# Euler product

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The infinite product

$$\prod_p\left(1-\frac{1}{p^s}\right)^{-1},$$

where $s$ is a real number and $p$ runs through all prime numbers. This product converges absolutely for all $s>1$. The analogous product for complex numbers $s=\sigma+it$ converges absolutely for $\sigma>1$ and defines in this domain the Riemann zeta-function

$$\zeta(s)=\prod_p\left(1-\frac{1}{p^s}\right)^{-1}=\sum_{n=1}^\infty\frac{1}{n^s}.$$

#### Comments

See also Euler identity and Zeta-function.

How to Cite This Entry:
Euler product. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Euler_product&oldid=33842
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article