Difference between revisions of "Euler product"
From Encyclopedia of Mathematics
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The infinite product | The infinite product | ||
| − | + | $$\prod_p\left(1-\frac{1}{p^s}\right)^{-1},$$ | |
| − | where | + | 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 |
| − | + | $$\zeta(s)=\prod_p\left(1-\frac{1}{p^s}\right)^{-1}=\sum_{n=1}^\infty\frac{1}{n^s}.$$ | |
Revision as of 18:55, 17 April 2014
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://encyclopediaofmath.org/index.php?title=Euler_product&oldid=12289
Euler product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_product&oldid=12289
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article