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# values of the Riemann zeta function in terms of Bernoulli numbers

###### Theorem.

Let $k$ be an even integer and let $B_{k}$ be the $k$th Bernoulli number. Let $\zeta(s)$ be the Riemann zeta function. Then:

$\zeta(k)=\frac{2^{{k-1}}|B_{k}|\pi^{k}}{k!}$ |

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

$\zeta(1-n)=\frac{(-1)^{{n+1}}B_{n}}{n}$ |

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

$\zeta(1-k)=-\frac{B_{k}}{k}.$ |

###### Remark.

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.

Related:

BernoulliNumber, ValueOfTheRiemannZetaFunctionAtS2

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

11M99*no label found*

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## Comments

## about Alozano's Riemann Z function entry

It is precise to clarify that the zeroes of Riemann Z function which it appear in the Alozano's entry, are all zeros trivial ones.

## Re: about Alozano's Riemann Z function entry

Thanks perucho, just to be clear, I added a remark in the entry itself about it.

Alvaro