A note on the mean square of the greatest divisor of n which is coprime to a fixed integer k

Jun Furuya (), T. Makoto Minamide () and Miyu Nakano ()
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Jun Furuya: Hamamatsu University School of Medicine
T. Makoto Minamide: Yamaguchi University
Miyu Nakano: Yamaguchi University

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 4, 990-1003

Abstract: Abstract Denote by $$\delta _{k}(n)$$ δ k ( n ) the greatest divisor of a positive integer n which is coprime to a given $$k\ge 2$$ k ≥ 2 . In the case of $$k=p$$ k = p (a prime) Joshi and Vaidya studied $$E_{p}(x):=\sum _{n\le x}\delta _{p}(n)-\frac{p}{2(p+1)}x^{2}$$ E p ( x ) : = ∑ n ≤ x δ p ( n ) - p 2 ( p + 1 ) x 2 (as $$x\rightarrow \infty $$ x → ∞ ) and obtained $$E_{p}(x)=\Omega _{\pm }(x)$$ E p ( x ) = Ω ± ( x ) by an elementary and beautiful approach. Here we study $$R_{p}^{(2)}(x):=\sum _{n\le x}\delta _{p}^{2}(n)-\frac{p^{2}}{3(p^{2}+p+1)}x^{3}+\frac{p}{6}x$$ R p ( 2 ) ( x ) : = ∑ n ≤ x δ p 2 ( n ) - p 2 3 ( p 2 + p + 1 ) x 3 + p 6 x and show $$R_{p}^{(2)}(x)=\Omega _{\pm }(x^{2})$$ R p ( 2 ) ( x ) = Ω ± ( x 2 ) . Moreover, using a method of Adhikari and Balasubramanian we consider a bound of $$|R_{k}^{(2)}(x)|/x^{2}$$ | R k ( 2 ) ( x ) | / x 2 for any square-free integer k.

Keywords: The greatest divisor; A mean square of an arithmetical function; Asymptotic behaviour; Estimates of error terms; Omega-results; 11A25; 11A99; 11N37 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00103-x

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