### Number of primes...

Posted:

**Wed Feb 25, 2009 8:36 pm UTC**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!

[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!