How many Primes?
in [Math]
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From: Johann Wiesenbauer on 7 Aug 2005 08:31 > On 7 Aug 2005 04:49:44 -0700, "Apple Pi" > <apple3.1415926535897932384626(a)gmail.com> wrote: > > >Is there a formula that gives me the amount of Prime > numbers in a > >Finite subset of N? > > > >In other words, is there a formula that, for every n > (Natural number), > >gives us the number of primes < n? > > One "formula" for this is just pi(n). But that's not > what you > want, that's just a definition. Is there an > "algebraic" formula > for pi(n)? No. > > There is a well-known formula that gives a good > _approximation_ > to pi(n) for large n; that formula is n/log(n). The > "prime number > theorem" says that > > pi(n) ~ n/log(n) > > as n -> infinity (here f(n) ~ g(n) means that > f(n)/g(n) -> 1.) > > >Thanks in advance, > >Apple Pi > > > ************************ > > David C. Ullrich Well, believe it or not, there is an "algebraic" formula for pi(n), involving al the zeros of Riemann's zeta function in the critical strip. See a nice animation at http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm using that formula. Johann
From: Ioannis on 11 Aug 2005 16:20 ý "Johann Wiesenbauer" <j.wiesenbauer(a)tuwien.ac.at> ýýýýýý ýýý ýýýýýý news:3955702.1123432329372.JavaMail.jakarta(a)nitrogen.mathforum.org... [snip] > Well, believe it or not, there is an "algebraic" formula for pi(n), involving al the zeros of Riemann's zeta function in the critical strip. See a nice animation at > > http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm > > using that formula. This is simply fantastic! > Johann -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable
From: David C. Ullrich on 11 Aug 2005 17:07 On Sun, 07 Aug 2005 12:31:39 EDT, Johann Wiesenbauer <j.wiesenbauer(a)tuwien.ac.at> wrote: >> On 7 Aug 2005 04:49:44 -0700, "Apple Pi" >> <apple3.1415926535897932384626(a)gmail.com> wrote: >> >> >Is there a formula that gives me the amount of Prime >> numbers in a >> >Finite subset of N? >> > >> >In other words, is there a formula that, for every n >> (Natural number), >> >gives us the number of primes < n? >> >> One "formula" for this is just pi(n). But that's not >> what you >> want, that's just a definition. Is there an >> "algebraic" formula >> for pi(n)? No. >> >> There is a well-known formula that gives a good >> _approximation_ >> to pi(n) for large n; that formula is n/log(n). The >> "prime number >> theorem" says that >> >> pi(n) ~ n/log(n) >> >> as n -> infinity (here f(n) ~ g(n) means that >> f(n)/g(n) -> 1.) >> >> >Thanks in advance, >> >Apple Pi >> >> >> ************************ >> >> David C. Ullrich > > >Well, believe it or not, there is an "algebraic" formula for pi(n), involving al the zeros of Riemann's zeta function in the critical strip. I believe that, having seen such formulas. I wouldn't call an infinite sum over the zeroes of the zeta function an "algebraic formula". Which is certainly not to say that _you_ shouldn't call it that. But it would be interesting to see an definition of "algebraic formula" that makes that series an algebraic formula, such that neither of the following two simpler formulas is "algebraic": (i) pi(x) (ii) sum_p L(p,x), where L(x,y) = 0 or 1 depending on whether x < y. >See a nice animation at > >http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm > >using that formula. > >Johann ************************ David C. Ullrich
From: Narcoleptic Insomniac on 11 Aug 2005 13:43 On Aug 7, 2005 7:39 AM, David C. Ullrich wrote: > On 7 Aug 2005 04:49:44 -0700, "Apple Pi" > <apple3.1415926535897932384626(a)gmail.com> wrote: > > >Is there a formula that gives me the amount of Prime > >numbers in a Finite subset of N? > > > >In other words, is there a formula that, for every n > >(Natural number), gives us the number of primes < n? > > One "formula" for this is just pi(n). But that's not > what you want, that's just a definition. Is there an > "algebraic" formula for pi(n)? No. > > There is a well-known formula that gives a good > _approximation_ to pi(n) for large n; that formula is > n/log(n). The "prime number theorem" says that > > pi(n) ~ n/log(n) > > as n -> infinity (here f(n) ~ g(n) means that > f(n)/g(n) -> 1.) I don't know what's more amazing: the formula itself or the fact that Gauss realized this when he was a teenager! ~Kyle
From: Narcoleptic Insomniac on 11 Aug 2005 14:00
On Aug 11, 2005 3:20 PM, Ioannis wrote: > Ÿ "Johann Wiesenbauer" <j.wiesenbauer(a)tuwien.ac.at> > ³Á±Èµ ÃÄ¿ ¼®½Å¼± > news:3955702.1123432329372.JavaMail.jakarta(a)nitrogen.m > athforum.org... > [snip] > > > Well, believe it or not, there is an "algebraic" > > formula for pi(n), > > involving al the zeros of Riemann's zeta function in > > the critical strip. See a nice animation at > > > > http://www.maths.ex.ac.uk/~mwatkins/zeta/pianim.htm > > > > using that formula. > > This is simply fantastic! It's definately breathtaking that the distribution of the primes can be related to the nontrivial roots of zeta. This is exactly what makes the Riemann hypothesis and number theory so interesting (to me at least). It makes me wonder what relationships there are (if any) between nontrivial roots of other L-series and the prime numbers. ~Kyle |