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
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $12$