= Solution
The <Möbius function> satisfies $\mu(p^e)^2=1$ for $e=0,1$ and is zero for $e\ge2$. Therefore
$$
H_1(s)=\prod_p(1+p^{-s})=\frac{\zeta(s)}{\zeta(2s)}.
$$
The inverse local factor is $(1+p^{-s})^{-1}=\sum_{e\ge0}(-1)^ep^{-es}$. Consequently the inverse under <Dirichlet convolution> is the <Liouville function>
$$
\boxed{h_2(n)=(-1)^{\Omega(n)},}
$$
where $\Omega(n)$ counts prime factors with multiplicity. Both functions are <multiplicative functions>. At $p^e$ with $e\ge1$, their convolution is $(-1)^e+(-1)^{e-1}=0$, while its value at one is one. Multiplicativity then proves $h_1*h_2=I$ on every positive <integer>. Its <Dirichlet series> is $H_2(s)=\zeta(2s)/\zeta(s)$ for $\operatorname{Re}s>1$.
Back to article page
