Product of Dirichlet Series

Theorem

Let f and g be arithmetic functions and consider the Dirichlet series'

n=1f(n)nsandn=1g(n)ns.

Formally, their product is given by

(n=1f(n)ns)(n=1g(n)ns)=k=1(fg)(k)ks

and this product converges absolutely if both series converge absolutely.

Proof

Formally,

(n=1f(n)ns)(n=1g(n)ns)=(m,n)Z>02f(n)g(m)(nm)s=k=1(m,n)Z>02mn=kf(n)g(m)(nm)s=k=11ks(m,n)Z>02mn=kf(n)g(m)=k=11ksnkf(n)g(kn)=k=1(fg)(k)ks.

If both series converge absolutely, then the convergence of the formal series follows from this result.