Product of Dirichlet Series

Theorem

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

∑n=1∞f(n)nsand∑n=1∞g(n)ns.

Formally, their product is given by

(∑n=1∞f(n)ns)(∑n=1∞g(n)ns)=∑k=1∞(f∗g)(k)ks

and this product converges absolutely if both series converge absolutely.

Proof

Formally,

(∑n=1∞f(n)ns)(∑n=1∞g(n)ns)=∑(m,n)∈Z>02f(n)g(m)(nm)s=∑k=1∞∑(m,n)∈Z>02mn=kf(n)g(m)(nm)s=∑k=1∞1ks∑(m,n)∈Z>02mn=kf(n)g(m)=∑k=1∞1ks∑n∣kf(n)g(kn)=∑k=1∞(f∗g)(k)ks.

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