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.
|||K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968)|
|||S. Lang, "Introduction to modular forms" , Springer (1976)|
|[a1]||E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)|
Euler identity. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Euler_identity&oldid=11612