Asymptotic density of k-almost primes

CRGreathouse · 2010-08-16, 20:12

Of course this is
$\pi_k(n)=\frac{n(\log\log n)^{k-1}}{(k-1)!\log n}+o\left(\frac{n(\log\log n)^{k-1}}{\log n}\right)$
for any fixed k, but are there better Li-type estimates, or at least more asymptotic terms (as Cipolla 1902 gives for $\pi(n)$ )?

For extra points:

Are Dusart-style absolute error bounds known?
Are there better asymptotic -- or better, Shoenfeld-style -- bounds known on the RH?

This comes out of comparing the number of n-bit semiprimes to the number predicted by the formula with k = 2, where I noticed what seemed to be a consistent relative (reciprocal density) bias of about 0.5 over the range 15-50 bits.

wblipp · 2010-08-22, 23:47

You might try an Li version motivated by comparison to the prime case. When k=1 this is n/log(n), which we know to improved upon by integral (1/log(x)). This can be motivated by the argument that 1/log(x) is correct density. The analogous adjustment would be the integral of

$\frac{(\log\log x)^{k-1}}{(k-1)!\log x}$

2010-08-16, 20:12	#1
CRGreathouse Aug 2006 5979₁₀ Posts	Asymptotic density of k-almost primes Of course this is $\pi_k(n)=\frac{n(\log\log n)^{k-1}}{(k-1)!\log n}+o\left(\frac{n(\log\log n)^{k-1}}{\log n}\right)$ for any fixed k, but are there better Li-type estimates, or at least more asymptotic terms (as Cipolla 1902 gives for $\pi(n)$ )? For extra points: Are Dusart-style absolute error bounds known? Are there better asymptotic -- or better, Shoenfeld-style -- bounds known on the RH? This comes out of comparing the number of n-bit semiprimes to the number predicted by the formula with k = 2, where I noticed what seemed to be a consistent relative (reciprocal density) bias of about 0.5 over the range 15-50 bits.

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2010-08-22, 23:47	#2
wblipp "William" May 2003 New Haven 2⁶·37 Posts	You might try an Li version motivated by comparison to the prime case. When k=1 this is n/log(n), which we know to improved upon by integral (1/log(x)). This can be motivated by the argument that 1/log(x) is correct density. The analogous adjustment would be the integral of $\frac{(\log\log x)^{k-1}}{(k-1)!\log x}$