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数学进展
研究论文
论Schinzel的一个问题
ON A PROBLEM OF SCHINZEL

邵品琮;
SHAO PIN-TSUNG

北京大学数学力学系四年级 中国科学院数学研究所
(Peking University

收稿日期: 1956-12-25
出版日期: 1956-11-15

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摘要  A.Schinzel和W.Sierpinski在他们的谕文里,对于Euler 函数(n)及除数和函数б(n)=∑d的性质,会利用了极其简单而初等的方法获得了一些有趣d/n的结果。华罗庚教授就Euler函数的性质,首先指出採用Brun方法,可以获得相当精密的枯果,中国科学院数季研究所王元同志,根据这一指示,对于的值的分布问题的讨论作出了很好的结论。闵嗣鹤教授指出採用Brun筛法对除数函数τ(n)=∑1,可以解决Schinzel
Abstract:Using elementary methods, A. Schinzel and W. Sierpinski obtained some results concerning the distribution of the Euler function (n) and the function σ(n) which represents the sum of divisors of n. Professor Hua Loo-Keng pointed firstly that about the function (n), we may obtain much more elaborate results by Brun’s method. According to his suggestion, Wang Yuan obtained the results about the function (n) which is better than the results of A. Schinzel and W. Sierpinski. Professor Min Szu-Hoa pointed to me that the Brun’s method may also be used to solve the problem of Schinzel about the function r(n) We obtained Theorem l. For every positive integer k and every sequence of positive integers δ_0, δ_1, … ,δ_(k-1), there exists a natural number n such that δ_v≤(n+v)≤c_0δ_v (v=0, 1, …, k-1) (1) where c_0=c_0(k) is a positive constant. More precisely, if we denote by N(X) the number of integers n in the interval 1 ≤ n≤ X satisfying (1), then for sufficiently large X, we haveN(X)>c'x/log~k X,(2) where c'=c'(k; δ_0, … , δ_(k-1)) is a positive constant. Theorem 2. For every positive integer k and every sequence of positive real numbers B_0,B_1,…, B(k-1), there exists a natural number n such that c_1B_v≤r(n+v)/r(n+v+1)≤ c_2 B_v(v=0, 1,…,k-1),(3) where c_1=c_1(k) and c_2=c_2(k) are positive constants. More precisely, if we denote by N~*(X)the number of integers n in the interval 1 ≤n≤X satisfying(3), then for sufficiently large X, we have N~*(X)>c~* X/log~(k+1)X, where c~*=c~*(k; B_0, B_1, …, B(k-1)) is a positive constant. I have been informed that these theorems were also proved by Schinzel independently and Wang Yung and Schinzel will publish a joint paper (to be appear in Bulletin de l’Academie Polonaise des Sciences), containing these theorems. The method of this paper may be applied to the function Ts(n)=
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