Evaluate $\zeta(0)$

Question

I am using the equation $$\zeta(s) = \frac{\pi^{s/2}}{2 \Gamma(s/2)} \int_1^\infty (\theta(t)-1)\left(t^{s/2} + t^{(1-s)/2}\right) \frac{dt}{t} + \frac{\pi^{s/2}}{(s-1)\Gamma(s/2)} - \frac{\pi^{s/2}}{s\Gamma(s/2)}$$

to evaluate $\zeta(0)$.

Because $\lim_{s \to 0}\Gamma(s) = \infty$, the term in front of the integral is equal to $0$ so we need only evaluate $$\frac{\pi^{s/2}}{(s-1)\Gamma(s/2)} - \frac{\pi^{s/2}}{s\Gamma(s/2)}$$ at $s = 0$ which gives $$\frac{\pi^0}{(0-1)\Gamma(0)} - \frac{\pi^0}{s\Gamma(0)}.$$

It is at this point where I am unsure how to proceed. I know I need to show the answer is $-1/2$ but I can't see how to manipulate this expression.

Answer 1

$\Gamma$ has a simple pole at $s=0$ with residue $1$, and so $\lim\limits_{s\to 0}s\Gamma(s)=1$. Thus:

$\lim\limits_{s\to 0}\frac{\pi^{\frac{s}{2}}}{(s-1)\Gamma(\frac{s}{2})}=\frac{1}{(-1)\cdot\infty}=0$

$\lim\limits_{s\to 0}\frac{\pi^{\frac{s}{2}}}{s\Gamma(\frac{s}{2})}=\lim\limits_{s\to 0}\frac{1}{2}\cdot\frac{\pi^{\frac{s}{2}}}{\frac{s}{2}\cdot\Gamma(\frac{s}{2})}=\frac{1}{2}\cdot\frac{1}{1}=\frac{1}{2}.$