# §27.4 Euler Products and Dirichlet Series

The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. Every multiplicative satisfies the identity

if the series on the left is absolutely convergent. In this case the infinite product on the right (extended over all primes ) is also absolutely convergent and is called the Euler product of the series. If is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes

Euler products are used to find series that generate many functions of multiplicative number theory. The completely multiplicative function gives the Euler product representation of the Riemann zeta function 25.2(i)):

27.4.3.

The Riemann zeta function is the prototype of series of the form

called Dirichlet series with coefficients . The function is a generating function, or more precisely, a Dirichlet generating function, for the coefficients. The following examples have generating functions related to the zeta function:

In (27.4.12) and (27.4.13) is the derivative of .