--On Thursday, August 07, 2003 1:21 AM -0700 Paul Leyland

<

pleyland@...> wrote:

>

>> From: ansamanta [mailto:ocsurfer19@...]

>

>> does the series 1/p, where p are prime numbers, converge or

diverge?

>>

>> please cc the answer to ocsurfer19@...

>>

>> keep up the good work,

>

> It diverges, though very slowly.

>

> This fact alone provides a proof that there are an infinite

number of primes.

>

>

> Paul

And that the proportion of numbers that are prime decreases more

slowly than n^m, where m < -1.

I vaguely remember that for m < -1, the sum is approximated by

(picture an integral sign instead of an "S")

oo oo

S n^m dn = [ 1/m * n^(m-1) ]

1 1

(note that at m=-1 this is a log instead, and log(oo)= oo , so

it diverges, and for 0>m>-1 it diverges since each term is

greater than the corresponding term with m = -1, I'm not sure

how to justify it in terms of the calculus though).

Anyhow, we have a converging series for all m < -1, which means

that the primes decrease in density at a speed only slightly

less than linear.

Sorry for rambling, it's been 3 years since I took any formal

math besides boolean logic and related fields.

Nathan