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MathGroup Archive 2006

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Re: Problem with a limit.

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66837] Re: Problem with a limit.
  • From: Roger Bagula <rlbagulatftn at yahoo.com>
  • Date: Thu, 1 Jun 2006 06:54:34 -0400 (EDT)
  • References: <200605291005.GAA07572@smc.vnet.net> <e5h5f7$eoi$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

danl at wolfram.com wrote:
>>This sum works really well:
>>
>>a = Sum[(PrimePi[k + 1] - PrimePi[k])/2^k, {k, 1, Infinity}]
> 
> 
> Possibly. Depends on what you want it to do, I guess.
> 
> 
> 
>>I got the idea to look at it from the other end as
>>primes dominate the low end of the Integers:
>>
>>  Limit[Sum[(PrimePi[k + 1] - PrimePi[k])/2^(n - k), {k, 1, n}], n ->
>>Infinity]
> 
> 
> This limit does not exist (so maybe the subject should have been "Problem
> without a limit"). The lim inf is easily seen to be 0 (because there are
> arbitrarily long runs of composites. The lim sup is at least 1 (take n+1
> to be prime). I am not sure but believe it is not as yet proven to be
> strictly larger than 1. Show the lim sup is >=5/4 and you get to claim a
> big prize.
> 
> 
> 
>>So I tried:
>>
>>Table[N[Limit[Sum[(PrimePi[ k + 1] - PrimePi[k])/2^(n - k), {k, 1, n}],
>>n ->10^m], {m, 1, 10}]
> 
> 
> You appear to be taking 10 sums (Limit won't help here). You will notice
> the 5th one is slightly larger than 1/1000. That's because there is a
> prime around 10^5-10. Also note that this can be rewritten to be
> incremental, that is, not start from scratch in order to increase n.
> 
> 
> 
>>It's just an interesting problem in how the primes are distributed.
> 
> 
> Yes. How the primes are distributed has been described, among other
> things, as "an interesting problem".
> 
> 
> Daniel Lichtblau
> Wolfram Research
> 
> 
Daniel Lichtblau,
Thanks for your reply.
I also want to thank Peter Pein and Jean-Marc Gulleitt
for their help.
Roger


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