seminar:at_bernoulli

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==== Abstract ==== | ==== Abstract ==== | ||

- | The Bernoulli numbers $B_n$ appear as coefficients in a Maclaurin series expansion of the function $x\mapsto x/(e^x-1).$ Some of their number-theoretic properties have a connection with the Euler-Maclaurin-sum formula:For $k\ge 0$ and $n\ge 2,$ | + | The Bernoulli numbers $B_n$ appear as coefficients in a Maclaurin series expansion of the function $x\mapsto x/(e^x-1).$ Some of their number-theoretic properties have a connection with the Euler-Maclaurin-sum formula: For $k\ge 0$ and $n\ge 2,$ |

$$1^k + 2^k + \cdots + (n-1)^k = \sum_{r=0}^k \frac{1}{k+1-r}{k\choose r}n^{k+1-r}B_r.$$ Their fractional parts are realized by a celebrated theorem of von Staudt and Clausen (1840) which states the following: For $k\ge 1,$ | $$1^k + 2^k + \cdots + (n-1)^k = \sum_{r=0}^k \frac{1}{k+1-r}{k\choose r}n^{k+1-r}B_r.$$ Their fractional parts are realized by a celebrated theorem of von Staudt and Clausen (1840) which states the following: For $k\ge 1,$ | ||

$$B_{2k}+\sum_{\substack{p\, | $$B_{2k}+\sum_{\substack{p\, |

Last modified: 2017/05/17 08:23