I'm trying to solve an exercise in my complex analysis text. The problem is :
If $Re(z)>1$ and $\zeta(z)$ is the Riemann zeta function, then show that $\zeta(z)\zeta(z-1)=\sum_{k=1}^{\infty} \frac{\sigma(k)}{k^z}$.
I've shown that the formula holds for $Re(z)>2$. But I cannot show for $1<Re(z)≤2$. How can I show it?