On The Maximum And Minimum Component Of The Standard Compositon Of An Integer

For each positive integer n, let the standard composition formula be n =p_<sup>α₁p₂^α₂…p_s^α_s, then the maximum component function M(n) and the minimum com-ponent function m(n) can be defined as follows: The main aim of the first chapter is to improve and establish the previousresults. There are three main results as following:Theorem 1 Suppose f(k) is an arbitrary arithmetic function,In the above-mentioned formula, the improvement of the exponent 21 depends onthe non-zero region study of theζ. The Theorem 3 can be seen the form of the short interval case in the Theorem1. In addition, the result is unrelated to the non-zero region ofζ.Theorem 3 Suppose f(k) is an arbitrary arithmetic function. There existsβIn the second part of this paper, we study a short interval result for the func-tionμ(e).Let n > 1 be an integer of canonical from n = is=1 pia i. The integer n =i=1 pbi iis called an exponential divisor of n if bi|ai for every i∈{1, 2, ..., s}, nota-tion: d|en. Let 1|e1.Letμ^e(n) =μ(a₁)μ(a_r) here n = (?). Observe that |μ^(e)| = 0 or|μ^(e)| = 1, according as n is e-squarefree or not. The properties of the functionμ(e)is investigated by many authors. An asymptotic formula for A(x):=(?)was established by M. V. Subarao[2] and improved by J. Wu[1]. Recently L′aszl′oT′oth[3] proved thatand 0 < < 9/25 and c > 0 are fixed constants. The aim of the chapter 2 is to study the short interval case and prove thefollowing...