Error term for Converting $\pi(x)$ to $\textrm{Li}(x)$ in the sums.

Question

$$\pi^{(k)}(x) \sim \Sigma_{n'\le x}{\pi \left(\frac{x}{n'} \right)} + O(E_1(x))$$

$$\Sigma_{n'\le x}{\pi \left(\frac{x}{n'}\right)} \sim \Sigma_{n'\le x}{\textrm{Li}\left(\frac{x}{n'}\right)} + O(E_2(x))$$

$\pi^{(k)}$ being the count of square free numbers with $k$ prime divisors ($\mathbb{P}_k(\mathbb{Z^+})$) and $n' \in \mathbb{P}_{k-1}(\mathbb{Z}^+)$ the set of positive integers with $k-1$ distinct prime divisors. I'm looking for a way to overcome the $O\left(\frac{x (\log\log x)^{k-2}}{\log x}\right)$ barrier, which is necessary to study compound arithmetic progressions, and some existential problems involving the primes.

These are the errors I need to study, before I explore any expressions $\pi_{k, N} + O(E_{k,N})$, where $E_{k,N}$ is the error in terms of $k$, the number of prime divisors in the set to estimate, and $N$, the number of terms in the summation expansion of the logarithmic integral. This can be explored with Abel Summation and Taylor Series expansions, but it is useful to know where to stop deriving $\pi_{k, N} + O(E_{k,N})$, and I have no clue how to get these terms.

Is there a newer source or a way where I don't have to assume the Riemann Hypothesis to find bounds?

UPDATE: Quoting H.M. Edwards, de la Valleé Poussin's result implies

$\pi(x) = $Li$(x) + O\bigg( e^{-\sqrt{c_1 log(x)}} \bigg)$

But I'm still not sure how to apply that to $\pi^{(k)}$ for $k$ > 1.

Also, the Wolfram Mathworld reference for almost primes might be useful as well. Does that imply $E_1(x)$ is related to the Bernoulli polynomial?

Answer 1

See this question on MathOverflow, and in particular this answer of Micah Milinovich:

In Tenenbaum's book "Introduction to analytic and probabilistic number theory" he uses the Selberg-Delange method to prove that the estimate $$\pi_k(x):=\sum_{n\leq x, \ \omega(n)=k} 1 = \frac{x}{\log x} \sum_{j=0}^N \frac{P_{j,k}(\log\log x)}{(\log x)^j} + O_A\left(\frac{x(\log\log x)^k}{k! \log x} R_N(x) \right) $$ holds uniformly for $x\geq 3$, $1\leq k \leq A \log \log x$, and $N\geq 0$ where $P_{j,k}$ is a polynomial of degree at most $k-1$. $$R_N(x) = e^{-c_1\sqrt{\log x}} + \left(\frac{c_2 N+1}{\log x}\right)^{N+1},$$ and $c_1$ and $c_2$ are positive constants which may depend on $A$. This is Theorem $4$ of Chapter $6$.

A similar result holds for $$N_k(x)=\sum_{\begin{array}{c} n\leq x\\ \Omega(n)=k \end{array}} 1,$$ the number of integers $n\leq x$ with $k$ not necessarily distinct prime factors, where the polynomials $P_{j,k}$ are replaced with different polynomials $Q_{j,k}$ which are also of degree at most $k-1$, however we must insist that $k<2\log \log x$, since the powers of two affect things too much otherwise.