PlanetMath: values of the Riemann zeta function in terms of Bernoulli numbers

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values of the Riemann zeta function in terms of Bernoulli numbers

(Theorem)

Theorem 1 Let be an even integer and let be the th Bernoulli number. Let $\zeta(s)$ be the Riemann zeta function. Then:

$\displaystyle \zeta(k)=\frac{2^{k-1}\vert B_k\vert\pi^k}{k!}$

Moreover, by using the functional equation , one calculates for all $n\geq 1$ :

$\displaystyle \zeta(1-n)=\frac{(-1)^{n+1}B_n}{n}$

which shows that $\zeta(1-n)=0$ for $n\geq 3$ odd. For $k\geq 2$ even, one has:

$\displaystyle \zeta(1-k)=-\frac{B_k}{k}.$

Remark 1 The zeroes of the zeta function shown above, $\zeta(1-n)=0$ for $n\geq 3$ odd, are usually called the trivial zeroes of the Riemann zeta function, while the non-trivial zeroes are those in the critical strip.

"values of the Riemann zeta function in terms of Bernoulli numbers" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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Attachments:: proof of values of the Riemann zeta function in terms of Bernoulli numbers (Proof) by rm50

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Cross-references: critical strip, even, odd, calculates, Riemann zeta function, Bernoulli number, even integer
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This is version 4 of values of the Riemann zeta function in terms of Bernoulli numbers, born on 2005-04-20, modified 2006-10-02.
Object id is 6960, canonical name is ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers.
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Classification:

AMS MSC:

11M99 (Number theory :: Zeta and $L$-functions: analytic theory :: Miscellaneous)

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