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

Autor: T. Makoto Minamide, Miyu Nakano, Jun Furuya
Rok vydání: 2021
Předmět:
Zdroj: Indian Journal of Pure and Applied Mathematics. 52:990-1003
ISSN: 0975-7465
0019-5588
Popis: Denote by $$\delta _{k}(n)$$ the greatest divisor of a positive integer n which is coprime to a given $$k\ge 2$$ . In the case of $$k=p$$ (a prime) Joshi and Vaidya studied $$E_{p}(x):=\sum _{n\le x}\delta _{p}(n)-\frac{p}{2(p+1)}x^{2}$$ (as $$x\rightarrow \infty $$ ) and obtained $$E_{p}(x)=\Omega _{\pm }(x)$$ by an elementary and beautiful approach. Here we study $$R_{p}^{(2)}(x):=\sum _{n\le x}\delta _{p}^{2}(n)-\frac{p^{2}}{3(p^{2}+p+1)}x^{3}+\frac{p}{6}x$$ and show $$R_{p}^{(2)}(x)=\Omega _{\pm }(x^{2})$$ . Moreover, using a method of Adhikari and Balasubramanian we consider a bound of $$|R_{k}^{(2)}(x)|/x^{2}$$ for any square-free integer k.
Databáze: OpenAIRE
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