Euler's Product form of Riemann Zeta Function

Theorem

Let $s \in \R: s > 1$.

Then:

$\ds \sum_{k \mathop \in \N_{>0} } \dfrac 1 {k^s} = \prod_{p \mathop \in \Bbb P} \dfrac 1 {1 - 1 / p^s}$

where $\Bbb P$ denotes the set of all prime numbers.

Proof

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Source of Name

This entry was named for Leonhard Paul Euler.

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler