Page 201 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

For x> 1, the series converges. The Riemann zeta function can also be defined for other values of x
using complex analysis.

Riemann proved that the zeta function is also equal to the infinite product

∞
1
ζ(x)= ,
1 − p −x
n=1 n
where

∞
P = {p 1 ,p 2 ,...,p n ,...} = {p n } n=1

is the set of all prime numbers. Similar to a series, if the sequence of partial products,

N
1
∞
{P N } N=1 , P N = 1 − p −x ,
n=1 n

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