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• ## Re: [PrimeNumbers] The most accurate equation for Pi(N)

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• ... There are several formulas that are more accurate, for instance ones by Riemann, Legendre, and Tchebecev (sp?) I think. x/(ln(x)-1) is more accurate, and
Message 1 of 5 , Mar 6, 2001
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At 01:00 AM 3/7/2001 +0000, prime_bo@... wrote:
>Is there a accurate way of calculating Pi(N)? I have heard of
>X/(LN(X)), but it is not entirely accurate. Is there a more precise
>equation or method other than counting? Is it possible, has it been
>done already? What is the current standing or theory? I would
>appreciate any insite or information.

There are several formulas that are more accurate, for instance ones by
Riemann, Legendre, and Tchebecev (sp?) I think. x/(ln(x)-1) is more
accurate, and x/ln(x)*(1+1/ln(x)) is more accurate. Riemann's method is
accurate to about sqrt(x), but it is a little more involved.

+--------------------------------------------------------+
| Jud McCranie |
| |
| 137*2^261147+1 is prime! (78,616 digits, 5/2/00) |
+--------------------------------------------------------+
• ... That gives an exact count, but it takes a lot of computation. +--------------------------------------------------------+ ...
Message 1 of 5 , Mar 6, 2001
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At 12:35 PM 3/7/2001 +0900, S.Tomabechi wrote:
>Do you mean Pi(N) = #{ prime numbers < N } ?
>If it is so, there is an accurate method so called Meissel-Lehmer method.

That gives an exact count, but it takes a lot of computation.

+--------------------------------------------------------+
| Jud McCranie |
| |
| 137*2^261147+1 is prime! (78,616 digits, 5/2/00) |
+--------------------------------------------------------+
• Do you mean Pi(N) = #{ prime numbers
Message 1 of 5 , Mar 6, 2001
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Do you mean Pi(N) = #{ prime numbers < N } ?
If it is so, there is an accurate method so called Meissel-Lehmer method.
I heard that calucation was done for N <10^17 several years ago.
You can find the topics about Pi(N) at Hans Riesel's book
Prime Numbers and Computer Methods for Factorization
(Progress in Mathematics, Vol 126) Birkhaeuser

Satoshi Tomabechi

On Wed, 07 Mar 2001 01:00:16 -0000
prime_bo@... wrote:
> Is there a accurate way of calculating Pi(N)? I have heard of
> X/(LN(X)), but it is not entirely accurate. Is there a more precise
> equation or method other than counting? Is it possible, has it been
> done already? What is the current standing or theory? I would
> appreciate any insite or information.
>
>
• ... Mathematica s Implementation Notes:
Message 1 of 5 , Mar 6, 2001
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On Tue, 06 March 2001, Jud McCranie wrote:
> At 12:35 PM 3/7/2001 +0900, S.Tomabechi wrote:
> >Do you mean Pi(N) = #{ prime numbers < N } ?
> >If it is so, there is an accurate method so called Meissel-Lehmer method.
>
> That gives an exact count, but it takes a lot of computation.

Mathematica's Implementation Notes:
<<<
Prime and PrimePi use sparse caching and sieving. For large , the Lagarias�Miller�Odlyzko algorithm for PrimePi is used, based on asymptotic estimates of the density of primes, and is inverted to give Prime.
>>>

I think that this algorithm is closer to the state of the art presently than the one you mention. However, it appears that everyone's joined the party, and the ultimate algorithm is a hotch-potch from about 7 contributors (including all of the above). We're all behind the times guys, look at this page:

http://numbers.computation.free.fr/Constants/Primes/countingPrimes.html

Jaw drops...

Phil

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