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Difference between revisions of "Euler identity"

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The relation
 
The relation
 
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$$
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\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" />.
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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
 
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$$
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\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.
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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
 
Another generalization of the Euler identity is the formula
 
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$$
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\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]]
 
+
$$
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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
 
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$$
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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.
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of weight $2k$, which are the eigen functions of the Hecke operator.
  
 
====References====
 
====References====
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====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>
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\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>

Revision as of 17:08, 12 October 2014

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=33574
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article