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The greatest value of $|R_n(s)|$ in the region $\operatorname{Re}(s)\ge 1+a$ is achieved at the boundary point $s=(1+a)+0i$. This is proven by chaining two inequalities: first, the triangle inequality bounds the complex modulus by the real tail, $|R_n(\sigma+it)|\le R_n(\sigma)$; second, the term-by-term comparison for real arguments shows that $R_n(\sigma)\le R_n(1+a)$ when $\sigma\ge 1+a$.

Conditions: The region is the closed half-plane $\operatorname{Re}(s)\ge 1+a$ with $a>0$.; $s = \sigma + it$ is a complex variable.; $R_n(s) = \sum_{k=n+1}^{\infty} \frac{1}{k^s}$ is the tail sum of the zeta series.

The greatest value of $|R_n(s)|$ in the region $\operatorname{Re}(s)\ge 1+a$ is achieved at the boundary point $s=(1+a)+0i$. This is proven by chaining two inequalities: first, the triangle inequality bounds the complex modulus by the real tail, $|R_n(\sigma+it)|\le R_n(\sigma)$; second, the term-by-term comparison for real arguments shows that $R_n(\sigma)\le R_n(1+a)$ when $\sigma\ge 1+a$.

Conditions: The region is the closed half-plane $\operatorname{Re}(s)\ge 1+a$ with $a>0$.; $s = \sigma + it$ is a complex variable.; $R_n(s) = \sum_{k=n+1}^{\infty} \frac{1}{k^s}$ is the tail sum of the zeta series.

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.