Mean value of a Smarandache-Type Function
نویسنده
چکیده
In this paper, we use analytic method to study the mean value properties of Smarandache-Type Multiplicative Functions Km(n), and give its asymptotic formula. Finally, the convolution method is used to improve the error term. Suppose m ≥ 2 is a fixed positive integer. If n = p α1 1 p α2 2 ...p α k k , we define K m (n) = p β1 1 p β2 which is a Smarandache-type multiplicative function. Yang Cundian and Li Chao proved in [1] that n≤x K m (n) = x 2 2ζ(m) p 1 + 1 (p m − 1)(p + 1) + O(x 3 2 +). In this paper, we shall use the convolution method to prove the following Theorem. The asymptotic formula n≤x K m (n) = x 2 2ζ(m) p 1 + 1 (p m − 1)(p + 1) + O(x 1+ 1 m e −c0δ(x)) holds, where c 0 is an absolute positive constant and δ(x) = (log x) 3/5 (log log x) −1/5 .
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