Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

Question

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation:

\begin{align} \sum_{l=1}^{\infty}\left(l+1/2\right)^0=\sum_{l=1}^{\infty}\frac{1}{\left(l+1/2\right)^0}-1=\zeta(0,1/2)-1 \end{align}

But I can't understand the first step. Where does that $-1$ come from?

Answer 1

Observe that the Hurwitz zeta function is given by $$ \zeta(z,q):=\sum_{n=0}^{\infty}\frac{1}{\left(n+q\right)^z} $$ The index starts at $0$. It explains formally why you have $-1$.