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单词 ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers
释义

values of the Riemann zeta function in terms of Bernoulli numbers


Theorem.

Let k be an even integer and let Bk be the kth Bernoulli numberDlmfDlmfMathworldPlanetmathPlanetmath. Let ζ(s) be the Riemann zeta functionDlmfDlmfMathworldPlanetmath. Then:

ζ(k)=2k-1|Bk|πkk!

Moreover, by using the functional equation (http://planetmath.org/RiemannZetaFunction), one calculates for all n1:

ζ(1-n)=(-1)n+1Bnn

which shows that ζ(1-n)=0 for n3 odd. For k2 even, one has:

ζ(1-k)=-Bkk.
Remark.

The zeroes of the zeta functionMathworldPlanetmath shown above, ζ(1-n)=0 for n3 odd, are usually called the trivial zeroes of the Riemann zeta function, while the non-trivial zeroes are those in the critical stripMathworldPlanetmath.

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更新时间:2025/5/4 21:42:04