Consider the function $g(z)= \sum^{\infty}_{n=1}\int^{n+1}_{n}\frac{t-n}{t^{z+1}}dt$.
I want to show that this defines an analytic function for Re(z)>0. Later I will use it to describe the Riemann Zeta function by $\zeta(z)=\frac{z}{z-1}-zg(z)$. I thought about proving that $g$ is holomorphic, for this I have several ways, but I didn't find a good one until now. I thought about using Morera's theorem, just trying to compute the derivative or at least show that the derivative exists. Another approach was to compute the integral and then go on with just the sum, but that does also not work well. What would be a good way to prove this?