I want to ask some question related with the sieve of Eratosthenes.

The sieve of Eratosthenes: write it as $E_1(x) (=\pi(x)-\pi(\sqrt x)+1)$.

Then we have an obvious result $$E_1(x)/x\ln^{-1}x = 1,$$ as $x\rightarrow \infty$ by PNT.

The question comes, we can think weight $a$ (positive integer) to each summation of the series, and write it as $E_a(x)$. (It is not a "sieve" when $a>1$.) In detail, $$E_a(x):=x- a \sum \lfloor \frac{x}{p_i} \rfloor + a^2 \sum \lfloor \frac{x}{p_i p_j} \rfloor - \cdots ,$$ for same index of the sieve of Eratosthenes.

Then the question is that : Are there some constants $c_a$ such that satisfies $$E_a(x)/x\ln^{-a}x = c_a ?$$ And, are there any papers or discussions on this function?

I'd been searched on it, but nothing found.

Thanks for reading.