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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

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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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.

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

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