Re: Problem with a limit.

*To*: mathgroup at smc.vnet.net*Subject*: [mg66789] Re: [mg66759] Problem with a limit.*From*: danl at wolfram.com*Date*: Tue, 30 May 2006 05:48:53 -0400 (EDT)*References*: <200605291005.GAA07572@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

> 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

**References**:**Problem with a limit.***From:*Roger Bagula <rlbagulatftn@yahoo.com>