Some remarks on Euler's $\phi$ function and some related problems
Some remarks on Euler's $\phi$ function and some related problems
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关于欧拉$phi$函数的一些评论及一些相关问题
DOI：
10.1090/s0002-9904-1945-08390-6
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发表时间：
1945
影响因子：
1.3
通讯作者：
P. Erdös
中科院分区：
文献类型：
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作者：
P. Erdös
函数<i>（n）被定义为与n互素的整数的个数，并且<t>（n）~ n 'JjLp\n（l-p~）在以前的文章中，我证明了以下结果：（1）对任意e > 0，<ε（#）= m有解的整数的个数w i ~ n为0（w[log n'””）. (2)存在无穷多个整数m g n使得方程<f>（x）~m对某些c > 0有多于m个解。在本文中，我们将证明使（x）~m有解的整数m g n的个数<j>大于en（log n）log log n.用同样的方法，我们可以证明<t>对于每一个k，使得（x）~tn有解的整数mSn的个数大于n（log n^ilog log n）。更清晰结果的证明遵循同样的思路，但要复杂得多。如果我们用f（n）表示整数m^n的个数，其中<j>（x）~m有解，我们有以下不等式
The function <i>(n) is defined to be the number of integers relatively prime to n, and <t>(n)~n'JjLp\n(l—p~~)In a previous paper I proved the following results : (1) The number of integers w i « for which <£(#) = m has a solution is 0(w[log n]'"") for every e > 0 . (2) There exist infinitely many integers m g n such that the equation <f>(x) ~m has more than m solutions for some c > 0 . In the present note we are going to prove that the number of integers m g n for which <j>{x)~m has a solution is greater than en (log n)log log n. By the same method we could prove that the number of integers mSn for which <t>(x)~tn has a solution is greater than n(log n^ilog log n) for every k. The proof of the sharper result follows the same lines, but is much more complicated. If we denote by f(n) the number of integers m^n for which <j>(x)~m has a solution we have the inequalities