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Old 2010-08-16, 20:12   #1
CRGreathouse's Avatar
Aug 2006

597910 Posts
Default 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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Old 2010-08-22, 23:47   #2
wblipp's Avatar
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}
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