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MathGroup Archive 2006

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Re: Problem with Infinite products

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65335] Re: Problem with Infinite products
  • From: Maxim <m.r at inbox.ru>
  • Date: Mon, 27 Mar 2006 06:56:01 -0500 (EST)
  • References: <dvrbsp$a3a$1@smc.vnet.net> <e035oq$2d8$1@smc.vnet.net> <e05re8$3r5$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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:

> Maxim Rytin,
> Actually does run better,
> still gives the wrong answers:
>
> Clear[f, zta]
> f[n_Integer?Positive, 1] := If[Mod[Prime[n], 12] - 1 == 0, Prime[n], 1/2]
> f[n_Integer?Positive, 2] := If[Mod[Prime[n], 12] - 5 == 0, Prime[n], 1/2]
> f[n_Integer?Positive, 3] := If[Mod[Prime[n], 12] - 7 == 0, Prime[n], 1/2]
> f[n_Integer?Positive, 4] := If[Mod[Prime[n], 12] - 11 == 0, Prime[n],  
> 1/2]
> zta[x_, m_] := Module[{n}, Product[f[n, m]^(x)/(-1 + f[n, m]^(x)), {n, 1,
>      Infinity}]]
> zta[2, 1]
> N[%]
> zta[2, 2]
> N[%]
> zta[2, 3]
> N[%]
> zta[2, 4]
> N[%]
> N[Product[zta[2, n], {n, 1, 4}]/Zeta[2]]
> Maxim wrote:
>
>>
>> zta[x_, m_] := Module[{n},
>>    Product[f[n, m]^(x)/(-1 + f[n, m]^(x)), {n, 1, Infinity}]]
>>
>> Then Product[zta[2, n], {n, 1, 4}] won't evaluate to zero.
>>
>> Maxim Rytin
>> m.r at inbox.ru
>>
>


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