Bsrgvty

I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less or equal than $x$ that are square free and with exactly $k$ primes on its decomposition. That is the cardinal of the following set
$$
mathcalJ_T(x,k) = n in mathbbN : n le x, Omega(n)=k , n mbox is square free .$$
Other way to describe this cardinal, using the Möbius function, is
$$ |mathcalJ_T(x,k) | = sum_Omega(n) = k, n le x |mu(n)|.$$

I am looking for the asymptotic behaviour on $x$, but this will depend also on $k$ in some way. The bound given using
$$
sum_Omega(n) = k, n le x |mu(n)| le sum_n le x |mu(n)| le frac6xpi^2 + O(sqrtx) ll x,
$$

is not good enought for my purposes.

Thanks in advanced, any reference or idea is helpful

It hasn't been pointed out yet that you can derive the answer simply directly from the statement of the Selberg–Sathe theorem, which gives (for fixed $k$) the asymptotic formulas
beginalign*
# nle xcolon omega(n) = k &sim frac xlog x frac(loglog x)^k-1(k-1)! \
# nle xcolon Omega(n) = k &sim frac xlog x frac(loglog x)^k-1(k-1)!.
endalign*
(Here $omega$ and $Omega$ count the prime factors of $n$ without and with multiplicity, respectively.)

Note that any nonsquarefree integer $n$ with $Omega(n) = k$ satisfies $omega(n) = j$ for some $1le jle k-1$. Therefore the bounds
beginalign*
# nle xcolon Omega(n) = k,, ntext is squarefree &le # nle xcolon Omega(n) = k \
# nle xcolon Omega(n) = k,, ntext is squarefree &ge # nle xcolon Omega(n) = k - sum_j=1^k-1 # nle xcolon omega(n) = j ,
endalign*
together with the Selberg–Sathe asymptotic formulas above, immediately imply that
$$
# nle xcolon Omega(n) = k,, ntext is squarefree sim frac xlog x frac(loglog x)^k-1(k-1)!.
$$

This argument can be made to work with some uniformity in $k$ as well (I think $k=o(loglog x)$ is what is needed).

You can derive a very precise asymptotic expansion of your quantity by the Selberg-Delange method.

I recommend that you adapt, to your situation, the arguments of Section II.6.1 of Tenenbaum: Introduction to analytic and probabilistic number theory. The starting point of your analysis should be the formula
$$sum_text$n$ square-freez^omega(n)n^-s=prod_pleft(1+fraczp^sright).$$
Then you will need to "factor out" $zeta(s)^z$ and proceed as in the mentioned chapter, where the analysis is carried out without the restriction that $n$ is square-free.