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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 $\operatorname{Re}(s)>1$.

The tail sum, or remainder sum, $R_n(s)$ is defined as the difference between the full Riemann zeta function $\zeta(s)$ and its $n$-th partial sum $f_n(s)$. Algebraically, it is the infinite series starting from $k=n+1$: $R_n(s) = \zeta(s) - f_n(s) = \sum_{k=n+1}^{\infty} \frac{1}{k^s}$.

Conditions: $s$ is a complex number with $\operatorname{Re}(s)>1$.; $\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^s}$ is the Riemann zeta function.; $f_n(s) = \sum_{k=1}^{n} \frac{1}{k^s}$ is the $n$-th partial sum.

The tail sum, or remainder sum, $R_n(s)$ is defined as the difference between the full Riemann zeta function $\zeta(s)$ and its $n$-th partial sum $f_n(s)$. Algebraically, it is the infinite series starting from $k=n+1$: $R_n(s) = \zeta(s) - f_n(s) = \sum_{k=n+1}^{\infty} \frac{1}{k^s}$.

Conditions: $s$ is a complex number with $\operatorname{Re}(s)>1$.; $\zeta(s) = \sum_{k=1}^{\infty} \frac{1}{k^s}$ is the Riemann zeta function.; $f_n(s) = \sum_{k=1}^{n} \frac{1}{k^s}$ is the $n$-th partial sum.