An Upper Bound for the Moments of a G.C.D. related to Lucas Sequences

Abstract

Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell_u(m)=lcm(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that $$\sum_{\substack{m>x\\ (m,a_2)=1}}\frac{1}{\ell_u(m)}\leq \exp(-(1/\sqrt{6}-\varepsilon+o(1))\sqrt{(\log x)(\log \log x)}),$$ when $x$ is sufficiently large in terms of $\varepsilon$, and where the $o(1)$ depends on $u$. Moreover, if $g_u(n)=\gcd(n,u_n)$, we will show that for every $k\geq 1$, $$\sum_{n\leq x}g_u(n)^{k}\leq x^{k+1}\exp(-(1+o(1))\sqrt{(\log x)(\log \log x)}),$$ when $x$ is sufficiently large and where the $o(1)$ depends on $u$ and $k$. This gives a partial answer to a question posed by C. Sanna. As a by-product, we derive bounds on $#\{n\leq x: (n, u_n)>y\}$, at least in certain ranges of $y$, which strengthens what already obtained by Sanna. Finally, we start the study of the multiplicative analogous of $\ell_u(m)$, finding interesting results.