Arithmetic functions with linear recurrence sequences
Let \(u= (u(x))\) be a linear recurrence sequence of integers satisfying \[ u(x+ n)= a_1u(x+ n-1)+\cdots+ a_{n-1} u(x+ 1)+ a_n u(x), \] \(x=1,2,\dots\), such that \(u\) is nondegenerate, \(u(x)\neq 0\) and no \(d\geq 2\) divides every element of \(u\). Let \(\theta(k)\) be an arithmetic function, defined for \(k\leq 0\) by \(\theta(0)= 0\), \(\theta(k)= \theta(|k|)\), and write \[ \theta(k)= \sum_{d|k|} {\xi(d)\over d} \] for \(\xi(d)\) an arithmetic function. The objective of the present paper is to study sums of the form \[ S_\nu(\theta, M,N)={1\over N} \sum^{M+N}_{x= M+1} (\theta(u(x)))^\nu. \] Similar sums have been investigated previously in the literature; see, for example, [Mean behavior of arithmetic functions on certain thin sequences. Schwarz, Wolfgang (ed.) et al., Elementare und analytische Zahlentheorie. Tagungsband. Stuttgart: Franz Steiner Verlag. Schriften der Wissenschaftlichen Gesellschaft an der Johann-Wolfgang-Goethe-Universität Frankfurt am Main 20, 65-76 (2006; Zbl 1175.11056)] by \textit{K. H. Indlekofer} and \textit{L. G. Lucht}. Theorem 1 of the current paper establishes that when \(\xi(k)= O(1)\) there is a positive constant \(\Gamma_\nu\) such that \[ S_\nu(\theta, M,N)= \Gamma_\nu+ O\Biggl({(\log(M+ N))^\nu\over N}+ {(\log\log N)^\nu\over\log N}\Biggr) \] for \(\nu= 1,2,\dots\)\ . In Theorem 2 the authors consider the sum \(S_\nu(\theta,M,N)\) when \(u(x)= ag^x+ b\), where \(abg\neq 0\), \(|g|>1\), \(\text{gcd}(ag, b)= \text{gcd}(g- 1,a+ b)= 1\), and obtain a stronger result where the error term becomes \(O({(\log(M+ N))^\nu\over N})\). Then they apply their results to the functions \({\alpha(n)\over n}\), \({\alpha(n)\over\varphi(n)}\), where \[ \alpha(n)+{1\over n} \sum_{d|n} d\varphi(d); \] these functions were studied, for example, by \textit{F.Luca} in [Ramanujan J. 9, No. 1--2, 33--44 (2005; Zbl 1155.11344)].
scientific article; zbMATH DE number 5190753
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