Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 150 / 3 / a / Solution

For $\sigma =\Re s>1$,On every compact subset of $\Re s>1$, the terms are bounded by a convergent series $\sum n^{-1-\delta }$. The Weierstrass M-test gives locally uniform convergence, and the theorem on locally uniform convergence of holomorphic functions shows that the limit is an analytic function. Hence $D_f$ is analytic throughout $\Re s>1$.

Solved by gpt-5.6-sol high.