| Riemann Zeta函数的七阶和式 |
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| 引用本文： | 徐策,程金发.Riemann Zeta函数的七阶和式[J].数学学报,2016,59(2):151-162. |
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| 作者姓名： | 徐策 程金发 |
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| 作者单位： | 厦门大学数学科学学院 厦门 361005 |
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| 基金项目： | 中央高校基本科研基金资助项目(20720150006);福建省自然科学基金资助项目(2011J01021) |
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| 摘 要： | 通过构造一个Riemann Zeta函数ζ(k)的部分和ζ_n(k)的幂级数函数,利用牛顿二项式展开及柯西乘积公式可以计算出一些重要的和式.再将该幂级数函数由一元推广到二元甚至多元,由此得到Riemann Zeta函数的高次方和式之间的关系.并利用对数函数与第一类Stirling数之间的关系式及ζ(k)函数满足的相关等式,可得出Riemann Zeta函数的18个七阶和式,以及其它一些高次方的和式.
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| 关 键 词： | Riemann Zeta函数 二项式展开 柯西乘积 Stirling数 |
The 7-Order Sums of Riemann Zeta Function |
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| Ce XU,Jin Fa CHENG.The 7-Order Sums of Riemann Zeta Function[J].Acta Mathematica Sinica,2016,59(2):151-162. |
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| Authors: | Ce XU Jin Fa CHENG |
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| Institution: | School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China |
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| Abstract: | In this paper, by constructing a partial sum of ζ(k) on the Riemann Zeta function, using the ζn(k) power series function, Binomial expression and Cauchy product formula, some important sums are calculated. By extending the power series function from one variable to multivariate, some relationships of high order sums on Riemann Zeta function are established. By using the relationship between the logarithmic function and Stirling numbers of the first kind, and some other related equations, 18 seven-order sums of Riemann Zeta function and some other high order sums are obtained. |
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| Keywords: | Riemann Zeta function binomial expression Cauchy product Stirling number |
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