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An upper bound for the moments of a GCD related to Lucas sequences. (English) Zbl 1418.11027

Summary: 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 \(\ell_u(m)=\mathrm{lcm}(m, z_u(m))\), 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.

MSC:

11B37 Recurrences
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11N64 Other results on the distribution of values or the characterization of arithmetic functions
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References:

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