Zeta of 2 as Product of Fractions with Prime Numerators
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Theorem
\(\ds \map \zeta 2\) | \(=\) | \(\ds \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 2 1 \times \dfrac 2 3 \times \dfrac 3 2 \times \dfrac 3 4 \times \dfrac 5 4 \times \dfrac 5 6 \times \dfrac 7 6 \times \dfrac 7 8 \times \dfrac {11} {10} \times \dfrac {11} {12} \times \dfrac {13} {12} \times \dfrac {13} {14} \times \cdots\) |
where:
- $\zeta$ denotes the Riemann zeta function
- $\ds \prod_p$ denotes the continued product over all prime numbers.
Proof
From Sum of Reciprocals of Powers as Euler Product:
- $\ds \map \zeta z = \prod_p \frac 1 {1 - p^{-z} }$
where $p$ ranges over the prime numbers.
Thus:
\(\ds \map \zeta 2\) | \(=\) | \(\ds \prod_p \frac 1 {1 - p^{-2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \prod_p \frac {p^2} {p^2 - 1}\) | multiplying top and bottom by $p^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \prod_p \frac {p^2} {\paren {p - 1} \paren {p + 1} }\) | Difference of Two Squares | |||||||||||
\(\ds \) | \(=\) | \(\ds \prod_p \paren {\frac p {p - 1} } \paren {\frac p {p + 1} }\) |
which is the result required.
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 64493 \, 4066 \ldots$