The Probability of Relative Primality of Gaussian Integers
The Probability of Relative Primality of Gaussian Integers
复制标题
高斯整数相对素数的概率
DOI：
--
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
Jeremy R. Johnson
中科院分区：
文献类型：
--
作者：
G. Collins;Jeremy R. Johnson
本文将给出两个整数互素概率的定理推广到任意数域。两个整数互质的概率是1/π（2），其中π是黎曼zeta函数，1/π（2）=6/π2。任意数域的定理指出，两个理想互素的概率是数域的zeta函数的倒数。特别地，由于高斯整数是唯一的因子分解域，我们得到两个高斯整数互质的概率是1/G（2），其中G是与高斯整数相关的zeta函数。
In this paper we generalize, to an arbitrary number field, the theorem which gives the probability that two integers are relatively prime. The probability that two integers are relatively prime is 1/ζ(2), where ζ is the Riemann zeta function and 1/ζ(2)=6/π2. The theorem for an arbitrary number field states that the probability that two ideals are relatively prime is the reciprocal of the zeta function of the number field evaluated at two. In particular, since the Gaussian integers are an unique factorization domain, we get the probability that two Gaussian integers are relatively prime is 1/ζ G (2) where ζ G is the zeta function associated with the Gaussian integers.