The moments of the logarithm of a G.C.D. related to Lucas sequences

Let $(u_n)_{n \geq 0}$ be a nondegenerate Lucas sequence satisfying $u_n = a_1 u_{n-1} + a_2 u_{n - 2}$ for all integers $n \geq 2$, where $a_1$ and $a_2$ are some fixed relatively prime integers; and let $g_u$ be the arithmetic function defined by $g_u(n) := \gcd(n, u_n)$, for all positive integers $n$. Distributional properties of $g_u$ have been studied by several authors, also in the more general context where $(u_n)_{n \geq 0}$ is a linear recurrence. We prove that for each positive integer $\lambda$ it holds \begin{equation*} \sum_{n \,\leq\, x} (\log g_u(n))^\lambda \sim M_{u,\lambda} x \end{equation*} as $x \to +\infty$, where $M_{u,\lambda} > 0$ is a constant depending only on $a_1$, $a_2$, and $\lambda$. More precisely, we provide an error term for the previous asymptotic formula and we show that $M_{u,\lambda}$ can be written as an infinite series.