Euler identity

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The relation

where is an arbitrary real number and the product extends over all prime numbers . The Euler identity also holds for all complex numbers with .

The Euler identity can be generalized in the form

which holds for every totally-multiplicative arithmetic function for which the series is absolutely convergent.

Another generalization of the Euler identity is the formula

for the Dirichlet series

corresponding to the modular functions

of weight , which are the eigen functions of the Hecke operator.


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


The product

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.


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