# Euler identity

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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://www.encyclopediaofmath.org/index.php?title=Euler_identity&oldid=36937
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article