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# The Riemann zeta-function $\zeta$ is defined by$\zeta(x) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^x}$and is used in number theory to study the distribution of prime numbers. What is the domain of $\zeta$ ?

## the set of real numbers $x$ such that the series is convergent.

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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### Video Transcript

so known is that if we replace X with peen, we get a more familiar looking piece. Aires. Now, from this section eleven point three, there's a fact labelled, one that states that this type of Siri's converges if P is bigger than one, diverges otherwise. So in our case, the domain of Zeta, it's pretty hard to draw, So I'll just write in dough. Man is it's the suitable X in the real numbers such that X is larger than one. If you want to write that as Interval, you could write that as one, not including the one because of the strict inequality over here.

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##### Kristen K.

University of Michigan - Ann Arbor

##### Samuel H.

University of Nottingham

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Join Bootcamp