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I have developed a formula for almost primes which is far more accurate asymptotically than Landau's well known

$$\pi_k(n) \sim \left( \frac{n}{\log n} \right) \frac{(\log\log n)^{k-1}}{(k - 1)!}$$

(Landau's is not good for high $n$, whereas the one I have been working on actually gets more accurate the higher $n$ becomes - see here.)

Is this of any significance?

Just out of interest, I have included some plots up to $n=9$:

enter image description here

where actual is green, Landau is blue, & mine is red.

(Note: I have changed the scale in each one.)

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  • $\begingroup$ I cannot understand what means "plots up to n=9" ? The first plot (n=1) is obvious prime counting function, i.e. k=1 and so on up to k=9 ? $\endgroup$ May 1 '17 at 20:23
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Yes, this is of significance. Why don't you write a paper and submit it to a journal?

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  • $\begingroup$ I have never written a paper before. How would I go about submitting it to a journal? I have no proof for it either, but have tested it fairly exhaustively. Does this matter? $\endgroup$
    – martin
    Dec 15 '13 at 15:42
  • $\begingroup$ @martin yes, it does matter (since it means that your formula might not be more accurate for very large $n.$) I would writing up your ideas, and talking to a number theorist. You have to write things very carefully (include the results of your experiments), lest they take you for a crank. If they think you are serious, they will be happy to talk. $\endgroup$
    – Igor Rivin
    Dec 15 '13 at 15:47
  • $\begingroup$ Can you tell me where I might go about finding a number theorist who might look at the work? $\endgroup$
    – martin
    Dec 15 '13 at 15:50
  • $\begingroup$ @martin. I totally agree with what Igor Rivin told. This is worth to be published even if you do not have the complete proof. $\endgroup$ Dec 15 '13 at 15:50
  • $\begingroup$ @martin. Where are you currently ? $\endgroup$ Dec 15 '13 at 15:51

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