The moments of split greatest common divisors

Sequences of the form $(\gcd(u_n,v_n))_{n \in \mathbb N}$, with $(u_n)_n$, $(v_n)_n$ sums of $S$-units, have been considered by several authors. The study of $\gcd(n,u_n)$ corresponds, following Silverman, to divisibility sequences arising from the split algebraic group $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$; in this case, Sanna determined all asymptotic moments of the arithmetic function $\log\,\gcd (n,u_n)$ when $(u_n)_n$ is a Lucas sequence. Here, we characterize the asymptotic behavior of the moments themselves $\sum_{n \leq x}\,\gcd(n,u_n)^\lambda$, thus solving the moment problem for $\mathbb G_{\mathrm{a}} \times \mathbb G_{\mathrm{m}}$. We give both unconditional and conditional results, the latter only relying on standard conjectures in analytic number theory.