathit{u}={(u_n)}_{n\ge 0}$ be a Lucas sequence, that is, a sequence of integers satisfying \(u_0 = 0\) $u_0=0$ , \(u_1 = 1\) $u_1=1$ , and \(u_n = a_1 u_{n - 1} + a_2 u_{n - 2}\) $u_n=a_1{u}_{n-1}+a_2{u}_{n-2}$ for every integer \(n \ge 2\) $n\ge 2$ , where \(a_1\) $a_1$ and \(a_2\) $a_2$ are fixed nonzero integers. For each prime number p with \(p \not \mid 2a_2D_{\varvec{u}}\) $p\nmid 2a_2{D}_{\mathit{u}}$ , where \(D_{\varvec{u}}:= a_1^2 + 4a_2\) $D_{\mathit{u}}:=a_1^2+4a_2$ , let \(\rho _{\varvec{u}}(p)\) ${\rho }_{\mathit{u}}(p)$ be the rank of appearance of p in \(\varvec{u}\) $\mathit{u}$ , that is, the smallest positive integer k such that \(p \mid u_k\) $p\mid u_k$ . It is well known that \(\rho _{\varvec{u}}(p)\) ${\rho }_{\mathit{u}}(p)$ exists and that \(p \equiv \big (D_{\varvec{u}} \mid p \big ) \pmod {\rho _{\varvec{u}}(p)}\) $p\equiv (D_{\mathit{u}}\mid p)(mod{\rho }_{\mathit{u}}(p))$ , where \(\big (D_{\varvec{u}} \mid p \big )\) $(D_{\mathit{u}}\mid p)$ is the Legendre symbol. Define the index of appearance of p in \(\varvec{u}\) $\mathit{u}$ as \(\iota _{\varvec{u}}(p):= \left( p - \big (D_{\varvec{u}} \mid p \big )\right) / \rho _{\varvec{u}}(p)\) ${\iota }_{\mathit{u}}(p):=\left(p,-,(,D_{\mathit{u}},\mid ,p,)\right)/{\rho }_{\mathit{u}}(p)$ . For each positive integer t and for every \(x > 0\) $x>0$ , let \(\mathcal {P}_{\varvec{u}}(t, x)\) ${\mathcal{P}}_{\mathit{u}}(t,x)$ be the set of prime numbers p such that \(p \le x\) $p\le x$ , \(p \not \mid 2a_2 D_{\varvec{u}}\) $p\nmid 2a_2{D}_{\mathit{u}}$ , and \(\iota _{\varvec{u}}(p) = t\) ${\iota }_{\mathit{u}}(p)=t$ . Under the Generalized Riemann Hypothesis, and under some mild assumptions on \(\varvec{u}\) $\mathit{u}$ , we prove that \(\begin{aligned} \#\mathcal {P}_{\varvec{u}}(t, x) = A\, F_{\varvec{u}}(t) \, G_{\varvec{u}}(t) \, \frac{x}{\log x} + O_{\varvec{u}}\!\left( \frac{x}{(\log x)^2} + \frac{x \log \log (3x)}{\varphi (t) (\log x)^2}\right) , \end{aligned}\) $$\begin{array}{c}\hfill \#{\mathcal{P}}_{\mathit{u}}(t,x)=A{F}_{\mathit{u}}(t)G_{\mathit{u}}(t)\frac{x}{logx}+O_{\mathit{u}}\left(\frac{x}{{(logx)}^2},+,\frac{xloglog(3x)}{\phi (t){(logx)}^2}\right),\end{array}$$ for all positive integers t and for all \(x > t^3\) $x>t^3$ , where A is the Artin constant, \(F_{\varvec{u}}(\cdot )\) $F_{\mathit{u}}(·)$ is a multiplicative function, and \(G_{\varvec{u}}(\cdot )\) $G_{\mathit{u}}(·)$ is a periodic function (both these functions are effectively computable in terms of \(\varvec{u}\) $\mathit{u}$ ). Furthermore, we provide some explicit examples and numerical data.