1. The problem asks to show that the non-trivial zeros of the analytic continuation of the function $f(s) = \sum_{n=1}^\infty n^{-s}$, defined initially for $\mathrm{Re}(s) > 1$, lie on the critical line $\mathrm{Re}(s) = \frac{1}{2}$ in the complex plane. 2. This function $f(s)$ is the Riemann zeta function $\zeta(s)$, which can be analytically continued to the whole complex plane except for a simple pole at $s=1$. 3. The non-trivial zeros of $\zeta(s)$ are those zeros in the critical strip $0 < \mathrm{Re}(s) < 1$. 4. The Riemann Hypothesis conjectures that all such non-trivial zeros have real part exactly $\frac{1}{2}$, i.e., they are of the form $s = \frac{1}{2} + ix$ where $x \in \mathbb{R}$. 5. However, this is an open problem in mathematics and has not been proven yet. 6. Therefore, it is not possible to show this fact rigorously at this time; it remains a famous unsolved problem. 7. We can summarize that the function $f(s)$ extends analytically beyond $\mathrm{Re}(s) > 1$, and its non-trivial zeros lie in the critical strip, but whether they all lie on the line $\mathrm{Re}(s) = \frac{1}{2}$ is unproven.