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

Let (un)n≥0 be a nondegenerate Lucas sequence satisfying un=a1un−1+a2un−2 for all integers n≥2, where a1 and a2 are some fixed relatively prime integers; and let gu be the arithmetic function defined by gu(n):=gcd⁡(n,un), for all positive integers n. Distributional properties of gu have been studied by several authors, also in the more general context where (un)n≥0 is a linear recurrence. We prove that for each positive integer λ it holds∑n≤x(log⁡gu(n))λ∼Mu,λx as x→+∞, where Mu,λ>0 is a constant depending only on a1, a2, and λ. More precisely, we provide an error term for the previous asymptotic formula and we show that Mu,λ can be written as an infinite series.