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

Abstract

Let $(u_n{)}_{n\ge 0}$ 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 $$\sum _{\begin{array}{c}m>x\\ (m,a_2)=1\end{array}}\frac{1}{{\ell }_u(m)}\le \exp (-(1/\sqrt{6}-\epsilon +o(1))\sqrt{(\log x)(\log \log x)}),$$ when $x$ is sufficiently large in terms of $\epsilon$, and where the $o(1)$ depends on $u$. Moreover, if $g_u(n)=\gcd (n,u_n)$, we will show that for every $k\ge 1$, $$\sum _{n\le x}{g}_u(n{)}^k\le x^{k+1}\exp (-(1+o(1))\sqrt{(\log x)(\log \log x)}),$$ 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.

arXiv comment10 pages. The main result has been improved