## 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:

 [1] J.J. Alba González, F. Luca, C. Pomerance and I.E. Shparlinski, On numbers $$n$$ dividing the $$n$$th term of a linear recurrence, Proc. Edinb. Math. Soc. 55 (2012), 271-289. · Zbl 1262.11015 [2] R. André-Jeannin, Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart. 29 (1991), 364-366. · Zbl 0737.11003 [3] T. Apostol, Introduction to analytic number theory, Springer, 1976. · Zbl 0335.10001 [4] E.R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), 1-28. · Zbl 0513.10043 [5] G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence sequences, Math. Surveys and Monographs 104, Amer. Math. Soc., Providence, RI, 2003. · Zbl 1033.11006 [6] A. Ivić and C. Pomerance, Estimate for certain sums involving the largest prime factor of an integer, pp. 769-789 in Topics in classical number theory, vol. I (Budapest, 1981), Colloq. Math. Soc. János Bolyai 34, North-Holland, Amsterdam, 1984. [7] P. Leonetti and C. Sanna, On the greatest common divisor of $$n$$ and the $$n$$th Fibonacci number, Rocky Mountain J. Math. 48 (2018), 1191-1199. · Zbl 1437.11025 [8] F. Luca and E. Tron, The distribution of self-Fibonacci divisors, pp. 149-158 in Advances in the theory of numbers, Fields Inst. Commun. 77, Fields Inst. Res. Math. Sci., Toronto, 2015. [9] M. Renault, The period, rank, and order of the $$(a, b)$$-Fibonacci sequence mod $$m$$, Math. Mag. 86 (2013), 372-380. · Zbl 1293.11025 [10] C. Sanna, The moments of the logarithm of a G.C.D. related to Lucas sequences, J. Number Theory 191 (2018), 305-315. · Zbl 1444.11028 [11] —-, On numbers $$n$$ dividing the $$n$$th term of a Lucas sequence, Int. J. Number Theory 13 (2017), 725-734. · Zbl 1416.11029 [12] —-, On numbers $$n$$ relatively prime to the $$n$$th term of a linear recurrence, Bull. Malays. Math. Sci. Soc. 42 (2019), 827-833. · Zbl 1458.11028 [13] C. Sanna and E. Tron, The density of numbers $$n$$ having a prescribed G.C.D. with the $$n$$th Fibonacci number, Indag. Math. (N.S.) 29 (2018), 972-980. · Zbl 1417.11012 [14] C. Smyth, The terms in Lucas sequences divisible by their indices, J. Integer Seq. 13 (2010), article 10.2.4. · Zbl 1210.11025 [15] L. Somer, Divisibility of terms in Lucas sequences by their subscripts, pp. 515-525 in Applications of Fibonacci numbers, vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993. [16] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Stud. Advanced Math. 46, Cambridge Univ. Press, 1995.
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