1. The problem asks to show that the non-trivial zeros of the analytic continuation of the function $f(s) = \sum_{n=1}^\infty n^{-s}$, originally defined 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 states 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 a famous unsolved problem in mathematics; it has not been proven yet. 6. What can be shown is that the zeros are symmetric about the critical line and the real axis, and that trivial zeros lie at negative even integers. 7. The analytic continuation and functional equation of $\zeta(s)$ are key tools in studying these zeros, but a proof that all non-trivial zeros lie on $\mathrm{Re}(s) = \frac{1}{2}$ remains open. Final answer: The statement that all non-trivial zeros of $\zeta(s)$ lie on the line $\mathrm{Re}(s) = \frac{1}{2}$ is the Riemann Hypothesis, which is not yet proven.