question about Riemann zeta $\zeta (0)$

Question

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

Answer 1

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.