How many Primes?
in [Math]
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From: Gerry Myerson on 11 Aug 2005 19:30 In article <21478669.1123797650116.JavaMail.jakarta(a)nitrogen.mathforum.org>, Narcoleptic Insomniac <i_have_narcoleptic_insomnia(a)yahoo.com> wrote: > 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. Dirichlet L-series are related to primes in arithmetic progressions. -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Johann Wiesenbauer on 12 Aug 2005 04:31 > 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 I see your point as to the infinite sum, but this problem could be possibly circumvented. After all, you need the value of the infinite series only with an error < 1/2, since rounding to the next integer will give the exact value of pi(n) then. I don't know though, if a suitable error term for the partial sums of the infinite series at issue is known. Johann
From: David C. Ullrich on 12 Aug 2005 11:46
On Fri, 12 Aug 2005 08:31:31 EDT, Johann Wiesenbauer <j.wiesenbauer(a)tuwien.ac.at> wrote: >> 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 > > >I see your point as to the infinite sum, but this problem could be possibly circumvented. After all, you need the value of the infinite series only with an error < 1/2, since rounding to the next integer will give the exact value of pi(n) then. I don't know though, if a suitable error term for the partial sums of the infinite series at issue is known. Well, just for the record it is an interesting and important formula, etc. But the fact that it's an infinite sum is not the only thing about it that looks somewhat non-"algebraic" to me. Is the zeta function itself an "algebraic" function? Or, as I tried to suggest: why is summing whatever over all zeroes of the zeta function more "algebraic" than just summing _1_ over all p < x? >Johann ************************ David C. Ullrich |