Definition
In general, if is a multiplicative function, then the Dirichlet series
is equal to
where the product is taken over prime numbers, and is the sum
In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes .
An important special case is that in which is totally multiplicative, so that is a geometric series. Then
as is the case for the Riemann zeta-function, where, and more generally for Dirichlet characters.
Read more about this topic: Euler Product
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