# question about Riemann zeta $\zeta (0)$ [duplicate]

This question already has an answer here:

i know that

$$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$

so

$$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty$$

but actually

$$\zeta (0)=-0.5$$

where is the wrong

please help

thanks for all

## marked as duplicate by J. M. is a poor mathematician, Amzoti, Zander, Henry T. Horton, Shuhao CaoMay 30 '13 at 4:19

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## 1 Answer

I also struggled with this for a while. Your definition of the Riemann zeta function is only its definition when the real part of $m$ is greater than $1$.

The domain of $\zeta$ though is $\mathbb{C}$, so the question is: how do we move from $\{z \mid \Re(z) > 1 \}$ to $\mathbb{C}$? The answer is analytic continuation.

Using the functional equation for $\sum\limits_{n = 1}^\infty \frac{1}{n^z}$, we can extend the domain of $\zeta$ to the complex numbers.

If you want to know the details, I suggest looking at the Wikipedia page on the function.

• The domain of $\zeta$ excludes $1$, beware (there is a pole there). – Najib Idrissi May 19 '13 at 9:15