The Theorem of the Day: Euler’s Product Formula for ζ(s)

For any complex number s having real part Re(s) > 1, 

$ζ(s) = 1 + \dfrac{1}{2^s} + \dfrac{1}{3^s} + \dfrac{1}{4^s} + \dfrac{1}{5^s} + . . . $

$= (1 − \dfrac{1}{p_1^s})^{-1}−(1 − \dfrac{1}{p_2^s})^{-1}−(1 − \dfrac{1}{p_3^s})^{-1} · · · $

where $p_1, p_2, p_3, . . .$ are the prime numbers.

Πηγή: theoremoftheday