Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural
numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes
(finite or infinite). We given an elementary proof that the partial sums
$\sum_{n \in < \P >: n \leq x} \frac{\mu(n)}{n}$ are bounded in magnitude by 1.
With the aid of the prime number theorem, we also show that these sums converge
to $\prod_{p \in \P} (1 - \frac{1}{p})$ (the case when $\P$ is all the primes
is a well-known observation of Landau). Interestingly, this convergence holds
even in the presence of non-trivial zeroes and poles of the associated zeta
function $\zeta_\P(s) := \prod_{p \in \P} (1-\frac{1}{p^s})^{-1}$ on the line
$\{\Re(s)=1\}$.

As equivalent forms of the first inequality, we have $|\sum_{n \leq x:
(n,P)=1} \frac{\mu(n)}{n}| \leq 1$, $|\sum_{n|N: n \leq x} \frac{\mu(n)}{n}|
\leq 1$, and $|\sum_{n \leq x} \frac{\mu(mn)}{n}| \leq 1$ for all $m,x,N,P \geq
1$.