The Mean Values Of A Series Of Arithmetic Functions

This paper have two chapters and primarily establish some results of two e-squarefree e-divisor functions and short interval results for them.In chapter1, we establish the asymptotic formula of (Τ3(e)(n))-r and a short interval result for (Υ(e)(n))-r.Let are integers, the integer d is called an e-divisor of n if bi|a,i for every i g{1,2,...,s}, notation: d|en. By convention l|el.Let Τ(e)(n) denote the number of e-divisor of n. The function Τ(e)(n) is called the e-divisor function. This function is multiplicative and for every prime p, we have Τ(e)(p)=l,Τ(e)(p2)=Τ(e)(p3)=2,Τ(e)(p4)=3,….. Many authors have studied the properties of the functions (see, for example,[3],[8],[11],[13],[16],[18],[20]). M.V.Subbarao[3] first established an asymptotic formula for the mean value of the function Τ(e)(n), and proved that whereJ.Wu[5] improved the above result and got the following result: whereFor some positive integer r, M.V.Subbarao[3] also proved, whereLaszlo T6th[13] established a more precise asymopototic formula where P2r-2(t) is a polynomial of degree2r-2in t, ur=2r+1-1/2r+2+1.The aim of chapter1is to show the asymptotic formula of (Τ3(3)(n))-r by the Dirichlet convolution method and a short interval asymptotic formula of (Τ(e)(n)))-We shall prove the following result:Theorem1Suppose N is a positive number, then whereTheorem2If x1/5+2ε <y <x, then whereIn chapter2, we establish a asymptotic formula of (t(e)(n)-r and a short interval result for (t(e)(n)))r.The integer n>1is called e-squarefree of n if all exponents a1;…, as are suqarefree. The integer1is also considered to be e-squarefree. The integer d is an e-squarefree e-divisor of n if b1|a1,…, bs|as and b1,…, bs are squarefree. Note that the integer1is e-squarefree but is not an e-divisor for n>1. Let t(e)(n) denote the number of e-squarefree e-divisor of n. The function t(e)(n) is called the e-squarefree e-divisor function. This function is multiplicative and for every prime p, we have t(e)(n)=1,t(e)(p2)=t(e)(p3)=t(e)p4)=t(e)(p5)=2,Τ(e)(p6)=4,….Many authors have studied the properties of this functions. L.Toth proved that the estimation holds for every e>0, whereThe aim of chapter2is to show the asymptotic formula of (t(e)(n)))-r by the Dirichlet convolution method and a short interval asymptotic formula of (t(e)(n))-r we shall prove the following result:Theorem3Suppose the integer r>1and N>1, then where Theorem4If x1/5+2e <y <x, then where...