Re: Problem with Infinite products

*To*: mathgroup at smc.vnet.net*Subject*: [mg65352] Re: Problem with Infinite products*From*: Roger Bagula <rlbagulatftn at yahoo.com>*Date*: Tue, 28 Mar 2006 04:05:17 -0500 (EST)*References*: <dvrbsp$a3a$1@smc.vnet.net> <e035oq$2d8$1@smc.vnet.net> <e05re8$3r5$1@smc.vnet.net> <e08kk8$4b1$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Maxim Rytin, Thank you very much for your help! On that you are probably right, I picked the default 1/2 to work at Zeta one, It doesn't work at Zeta[2]. Thanks for pointing that out. It still should give a better answer. I arranged it so it missed a singularity each time. That got it: I used another "If" and it works: f[n_, 1] := If[Mod[Prime[n], 12] - 1 == 0, Prime[n], 0] f[n_, 2] := If[Mod[Prime[n], 12] - 5 == 0, Prime[n], 0] f[n_, 3] := If[Mod[Prime[n], 12] - 7 == 0, Prime[n], 0] f[n_, 4] := If[Mod[Prime[n], 12] - 11 == 0, Prime[n], 0] zeta[x_, m_] := Product[If[f[n, m] == 0, 1, f[n, m]^(x)/(-1 + f[n, m]^(x))], {n, 1, Infinity}] The results look like the results I got from the sums. Product values: baa={1.00734,1.04776,1.02578,1.01143} error is: (3/2)*Apply[Times, baa] - Pi^2/6 -0.00238175 Sum values at 1000000 terms assuming equal populations: ebb={1.02912,1.01745,1.0216,1.02518} (3/2)*Apply[Times, ebb] - Pi^2/6 0.0000108624 In any case it has been demonstrated that such product function factors do exit. As far as I know this is a new unique approach to the Zeta function. Maxim wrote: > Nothing really surprising here. Consider zta[2, 1]: by Dirichlet's > theorem, there is an infinite number of primes that equal 5, 7 or 11 > modulo 12; then for those primes the If condition in zta[2, 1] is false > and the corresponding factor equals (1/2)^2/(-1 + (1/2)^2) == -1/3. Since > there is an infinite number of factors equal to -1/3, the limit of the > sequence of factors cannot be 1 and the product doesn't converge. > > Maxim Rytin > m.r at inbox.ru > > On Sun, 26 Mar 2006 10:46:32 +0000 (UTC), Roger Bagula > <rlbagulatftn at yahoo.com> wrote: > >