Riemann Zeta Function of 1000

Theorem
To at least $100$ decimal places:


 * $\map \zeta {1000} \approx 1$

where $\zeta$ denotes the Riemann zeta function.

Proof
By definition of the general harmonic numbers:
 * $\ds \map \zeta r = \lim_{n \mathop \to \infty} H_n^{\paren r} = \sum_{k \mathop \ge 1} \frac 1 {k^r}$

From Sequence of General Harmonic Numbers Converges for Index Greater than 1:

which is $1 \cdot 000 \ldots$ to a good few hundred decimal places.