关于可加函数的大偏差
On large deviations of additive functions
摘要：
我们证明，如果两个添加剂功能（从某类）采取大的值大致相同的概率，那么他们必须是相同的。Kac-Kubilius模型表明，给定的加性函数的值的分布可以由随机变量的和来建模。我们表明，该模型是准确的（在一个大偏差意义上），当一个人正在寻找其平均值附近的加性函数的值，但失败，由一个常数倍，为大值的加性函数。我们认为，这种现象之所以会出现，是因为模型在“大”素数上的加性函数的值上崩溃了。在本文的第二部分，我们的动机是一个问题的埃利奥特，了解有多少分布的价值的添加剂功能的素数决定，并决定，分布的价值的添加剂功能的所有整数。例如，我们的主要定理，意味着一个积极的，强加性函数是大致泊松分布的整数当且仅当它是$1+o(1)$或$o(1)$对几乎所有的素数。
英文摘要：
We prove that if two additive functions (from a certain class) take large values with roughly the same probability then they must be identical. The Kac-Kubilius model suggests that the distribution of values of a given additive function can be modeled by a sum of random variables. We show that the model is accurate (in a large deviation sense) when one is looking at values of the additive function around its mean, but fails, by a constant multiple, for large values of the additive function. We believe that this phenomenon arises, because the model breaks down for the values of the additive function on the "large" primes. In the second part of the paper, we are motivated by a question of Elliott, to understand how much the distribution of values of the additive function on primes determines, and is determined by, the distribution of values of the additive function on all of the integers. For example, our main theorem, implies that a positive, strongly additive function is roughly Poisson distributed on the integers if and only if it is $1+o(1)$ or $o(1)$ on almost all primes.
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