Riemann Zeta Function of 1000

Theorem

To at least $100$ decimal places:

$\zeta \left({1000}\right) \approx 1$

where $\zeta$ denotes the Riemann zeta function.

Proof

By definition of the general harmonic numbers:

$\displaystyle \zeta \left({r}\right) = \lim_{n \mathop \to \infty} H_n^{\left({r}\right)} = \sum_{k \mathop \ge 1} \frac 1 {k^r}$
 $\displaystyle \zeta \left({1000}\right)$ $\le$ $\displaystyle \dfrac {2^{1000} } {2^{1000} - 1}$ $\displaystyle$ $=$ $\displaystyle \dfrac 1 {1 - 2^{-1000} }$

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

$\blacksquare$