Abstract
An asymptotic formula of a sum function involving gcd and characteristic function of the set of r–free numbers
Highlights
Let k and j be two positive integers
We note that the formula (2) has a lot of interesting applications
K≤x j=1 is given, where x is any real number greater than 1; s and r are any fixed positive integers; and μr is the characteristic function of the set of r-free numbers
Summary
Let k and j be two positive integers. We denote by gcd(k, j) the greatest common divisor of the integers k and j. For any two arithmetical functions f and g, let us consider the sum function The function given in (1) is a generalization of the following sum function Sf,g (k) := f (d) g (k/d) = (f ∗ g) (k) , d|k where the symbol “∗” denotes the Dirichlet convolution of arithmetic functions.
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