I am interested in the set $A$ of all positive integer numbers such that when factored into primes, the sum of the exponents is odd (I think of $A$ as the multiplicative odd numbers).


I want to know if it has positive upper density, more precisely
$$\bar d(A):=\limsup\_{n\to\infty}\frac{|A\cap[1,n]|}n$$
I think I read somewhere that it has density $1/2$ (and the $\lim$ exist, not just the $\limsup$), but I would be happy with a proof that $\bar d(A)>0$.



Density of the "multiplicative odd numbers"
Gerry has the right idea here: you are asking about the limiting behaviour of the sum
$$\frac{1}{x} \sum\_{n \leq x}{\frac{1 - (-1)^{\Omega(n)}}{2}}.$$
The arithmetic function $\lambda(n) = (-1)^{\Omega(n)}$ is known as Liouville's function. It is well-known (and equivalent to the prime number theorem!) that the summatory function of the Liouville function,
$$L(x) = \sum\_{n \leq x}{\lambda(n)},$$
satisfies the asymptotic
$$L(x) = o(x)$$
as $x$ tends to infinity. (In fact, one can probably improve this slightly in the usual way to get better error terms in the prime number theorem.) So it is indeed true that
$$d(A) = \lim\_{x \to \infty} \frac{1}{x} \sum\_{n \leq x}{\frac{1 - (-1)^{\Omega(n)}}{2}} = \frac{1}{2},$$
and with a little work you could actually say something slightly stronger about the rate at which this converges.


