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The lecture restricts the proof to the region $\operatorname{Re}(s)\ge 1+a$ (with a fixed $a>0$) to avoid the singularity at $\operatorname{Re}(s)=1$. Near this boundary, the zeta function blows up, which prevents the partial sums from converging uniformly across the entire open half-plane $\operatorname{Re}(s)>1$.

Conditions: $a$ is a fixed positive constant ($a>0$).; The region of interest is the closed half-plane $\operatorname{Re}(s)\ge 1+a$.; The full defining region of the zeta series is $\operatorname{Re}(s)>1$.

The lecture restricts the proof to the region $\operatorname{Re}(s)\ge 1+a$ (with a fixed $a>0$) to avoid the singularity at $\operatorname{Re}(s)=1$. Near this boundary, the zeta function blows up, which prevents the partial sums from converging uniformly across the entire open half-plane $\operatorname{Re}(s)>1$.

Conditions: $a$ is a fixed positive constant ($a>0$).; The region of interest is the closed half-plane $\operatorname{Re}(s)\ge 1+a$.; The full defining region of the zeta series is $\operatorname{Re}(s)>1$.