罗 淼, 谭千蓉

(攀枝花学院数学与计算机学院,四川攀枝花617000)

设m是一个正整数.记

为m能表示成8个平方数之和的表示方法数,其中#A表示集合A的基数,而Z表示所有整数的集合.

数学家C.G.Jacobi在1828年证明了如下的八平方和公式[1]:

其中,d跑遍m的所有正因子.

在本文中,将用二次型的解析理论中的Siegel公式来给出r(m)的一个表达式.这个表达式和Jacobi八平方和公式是等价的.

1 Siegel公式

2 β∞(m)和 βp(m)的计算

3 r(m)的表达式

很容易看出这些结果与用Jacobi八平方和公式算出的结果是一样的.

致谢攀枝花学院培育项目(2012PY08)对本文给予了资助,谨致谢意.

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