一个新数论函数的均值

A new arithmetic function and its mean value

数论函数的性质研究在数论中占有举足轻重的地位,很多函数的单个取值是没有规律的,但是其均值往往具有非常规则的渐近公式。美籍罗马尼亚著名数论专家F.Smarandache教授引入了简单数的概念。如果正整数n的所有真因子的乘积不超过n,称n为简单数。令A表示所有简单数集合,既有A={2,3,4,5,6,7,8,9,10,11,13,14,15,17,19,21,…}.容易看出n有4种情形,即n=p,n=p2,n=p3,n=pq,其中p,q是不同的素数。关于简单数的性质及相关的均值问题已有不少学者进行了研究,也获得了一系列有意义的研究成果。文中研究了一个类似欧拉函数φ(n)的新的Smarandache可乘数论函数J(n),其中J(n)为模n所有原Dirichlet特征的个数,即J(n)=n∏p|n(p-1)2.利用初等数论的方法解决了J(n)可乘数论函数在简单数序列中的均值问题,并给出了一个有趣的渐近式,即对任意x∈R,x≥3,有渐近式Σn≤x,n∈A J(n)=Dx4+Ox4ln lnx ln()x,其中D为可计算的常数。从而丰富了数论函数的内容。为以后更多的学者研究数论函数在特殊序列上的性质提供了参考依据。但是,文中只研究了此函数在特殊数列上的性质,是否在其它数列上也有简单的渐近公式值得更多的学者去讨论和探究。

Arithmetic function plays a significant role in the theory of the nature of the research, many of the functions of a single value is not regular,but the average is often very rules of asymptotic formula. Romania,a famous American number theory experts F. S marandache professor introduced the concept of simple Numbers. If the product of its proper divisors is less than or equal to n,then this positive integer n is called simple number. Make A for all simple several collections, both A = t2,3,4,5,6,7,8,9,10,11,13,14,15,17,19,21,}.We can see that there are four situations, namely,n = p,n = pZ ,n = p3 ,n = pq,that it is different prime Numbers. In this paper,the author study a new Smarandache muhiplicative arithmetic function similar to the Euler function J（n）, the J（n） for all the original characteristics of Dirichlet mold number,J（n）=nΠp｜ n（p-1）2.Using elementary number theory solution of J（n） arithmetic function problem,and an interestiong formula of arbitrary,namely, ∑n≤x,n∈AJ（n）=Dx4＋O（x4lnlnx/lnx）,neA in a simple number sequence of multiplicative asymptotic formula is given,Has asymptotic = Dx4＋ o｜x41n/lnx｜,where~ D is constant can be lnx 1 calculated, which enriched the content of number theory function. For more scholars research on number theory function in a particular sequence of properties provides a reference basis. However, this function is only studied in the nature of the special series,whether on the other series also has a simple asymptotic formula deserves more scholars to discuss and explore.