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