# Euler identity

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) |

#### Comments

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.

#### 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=11612