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.



Question on an arithmetic function with the sieve of Eratosthenes
Just as the original sieve of Eratosthenes formula is a rewriting of
$$
E\_1(x) = \sum\_{n\le x} \prod\_{\substack{p\le\sqrt x \\ p\mid n}} (1-1) = \sum\_{\substack{n\le x \\ p\mid n\Rightarrow p>\sqrt x}} 1,
$$
this modification is the same as
$$
E\_a(x) = \sum\_{n\le x} \prod\_{\substack{p\le\sqrt x \\ p\mid n}} (1-a) = \sum\_{n\le x} (1-a)^{\omega\_{\sqrt x}(n)},
$$
where $\omega\_{\sqrt x}(n)$ is the number of distinct primes up to $\sqrt x$ that divide $n$. This sum should be quite similar to $\sum\_{n\le x} (1-a)^{\omega(n)}$, which is classical even for $a$ a complex number (see Montgomery/Vaughan or Tenenbaum's books).


