羅 淼, 譚千蓉

(攀枝花學(xué)院數(shù)學(xué)與計(jì)算機(jī)學(xué)院,四川攀枝花617000)

設(shè)m是一個(gè)正整數(shù).記

為m能表示成8個(gè)平方數(shù)之和的表示方法數(shù),其中#A表示集合A的基數(shù),而Z表示所有整數(shù)的集合.

數(shù)學(xué)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(gè)表達(dá)式.這個(gè)表達(dá)式和Jacobi八平方和公式是等價(jià)的.

1 Siegel公式

2 β∞(m)和 βp(m)的計(jì)算

3 r(m)的表達(dá)式

很容易看出這些結(jié)果與用Jacobi八平方和公式算出的結(jié)果是一樣的.

致謝攀枝花學(xué)院培育項(xiàng)目(2012PY08)對(duì)本文給予了資助,謹(jǐn)致謝意.

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