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2019 An upper bound for the moments of a GCD related to Lucas sequences
Daniele Mastrostefano
Rocky Mountain J. Math. 49(3): 887-902 (2019). DOI: 10.1216/RMJ-2019-49-3-887

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.

Citation

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Daniele Mastrostefano. "An upper bound for the moments of a GCD related to Lucas sequences." Rocky Mountain J. Math. 49 (3) 887 - 902, 2019. https://doi.org/10.1216/RMJ-2019-49-3-887

Information

Published: 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07088341
MathSciNet: MR3983305
Digital Object Identifier: 10.1216/RMJ-2019-49-3-887

Subjects:
Primary: 11B37, 11B39
Secondary: 11A05, 11N64

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 3 • 2019
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