Asymptotic density of k-almost primes

Abstract

Landau's well known asymptotic formula $$N_k(x):=\mid \{n\le x:\Omega (n)=k\}\mid \sim \left(\frac{x}{\log x}\right)\frac{(\log \log x{)}^{k-1}}{(k-1)!}(x\to \infty ),$$ which also holds for $${\pi }_k(x):=\mid \{n\le x:\omega (n)=k\}\mid ,$$ is known to be fairly poor for $k>1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and ${\pi }_k(x)$ becomes very technical [1, Chapter II.6, $\S$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

Withdrawn by the author due to errors in the paper. This work was an early attempt at a number theory paper and has since been superseded by corrected and substantially revised research

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