The zeta function (for $\Re(s)>1$) is given by the definition:
- $\zeta(s) = \sum _{n=1}^{\infty} \left [\frac{1}{n^s} \right]$
The zeta function also can be given by the following functional equation:
- $\zeta(s) = 2^s\pi ^{s-1}\sin\left( \frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$
Using the above functional equation, we also know than for all negative even integers (i.e. $s=-2, -4, -6...$) the zeta function has trivial zeroes i.e. $\zeta(-2) = 0$, $\zeta(-4) = 0$, $\zeta(-6) = 0$ and so on.
Substituting $s=3$ in the functional equation we have:
a. $\zeta(3) = \sum _{n=1}^{\infty} \left [\frac{1}{n^3} \right]$
b. $\zeta(3) = 2^3\pi ^{3-1}\sin\left( \frac{\pi 3}{2}\right)\Gamma(1-3)\zeta(1-3)$
Since, $\zeta(1-3)=\zeta(-2)=0$, using this in the R.H.S of the functional equation (b.) seems to state that $\zeta(3) = 0$. But, using (a.) we know that the $\zeta(3)$ is convergent but not equal to zero.
So, these two values are different. What is it that I am missing in the functional definition and where is the discrepancy?
P.S. I am from a C.S. background and a beginner in analysis still trying to learn.
The function $\Gamma$ has a pole at every non-positive integer. The pole of $\Gamma$ at $-2$ "cancels out" the zero of $\zeta$ at $-2$ since they are both simple.