Estimation Involving a Class of Special Arithmetical Functions
Tingwei Yu *
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, China.
*Author to whom correspondence should be addressed.
Abstract
The mean value estimation of arithmetical function is closely related to many problems in number theory. Let f be an arithmetical function satisfying some conditions. Let [r] be the integral part of r. This paper proves that the asymptotic expression $$S_f(y):=\sum_{n \leq y} f([y / n]) /\left([y / n]^{k-1}\right)\left(k \in \mathbb{N}^{+}\right)$$ and the error term of this asymptotic formula is \(\Omega\)(y). The arithmetical function in this paper satisfies certain conditions, and the Dirichlet hyperbolic principle is used in the proof of the conclusion. With the different values of the independent variable of the function, the function value of the arithmetical function is often irregular, and the property of the mean value of the arithmetical function is more regular than that of the arithmetical function itself. Therefore, with the help of the mean value estimation results of the arithmetical function, we can have a deeper understanding of the nature of the arithmetical function itself, and then provide ideas for solving more problems.
Keywords: Arithmetical function, asymptotic formula, integral part, dirichlet hyperbolic principle