THE INTERMEDIATE PRIME DIVISORS OF INTEGERS
THE INTERMEDIATE PRIME DIVISORS OF INTEGERS
复制标题
整数的中质因数
DOI：
10.1090/s0002-9939-1987-0902529-x
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
Janos Galambos
中科院分区：
文献类型：
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作者：
J. D. Koninck;Janos Galambos
设p{ - j}随n趋于无穷大。我们证明了log log ^，当f '经过p为中间值的指数时，在自然密度意义下形成一个极限泊松过程。令Pi +00）。特别是，概率数论的最简单结果（见艾略特（2，导言））意味着，密度为1时，co（«）/loglog«渐近为1。因此，对于密度1，我们可以区分三种类型的素因子：如果j有界为n ->>+00，我们称pi为小因子;如果co-j保持有界，则称pj为大因子;所有其他因子都是中间因子。对于小素因子的研究，初等数论的工具就足够了。大素因子需要特殊的工具，但非常古老的结果（由于Dickman，参见De Koninck和Ivic（1）关于涉及大素因子的准确陈述和渐近公式）告诉我们，（log/7-）/logn福尔斯落入区间（a，B），0 < a < B < 1，对于j = co具有正密度。并且结果在性质上是相似的。这也许可以解释为什么在概率数论中，在r = r（N）处用（logr）/log N -* 0截断加法函数是“必要的”，而且是如此成功：它只是消除了大素因子的影响（见艾略特（3），特别是第12章）。当截断被放弃并获得新类型的结果时，它确实需要一种全新的攻击方法（再次参见（3））。截断方法，其中的中间素因子贡献了所有的影响，一个声明的有效性，已经表明，中间素因子的行为渐近作为独立的随机变量。事实上，这种渐近独立性甚至比从
Let p{ — j tend to infinity with n. We show that log log ^, as /' goes through the indices for which p. is intermediate, forms a limiting Poisson process in the sense of natural density. Let Pi +00). In particular, the simplest results of probabilistic number theory (see Elliott (2, Introduction)) imply that, with density one, co(«)/loglog« is asymptotically one. Hence, with density one, we can distinguish three types of prime divisors: we call pi small if j is bounded as n -» +00, pj large if co — j remains bounded, and all others intermediate. For the investigation of the small prime divisors, tools of elementary number theory suffice. Large prime divisors require special tools, but very old results (due to Dickman, see De Koninck and Ivic (1) for accurate statements and for asymptotic formulas involving large prime divisors) tell us that (log/7-)/logn falls into the interval (a, b), 0 < a < b < 1, with positive density for j = co. Extensions are also known for all large prime divisors, and the results are similar in nature. This perhaps explains why it was 'necessary' and so successful in probabilistic number theory to truncate additive functions at r = r(N) with (logr)/log N -* 0: it simply cancels the effect of the large prime divisors (see Elliott (3), particularly Chapter 12). It indeed required a completely new method of attack when the truncation was abandoned and new types of results were obtained (once again, see (3)). The truncation methods, in which the intermediate prime divisors contributed all the influence for the validity of a statement, already show that the intermediate prime divisors behave asymptotically as independent random variables. The fact that this asymptotic independence is even stronger than what follows from