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Values of ζ(s) known exactly at even integers, mysterious at odd ones
Table of zeta function values at even integers
| s | ζ(s) | exakte Form |
|---|
| 2 | 1,64493… | π²/6 |
| 3 | 1,20206… | unbekannt, Apéry |
| 4 | 1,08232… | π⁴/90 |
| 6 | 1,01734… | π⁶/945 |
| -2,-4,… | 0 | triviale Nullstellen |
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The critical strip and Riemann Hypothesis
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Euler product formula: primes and integers connected
ζ(s) = Σ 1/nˢ = Π (1−p⁻ˢ)⁻¹
Left: sum over all positive integers n. Right: product over all primes p.
The equality encodes the Fundamental Theorem of Arithmetic. Riemann extended ζ to complex s.
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Question
Berapa ζ(2)?
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