zeta(2), zeta(3), zeta(4) compared
z(3) sits between two values with known closed forms involving pi. Whether z(3) involves pi is still unknown.
Partial sums of 1+1/8+1/27+... converging to ζ(3)
The partial sums 1 + 1/8 + 1/27 + 1/64... approach ζ(3) ≈ 1.20206 from below. Convergence is slow: even at n=50 the sum is still 0.003 away.
Even integers: rational multiples of powers of π. Odd integers: unknown.
Table showing zeta at even integers known as pi fractions but odd integers unknown
| Gerade s: exakte Formeln | Ungerade s: Rätsel |
|---|---|
| ζ(2) = π²/6 | ζ(3) = 1,20206... |
| ζ(4) = π⁴/90 | irasional (Apéry 1978) |
| ζ(6) = π⁶/945 | ζ(5) = 1,03693... |
| ζ(8) = π⁸/9450 | irasional? unbekannt |
| Alle = rasional × π^s | Keine bekannte π-Verbindung |
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