Riemann Zeta Function of 1000

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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:

\(\ds \map \zeta {1000}\) \(\le\) \(\ds \dfrac {2^{1000} } {2^{1000} - 1}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {1 - 2^{-1000} }\)

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

$\blacksquare$


Sources