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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=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