Davenport关于富足数的一个定理的变形
Variations on a theorem of Davenport concerning abundant numbers
摘要：
设σ（n）= \sum_{d \mid n} d是通常的约数和函数。1933年，Davenport证明了n/σ（n）具有连续分布函数。换句话说，对于所有u \in [0，1]，极限D（u）：= \lim_{x\to\infty} \frac{1}{x} \sum_{n \leq x,~n/σ(n) \leq u} 1存在，并且随u连续变化。研究了复值乘性函数f的和\sum_{n \leq x,~n/σ(n) \leq u} f（n）的性质.我们的结果涵盖了许多更常见的函数，包括φ（n），τ（n）和μ（n）。它们也适用于两个平方和的表示函数，得到达文波特结果的以下模拟：对于所有u \in [0，1]，极限\[ \tilde{D}(u):= \lim_{R\to\infty} \frac{1}{πR}\#\{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{σ(x^2+y^2)} \leq u\} \]存在，并且\tilde{D}（u）在[0，1]上是连续的和严格增加的。
英文摘要：
Let σ(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/σ(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/σ(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/σ(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including φ(n), τ(n), and μ(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport's result: For all u \in [0,1], the limit \[ \tilde{D}(u):= \lim_{R\to\infty} \frac{1}{πR}\#\{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{σ(x^2+y^2)} \leq u\} \] exists, and \tilde{D}(u) is both continuous and strictly increasing on [0,1].
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