What is the definition of the tail sum $R_n(s)$ for the Riemann zeta function?

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.

Reasoning, step by step

1. Identify the full series $\zeta(s)$.
2. Identify the finite partial sum $f_n(s)$ up to $n$ terms.
3. Subtract the partial sum from the full series to isolate the remaining terms.
4. Express the result as an infinite sum starting from index $k=n+1$.

Example

The instructor then defines the partial sums $f_n(s) = \sum_{k=1}^n \frac{1}{k^s}$ and introduces the concept of tail sums, or remainder sums, denoted as $R_n(s)$. The tail sum is defined as the difference between the full zeta function and the partial sum: $R_n(s) = \zeta(s) - f_n(s) = \sum_{k=n+1}^\infty \frac{1}{k^s}$.

Common misconceptions

• Confusing the tail sum $R_n(s)$ with the partial sum $f_n(s)$.
• Thinking the tail sum starts at $k=n$ instead of $k=n+1$.

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