## prime number question

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• does the series 1/p, where p are prime numbers, converge or diverge? please cc the answer to ocsurfer19@yahoo.com keep up the good work, Anand
Message 1 of 4 , Aug 6, 2003
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does the series 1/p, where p are prime numbers, converge or diverge?

keep up the good work,

Anand
• Diverges. If you take all the primes p
Message 2 of 4 , Aug 7, 2003
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Diverges. If you take all the primes p<=x, and sum 1/p, the value is
loglog x+C, where C is some number which tends to a fixed value as x
goes to infinity.

This formula, and others, are elementary properties of the series of
primes, and can be found in practically any basic number theory book.

Andy

> does the series 1/p, where p are prime numbers, converge or diverge?
>
>
> keep up the good work,
>
> Anand
• ... It diverges, though very slowly. This fact alone provides a proof that there are an infinite number of primes. Paul
Message 3 of 4 , Aug 7, 2003
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> From: ansamanta [mailto:ocsurfer19@...]

> does the series 1/p, where p are prime numbers, converge or diverge?
>
>
> 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
• --On Thursday, August 07, 2003 1:21 AM -0700 Paul Leyland ... diverge? ... number of primes. ... And that the proportion of numbers that are prime decreases
Message 4 of 4 , Aug 7, 2003
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--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?
>>
>>
>> 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

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