On the density of some sequences of integers
On the density of some sequences of integers
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关于某些整数序列的密度
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发表时间：
1948
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影响因子：
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通讯作者：
P. Erdös
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作者：
P. Erdös
令a1 <a2<。. * 是任何整数序列，使得没有人整除任何其他整数，并设&<bz<。. * 是由至少能被一个a整除的那些整数组成的序列。人们曾经指出，B的序列必然具有密度。贝西科维奇'表明，这是不是这样的。后来Davenport和I2表明，B的序列总是具有对数密度，换句话说，存在Em，（l/log n）xbisn l/B，并且该对数密度也是B的较低密度。很容易看出，如果Ellai收敛，则B的序列具有密度。也很容易看出，如果每对a是互质的，则B的密度等于n（ll/a，），也就是说，当且仅当x1/a，发散时，密度为0。在本文中，我调查什么较弱的条件将确保B的有一个密度。设f（n）表示不超过n的a的个数。我证明了如果f（n）<en/log n，其中c是一个常数，那么B有一个密度。这个结果是最可能的，因为我们证明了如果g（n）是任何随n趋于无穷大的函数，则存在序列a，其中f（n）<n。(n)/log z，其中不存在B的密度。前一个结果将作为一个稍微精确的定理的结果而得到。令4（n; x; yr，y2，* . .，y，J通常表示不超过1 z的整数的数目，这些整数可被x整除但不可被yl，* 整除。.，y，，.那么B具有密度的充分必要条件是
Let al<a2< . . * be any sequence of integers such that no one divides any other, and let &<bz< . . * be the sequence composed of those integers which are divisible by at least one a. It was once conjectured that the sequence of b’s necessarily possesses a density. Besicovitch’ showed that this is not the case. Later Davenport and I2 showed that the sequence of b’s always has a logarithmic density, in other words that Em,,, (l/log n) xbisn l/b; exists, and that this logarithmic density is also the lower density of the b’s. It is very easy to see that if Ellai converges, then the sequence of b’s possesses a density. Also it is easy to see that if every pair of a’s is relatively prime, the density of the b’s equals n(ll/a,), that is, is 0 if and only if x.1/a; diverges. In the present paper I investigate what weaker conditions will insure that the b’s have a density. Let f(n) denote the number of a’s not exceeding n. I prove that if f(n) <en/log n, where c is a constant, then the b’s have a density. This result is best possible, since we show that if g(n) is any function which tends to infinity with n, then there exists a sequence a, withf(n) <n.#(n)/log z, for which the density of the b’s does not exist. The former result will be obtained as a consequence of a slightly more precise theorem. Let 4(n; x; yr, y2, * . . , y,J denote generally the number of integers not exceeding 1z which are divisible by x but not divisible by yl, * . . , y,,. Then a necessary and sufficient condition for the b’s to have a density is that