由Siegel公式導出一個整數表為8個平方數之和的表示數

羅 淼, 譚千蓉

(攀枝花學院數學與計算機學院,四川攀枝花617000)

設m是一個正整數.記

為m能表示成8個平方數之和的表示方法數,其中#A表示集合A的基數,而Z表示所有整數的集合.

數學家C.G.Jacobi在1828年證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,將用二次型的解析理論中的Siegel公式來給出r(m)的一個表達式.這個表達式和Jacobi八平方和公式是等價的.

1 Siegel公式

2 β∞(m)和 βp(m)的計算

3 r(m)的表達式

很容易看出這些結果與用Jacobi八平方和公式算出的結果是一樣的.

致謝攀枝花學院培育項目(2012PY08)對本文給予了資助,謹致謝意.

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