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:
>
>