An upper bound for the moments of a GCD 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 $\ell _u(m)={\rm lcm}(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that $$ \sum _{\substack {m>x\\ (m,a_2)=1}}\frac {1}{\ell _u(m)}\leq \exp \biggl (\!-\biggl (\frac 1{\sqrt {6}}-\varepsilon +o(1)\biggr )\sqrt {(\log x)(\log \log x)}\biggr ), $$ 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 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 upon $u$ 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 was already obtained by Sanna. Finally, we begin the study of the multiplicative analogs of $\ell _u(m)$, finding interesting results.