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

```

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