Namespaces
Variants
Actions

Difference between revisions of "Euler product"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Category:Number theory)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
{{TEX|done}}
 
The infinite product
 
The infinite product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365601.png" /></td> </tr></table>
+
$$\prod_p\left(1-\frac{1}{p^s}\right)^{-1},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365602.png" /> is a real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365603.png" /> runs through all prime numbers. This product converges absolutely for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365604.png" />. The analogous product for complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365605.png" /> converges absolutely for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365606.png" /> and defines in this domain the Riemann zeta-function
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036560/e0365607.png" /></td> </tr></table>
+
$$\zeta(s)=\prod_p\left(1-\frac{1}{p^s}\right)^{-1}=\sum_{n=1}^\infty\frac{1}{n^s}.$$
  
  
Line 11: Line 12:
 
====Comments====
 
====Comments====
 
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
 
See also [[Euler identity|Euler identity]] and [[Zeta-function|Zeta-function]].
 +
 +
[[Category:Number theory]]

Latest revision as of 18:50, 18 October 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
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article