Trivial zeros of the Riemann zeta function

[mrbohn1]

Clearly I am missing something obvious here, but how is it that negative even numbers are zeros of the Riemann zeta function?

For example:

$$\zeta (-2)=1+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+...=1+4+9+..$$

Which is clearly not zero. What is it that I am doing wrong?

 

[Santa1]

You are using the definition of zeta(s) for Re(s) > 1 with a number that has a real part smaller than or equal to 1.

 

[CRGreathouse]

What is it that I am doing wrong?

You need not the function you posted, but its analytic continuation.

 

[mrbohn1]

Thanks! It all becomes clear.