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Number of primes...

Posted: Wed Feb 25, 2009 8:36 pm UTC
by taby
Where [imath]p_k[/imath] is the [imath]k[/imath]th prime number, the following ratio [imath]x[/imath] seems to settle toward some non-zero number:

[imath]\lim_{k \to \infty} \frac{k}{p_k} = x[/imath].

For example:
[imath]k = 1e0[/imath], [imath]p_k = 2[/imath], [imath]x = 0.5[/imath],
[imath]k = 1e1[/imath], [imath]p_k = 29[/imath], [imath]x \approx 0.344828[/imath],
[imath]k = 1e2[/imath], [imath]p_k = 541[/imath], [imath]x \approx 0.184843[/imath],
...
[imath]k = 1e10[/imath], [imath]p_k = 252097800623[/imath], [imath]x \approx 0.039667[/imath],
[imath]k = 1e11[/imath], [imath]p_k = 2760727302517[/imath], [imath]x \approx 0.036222[/imath],
[imath]k = 1e12[/imath], [imath]p_k = 29996224275833[/imath], [imath]x \approx 0.033338[/imath].

That is... For the integers 1 through 2 inclusive, 0.5 of them are prime. Likewise, for the integers 1 through 29996224275833, approximately 0.03 of them are prime.

Is anyone familiar with what [imath]x[/imath] settles on as [imath]k \to \infty[/imath]? What is this constant called? Does anyone have access to [imath]k > 1e12[/imath]?

Thanks for any help!

Re: Number of primes...

Posted: Wed Feb 25, 2009 8:58 pm UTC
by Buttons
I believe the constant is called "zero", and its decimal expansion begins 0.00000....

It is odd that the ratio would appear to remain "high" for such large values, but primes do not have positive density on the naturals, so it cannot converge to a positive number.

Re: Number of primes...

Posted: Wed Feb 25, 2009 8:59 pm UTC
by stephentyrone
The Prime Number Theorem says that the relative error of approximating [imath]p_k[/imath] with [imath]k \log k[/imath] is asymptotically zero, so:
[math]\lim_{k\rightarrow\infty} \frac k {p_k} = \lim_{k\rightarrow\infty}\frac 1 {\log k} = 0.[/math]
It just goes to zero rather slowly.

Re: Number of primes...

Posted: Wed Feb 25, 2009 9:01 pm UTC
by taby
I see. Thank you stephentyrone for linking the concepts together for me.

Re: Number of primes...

Posted: Thu Feb 26, 2009 9:41 am UTC
by Matterwave1
I don't think k/pk is the number of primes in the first pk-integers that are prime. pk is the k'th prime number, not the number of primes below k...or have I missed something? :?

Re: Number of primes...

Posted: Thu Feb 26, 2009 1:59 pm UTC
by Buttons
Of the first pk integers, k of them are prime, namely p1 through pk.

Re: Number of primes...

Posted: Thu Feb 26, 2009 3:07 pm UTC
by taby
Here [imath]p_k[/imath] is the [imath]k[/imath]th prime number. Here [imath]n[/imath] would denote any positive integer. In terms of the prime counting function [imath]\pi(n)[/imath], where [imath]n = p_k[/imath], it is simplified to [imath]\pi(p_k) = k[/imath]. The "long" version would be:

[imath]\lim_{k \to \infty} \frac{\pi(p_k)}{p_k} = 0[/imath].

So the equation I wrote was basically a ratio involving the number of primes and the number of positive integers. As pointed out by the others, this ratio is 0 for an infinitely large [imath]n[/imath] (or [imath]p_k[/imath], in this case).

I'm not sure if that's what you're referring to.

Re: Number of primes...

Posted: Fri Feb 27, 2009 7:47 am UTC
by Matterwave1
Buttons wrote:Of the first pk integers, k of them are prime, namely p1 through pk.


oh, got it. thanks. :)

Re: Number of primes...

Posted: Sun Mar 01, 2009 5:33 pm UTC
by PM 2Ring
I recently posted a C program here that approximates the prime counting function, if you're interested. Can I have your prime number?

Here is the function compared to some actual values of Pi(x):
Spoiler:

Code: Select all

 
       The distribution of primes
       --------------------------
   x            Pi(x)      Riemann(x)

 1E1                4               5
 1E2               25              26
 1E3              168             168
 1E4            1,229           1,227
 1E5            9,592           9,587
 1E6           78,498          78,527
 1E7          664,579         664,667
 1E8        5,761,455       5,761,552
 1E9       50,847,534      50,847,455
 1E10     455,052,511     455,050,683