Euler identity
The relation
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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
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which holds for every totally-multiplicative arithmetic function for which the series
is absolutely convergent.
Another generalization of the Euler identity is the formula
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for the Dirichlet series
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corresponding to the modular functions
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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
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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) |
Euler identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_identity&oldid=11612