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## Re: [PrimeNumbers] Digest Number 3824

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• This reference only shows that I independently proved Mertens theorem (my formula 9). It still doesn t answer why we need to adjust the calculated
Message 1 of 2 , Feb 24, 2014
This reference only shows that I independently proved Mertens' theorem (my formula 9).

It still doesn't 'answer' 'why' we need to adjust the calculated probability with this constant to get the prime count. As mentioned, I am worried about this giant wave of prime multiples that is building up with this factor. But maybe it is all self-regulary by adjusting the dx.

Also, so far, I haven't seen anyone using the probability product in an integration (my formula 6), which is just the key to the accurate prime counting results.

From: "primenumbers@yahoogroups.com" <primenumbers@yahoogroups.com>
To: primenumbers@yahoogroups.com
Sent: Monday, February 24, 2014 2:36 PM
Subject: [PrimeNumbers] Digest Number 3824

There is 1 message in this issue.

Topics in this digest:

1a. Re: Probabilistic approach to prime counting
From:  djbroadhurst

Message
________________________________________________________________________
1a. Re: Probabilistic approach to prime counting
Posted by:  david.broadhurst@... djbroadhurst
Date: Sun Feb 23, 2014 4:34 am ((PST))

Chris de Corte wrote:

> The question of why we had to correct our probabilities
> with a factor alpha [i.e. exp(Euler)] from 2 onward remains
> open. And to be honest, we can’t give a good mathematical answer.

Answer:

http://mathworld.wolfram.com/MertensTheorem.html

F. Mertens,
"Ein Beitrag zur analytischen Zahlentheorie",
J. reine angew. Math. 78 (1874) 46--62.

http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN243919689_0078

David

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• ... You proved nothing. Rather you made a faulty heuristic and compared it with some data on small primes. Your heuristic contained one very obvious mistake:
Message 2 of 2 , Feb 24, 2014
Chris De Corte wrote:

> I independently proved Mertens' theorem

You proved nothing. Rather you made a faulty heuristic
and compared it with some data on small primes.

Your heuristic contained one very obvious mistake:
you sieved out primes less than x, while we
know that it is sufficient to sieve out primes less
than or equal to sqrt(x), to ensure that x is prime.

By this means you lost an obvious factor of

2 = log(x)/log(sqrt(x)).

Had you included this, your "fugde factor"
would have been the famous factor

2/exp(Euler) =~ 1.123

by which the sieve of Eratosthenes outperforms
a random sieve modelled by a Mertens product.

To prove this, simply combine Mertens' theorem (1874)
and the prime number theorem of Hadamard and
de la Vallée-Poussin (1896), neither of which
were proven in your "document".

The factor 2/exp(Euler) has been discussed several times on this list:

https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/21565

https://groups.yahoo.com/neo/groups/primenumbers/conversations/topics/20936

Hans Riesel discusses it on pp 66-67 of his book, following Theorem 3.1:

http://tinyurl.com/y9whej4

> the sieve of Eratosthenes sieves out numbers
> more efficiently than does a "random" sieve

David
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