# 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.

#### References

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