Re: Methods that count primes without counting primes or referring to them...
From: Luis A. Rodriguez (luiroto_at_yahoo.com)
Date: 10/10/04
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Date: 10 Oct 2004 08:20:15 -0700
incognito@fornow.com (Random Searcher) wrote in message news:<200410082250.i98MoVF29636@proapp.mathforum.org>...
> Hey all.
>
> I've been reading over about as much stuff as I can find about prime distribution and prime counting, and I've been left with the following question.
>
> Assuming I'm understanding what I'm reading correctly (I'm getting
this especially from John Derbyshire's book on the Riemann
Hypothesis), Riemann's paper on the distribution of primes provides a
function for generating the number of primes less than n that doesn't
require using the primes in any way - basically, the function provides
a way of seeing the primes as a result of another process or system of
patterns and thus shines light on why the primes are distributed the
way they are.
>
> Is that a correct reading of things?
>
> And, if so, are there any other functions or algorithms known that
return the count of primes without explicitly referring to them? I'm
finding I'm having a really hard time tracking this down - all of the
other methods that I can find for counting primes (primarily from
mathworld.com) seem to make heavy use of iterating through sums or
products over primes and combinations of primes as well as frequently
using values of the prime counting function for recursively smaller
values in their tallying.
>
> Thanks!
No, there is not a function that return exactly the nummber of
primes below
a given limit, except a Riemann formula that requires the knowledge
of the zeros of Z funtion. This is worst than applying the
Erathostenes Sieve !!!
There are long algorithms that permits count the primes, provided the
number below the square root of the limit be known. A species of
Erathostenes Sieve
disguised in the form of a computerized algorithm.
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