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27 Functions of Number TheoryMultiplicative Number Theory

§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 f satisfies the identity

27.4.1 n=1f(n)=p(1+r=1f(pr)),

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

27.4.2 n=1f(n)=p(1-f(p))-1.

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

27.4.3 ζ(s)=n=1n-s=p(1-p-s)-1,
s>1.

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

27.4.4 F(s)=n=1f(n)n-s,

called Dirichlet series with coefficients f(n). The function F(s) 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:

27.4.5 n=1μ(n)n-s =1ζ(s),
s>1,
27.4.6 n=1ϕ(n)n-s =ζ(s-1)ζ(s),
s>2,
27.4.7 n=1λ(n)n-s =ζ(2s)ζ(s),
s>1,
27.4.8 n=1|μ(n)|n-s =ζ(s)ζ(2s),
s>1,
27.4.9 n=12ν(n)n-s =(ζ(s))2ζ(2s),
s>1,
27.4.10 n=1dk(n)n-s =(ζ(s))k,
s>1,
27.4.11 n=1σα(n)n-s=ζ(s)ζ(s-α),
s>max(1,1+α),
27.4.12 n=1Λ(n)n-s =-ζ(s)ζ(s),
s>1,
27.4.13 n=2(logn)n-s =-ζ(s),
s>1.

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