- 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. - At 01:00 AM 3/7/2001 +0000, prime_bo@... wrote:
>Is there a accurate way of calculating Pi(N)? I have heard of

There are several formulas that are more accurate, for instance ones by

>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.

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) |

+--------------------------------------------------------+ - At 12:35 PM 3/7/2001 +0900, S.Tomabechi wrote:
>Do you mean Pi(N) = #{ prime numbers < N } ?

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

>If it is so, there is an accurate method so called Meissel-Lehmer method.

+--------------------------------------------------------+

| Jud McCranie |

| |

| 137*2^261147+1 is prime! (78,616 digits, 5/2/00) |

+--------------------------------------------------------+ - 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.

>

> - On Tue, 06 March 2001, Jud McCranie wrote:
> At 12:35 PM 3/7/2001 +0900, S.Tomabechi wrote:

Mathematica's Implementation Notes:

> >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.

<<<

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

Mathematics should not have to involve martyrdom;

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