- I meant sum over the primes.

The page in question uses a double-sided implies symbol, i.e. implies and is

implied by.

Going from left to right is no problem for me - any finite number of primes

has a finite sum.

But how does the statement that ak+b contains an infinite number of primes

imply that sum(1/p) is infinite (compare with Brun's constant)?

Jon Perry

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-----Original Message-----

From: Marcel Martin [mailto:znz@...]

Sent: 29 April 2002 23:34

To: PrimeNumbers@yahoogroups.com

Subject: [PrimeNumbers] 1/p = p, always, especially whan p is prime

I just read a deep criticism about the statement

For all p prime such that p=a mod q, Sum(1/p) = infinity

found at

http://algo.inria.fr/banderier/Seminar/Vardi/index.html

I cannot see what motivated the criticism, except maybe the fact

that for some people Sum(1/p) is the same that Sum(p).

Marcel Martin

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Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ - I would comment, except for the site seems to have disappeared.

http://algo.inria.fr/banderier/Seminar/Vardi/index.html

>I meant sum over the primes.

i.e. we sum 1/p for p in ak+b, i.e. p over ak+b.

>Your second error was to believe that INRIA can claim 'the sum

My third error must therefore have been to ....

>of an infinite numbers of positive increasing values is equal to

>infinity'.

While I have no problem believing that every sum of inverted primes from an

AP is infinite, I do not see how the RHS implies the LHS without further and

extensive work.

>At first glance, it doesn't look trivial

It's not. This would say a great deal about the distribution of primes in

AP's, and perhaps more importantly, offer insight as to the 'cut-off' point

that determines whether a sequence is finite or infinite.

Jon Perry

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-----Original Message-----

From: Marcel Martin [mailto:znz@...]

Sent: 30 April 2002 22:31

Cc: PrimeNumbers@yahoogroups.com

Subject: Re: [PrimeNumbers] 1/p = p, always, especially whan p is prime

>I meant sum over the primes.

Yes, I had understood. And that's one of your two errors. What is

summed is the inverses of the primes, not the primes themselves.

Your second error was to believe that INRIA can claim 'the sum

of an infinite numbers of positive increasing values is equal to

infinity'.

The INRIA (very roughly, "National Institut for Research in Computer

Science") is not a bozo club. When one finds what looks like an error

in what they published, there are two possibilities:

1) this is a typo;

2) the reader doesn't quite understand what they wrote.

>But how does the statement that ak+b contains an infinite number

That's precisely what they wrote, a proof. At first glance, it

>of primes imply that sum(1/p) is infinite?

doesn't look trivial. And, always at first glance, I am not sure

to be able to understand it. I think that my knowledge about

the tools they used could be written on a postage stamp. A small

one.

Marcel Martin

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Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ - DOES "THE SUM OF THESE PRIMES" MEAN "THE SUM OF THE INVERSES"?

That's precisely why you ironically wrote 'the immortal statement',

because if we sum the primes, in that case, of course, the sum is

trivially infinite.

It was a small, and fairly self-correcting typo, which I did correct in a

following post.

As for the RHS implying the LHS, I have yet to comprehend its subtleties.

Jon Perry

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-----Original Message-----

From: Marcel Martin [mailto:znz@...]

Sent: 01 May 2002 23:32

Cc: PrimeNumbers@yahoogroups.com

Subject: Re: [PrimeNumbers] 1/p = p, always, especially whan p is prime

>>I meant sum over the primes.

That's incredible. In your first post, you wrote this

>i.e. we sum 1/p for p in ak+b, i.e. p over ak+b.

>Apparently, the fact that there are an infinite number of primes in a+kq

^^^^^^^^^^^^^^^^^^^^^^^

>implies the sum of these primes is infinite.

DOES "THE SUM OF THESE PRIMES" MEAN "THE SUM OF THE INVERSES"?

That's precisely why you ironically wrote 'the immortal statement',

because if we sum the primes, in that case, of course, the sum is

trivially infinite.

You was wrong. Period. And now, as usual, you are trying to get

out of the mess by trying to change what you said. You did that

with me about Wilson theorem, you did that with Jud McCranie about

Godel theorem. You always do that.

Is it so difficult to say 'Ok, I was wrong. Next chapter.'?

Marcel Martin

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