Namespaces
Variants
Actions

Difference between revisions of "Euler identity"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (better)
 
(3 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
The relation
 
The relation
 
+
$$
<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/e036490/e0364901.png" /></td> </tr></table>
+
\sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \left({1 - \frac{1}{p^s} }\right)^{-1}
 
+
$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e0364902.png" /> is an arbitrary real number and the product extends over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e0364903.png" />. The Euler identity also holds for all complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e0364904.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e0364905.png" />.
+
where $s>1$ is an arbitrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma > 1$.
  
 
The Euler identity can be generalized in the form
 
The Euler identity can be generalized in the form
 
+
$$
<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/e036490/e0364906.png" /></td> </tr></table>
+
\sum_{n=1}^\infty f(n) = \prod_p \left({1 - \frac{1}{f(p)} }\right)^{-1}
 
+
$$
which holds for every totally-multiplicative arithmetic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e0364907.png" /> for which the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e0364908.png" /> is absolutely convergent.
+
which holds for every [[totally multiplicative function|totally multiplicative arithmetic function]] $f(n)$ for which the series $\sum_{n=1}^\infty f(n) $ is absolutely convergent.
  
 
Another generalization of the Euler identity is the formula
 
Another generalization of the Euler identity is the formula
 
+
$$
<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/e036490/e0364909.png" /></td> </tr></table>
+
\sum_{n=1}^\infty \frac{a_n}{n^s} = \prod_p \left({1 - a_p p^{-s} + p^{2k-1-2s} }\right)^{-1}
 
+
$$
 
for the [[Dirichlet series|Dirichlet series]]
 
for the [[Dirichlet series|Dirichlet series]]
 
+
$$
<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/e036490/e03649010.png" /></td> </tr></table>
+
F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}\ ,\ \ \ s = \sigma + it\ ,\ \ \ \sigma > 1
 
+
$$
 
corresponding to the modular functions
 
corresponding to the modular functions
 
+
$$
<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/e036490/e03649011.png" /></td> </tr></table>
+
f(z) = \sum_{n=1}^\infty a_n e^{2\pi i n z}
 
+
$$
of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036490/e03649012.png" />, which are the eigen functions of the Hecke operator.
+
of weight $2k$, which are the eigen functions of the Hecke operator.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Chandrasekharan,  "Introduction to analytic number theory" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Introduction to modular forms" , Springer  (1976)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Chandrasekharan,  "Introduction to analytic number theory" , Springer  (1968)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,  "Introduction to modular forms" , Springer  (1976)</TD></TR>
 +
</table>
  
  
Line 32: Line 35:
 
====Comments====
 
====Comments====
 
The product
 
The 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/e036490/e03649013.png" /></td> </tr></table>
+
\prod_p \left({1 - \frac{1}{p^s} }\right)^{-1}
 
+
$$
 
is called the Euler product. For Hecke operators in connection with modular forms see [[Modular form|Modular form]]. For totally-multiplicative arithmetic functions cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]].
 
is called the Euler product. For Hecke operators in connection with modular forms see [[Modular form|Modular form]]. For totally-multiplicative arithmetic functions cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of the Riemann zeta-function" , Clarendon Press  (1951)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of the Riemann zeta-function" , Clarendon Press  (1951)</TD></TR>
 +
</table>

Latest revision as of 19:13, 14 December 2015

The relation $$ \sum_{n=1}^\infty \frac{1}{n^s} = \prod_p \left({1 - \frac{1}{p^s} }\right)^{-1} $$ where $s>1$ is an arbitrary real number and the product extends over all prime numbers $p$. The Euler identity also holds for all complex numbers $s = \sigma + it$ with $\sigma > 1$.

The Euler identity can be generalized in the form $$ \sum_{n=1}^\infty f(n) = \prod_p \left({1 - \frac{1}{f(p)} }\right)^{-1} $$ which holds for every totally multiplicative arithmetic function $f(n)$ for which the series $\sum_{n=1}^\infty f(n) $ is absolutely convergent.

Another generalization of the Euler identity is the formula $$ \sum_{n=1}^\infty \frac{a_n}{n^s} = \prod_p \left({1 - a_p p^{-s} + p^{2k-1-2s} }\right)^{-1} $$ for the Dirichlet series $$ F(s) = \sum_{n=1}^\infty \frac{a_n}{n^s}\ ,\ \ \ s = \sigma + it\ ,\ \ \ \sigma > 1 $$ corresponding to the modular functions $$ f(z) = \sum_{n=1}^\infty a_n e^{2\pi i n z} $$ of weight $2k$, which are the eigen functions of the Hecke operator.

References

[1] K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968)
[2] S. Lang, "Introduction to modular forms" , Springer (1976)


Comments

The product $$ \prod_p \left({1 - \frac{1}{p^s} }\right)^{-1} $$ is called the Euler product. For Hecke operators in connection with modular forms see Modular form. For totally-multiplicative arithmetic functions cf. Multiplicative arithmetic function.

References

[a1] E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)
How to Cite This Entry:
Euler identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_identity&oldid=11612
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article