Xjyuk

I need to know, or at least have a good bound for, the asymptotic behaviour on $x$ of amount of integers less o 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

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.

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).