论文标题
关于某种类型整数集的自然密度
On natural densities of sets of some type integers
论文作者
论文摘要
令$ a_0 = b_0 = 0 $和$ 0 <a_1 \ leq b_1 <a_2 \ leq b_2 <\ ldots \ leq b_ {n} $是整数。令$ q \ left（x; \ bigcup_ {j = 1}^{n} [a_j，b_j] \ right）$是$ 1 $和$ x $之间的整数数量，使得所有质量分解中的所有指数均以$ \ bigCup_ in in BigCup_ {j = 1}}}}}}}^{n} n} [a_j，b_j，以下公式包含： $ \ lim_ {x \ to \ infty} {\ frac {q \ left（x; \ bigCup_ {j = 1}^{n} [a_j，b_j] \ right）}} {x}}} {x}} = \ prod \ limi ts_ {p} \ sum \ limits_ {i = 0}^{n} \ left（\ frac {1} {p^{a_ {a_ {i}}}}}} - \ frac {1} {p^{p^{b_ {b_ {i} +1} +1}}}}}}} \ right）在本文中，我们证明了这个结果，然后将其概括。
Let $a_0=b_0=0$ and $0<a_1\leq b_1<a_2\leq b_2<\ldots\leq b_{n}$ be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)$ be the number of integers between $1$ and $x$ such that all exponents in their prime factorization are in $\bigcup_{j=1}^{n}[a_j,b_j]$. The following formula holds: $$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_j,b_j]\right)}{x}}=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right).$$ In this paper, we prove this result and then generalize it.
