On the number of partitions into primes
On the number of partitions into primes
复制标题
关于素数的划分数
DOI：
10.1007/s11139-007-9037-5
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
R. Vaughan
中科院分区：
文献类型：
--
作者：
R. Vaughan
摘要
显然，有一种持续的信念，即在目前的知识状态下，当n很大时，不可能获得一个渐近公式来计算n分成素数的次数。本文给出了这样一个公式。由于素数的分布只能通过使用对数积分和黎曼ζ函数的零点之和来精确描述，所以不能期望主项只涉及初等函数。然而，当n被真实的变量代替时，得到的公式是不成立的。
${\mathcal{C}}^{\infty}$
并且容易看出是单调的。
Abstract
There is, apparently, a persistent belief that in the current state of knowledge it is not possible to obtain an asymptotic formula for the number of partitions of a number n into primes when n is large. In this paper such a formula is obtained. Since the distribution of primes can only be described accurately by the use of the logarithmic integral and a sum over zeros of the Riemann zeta-function one cannot expect the main term to involve only elementary functions. However the formula obtained, when n is replaced by a real variable, is in
${\mathcal{C}}^{\infty}$
and is readily seen to be monotonic.