关于Wilson's Theorem
Around Wilson's theorem
摘要：
我们研究的系列s（n，x），这是从1到n的乘积x Gamma（k）/k的正弦的平方的和，其中x是一个变量。利用Wilson定理证明了当x = Pi/2时，s（n，x）的整数部分是小于或等于n的素数的个数，并对x是Pi的有理数倍得到了类似的公式.我们证明了当n趋于无穷大时，几乎所有的勒贝格测度中的x，s（n，x）都等价于n/2，而对于几乎所有的Baire意义下的x，1/2是s（n，x）与小于或等于n的素数个数之比的一个极限点.
英文摘要：
We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in the Lebesgue measure s(n,x) is equivalent to n/2 when n tends to infinity, while for almost all x in the Baire sense, 1/2 is a limit point of the ratio of s(n,x) to the number of primes less or equal to n.
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