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 ${\sum }_{n\le x}\gcd (n,u_n{)}^{\lambda }$, 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.