20130313, 03:43  #1  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23A4_{16} Posts 
Problem E7 of Richard Guy's "Unsolved problems in number theory"
Quote:


20130313, 04:45  #2 
Romulan Interpreter
Jun 2011
Thailand
2^{2}×3×739 Posts 
When one approximates n/Pn with 1/log, then it seems that it converges. No idea how to prove, tho... You ask me to prove it if Erdos could not?! :surprised
Last fiddled with by LaurV on 20130313 at 04:46 
20130313, 07:51  #3  
"Gang aft agley"
Sep 2002
3,581 Posts 
Seems worthwhile to talk about.
Richard K. Guy replied: Quote:
Last fiddled with by only_human on 20130313 at 08:30 

20130313, 12:20  #4  
"Forget I exist"
Jul 2009
Dumbassville
8,369 Posts 
I'm not really mathematical enough to think about it to a great degree but:
Quote:
gives multiple formula for could these lead to a simplification that might have a easy way to solve ? Last fiddled with by science_man_88 on 20130313 at 12:23 

20130313, 18:42  #5  
"Gang aft agley"
Sep 2002
3,581 Posts 
To help frame this question about convergence, I've included the text as stated in Richard Guy's book:
Quote:
Last fiddled with by only_human on 20130313 at 18:54 Reason: to use block quote 

20130313, 19:25  #6 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·2,281 Posts 
I reposted the message from NMBRTHRY, because it looked like a fantastic exercise for any skill level (except for a very low one. Think of it as a projectEuleronsteroids problem,  only the answer is not known). For a technically minded audience (myself likely included; I have no relevant theoretical knowledge), there's an interesting aspect of summation with error accumulation estimate and control. The zero level thinking would be that "surely, the positive and negative errors will cancel on average"; one needs to depart from that at least one level higher, and get a bound for expected error, or else after a while the summation will become meaningless. The sequence grows very, very slowly.
I've tinkered for half an hour to see where I could get naïvely; ran some simpleminded calculation with a 40liner and I was barely able to see the value 0.0304 surpassed. (So, that's ways and ways from Oliveira e Silva's value! And it is growing so slow, that I cannot yet imagine what modeling he used...) I run summation in pairs of terms, with the parwise sum rewritten to minimize error. I left it for myslef for the weekend to hijack yafu sieve and refactor with the strength of yafu code behind this. Could be fun for anyone, I thought, ...and that's when I reposted. 
20130313, 19:35  #7 
"Gang aft agley"
Sep 2002
3,581 Posts 

20130313, 20:08  #8  
Nov 2003
2^{6}·113 Posts 
Quote:
showing that the series does not "diverge by oscillation". This, in turn, involves showing that there do not exist too many intervals in which the primes are somewhat denser than average, followed by intervals in which the primes are somewhat less dense than average. Quote:
The problem is showing that the errors are essentially a 'random walk'; that they oscillate positive and negative, and do not have long runs (when primes are too close (or sparse) on average) where the error remains either positive or negative for long periods. The answer depends on very subtle issues in the distribution of primes. 

20130313, 21:28  #9 
Nov 2012
Canada
3·7 Posts 
Pn
Questions regarding the distribution of primes are always interesting.
(As an analogy, looking at problems like this with a practiced eye helps. The example of Marion Tinsley shows that you should have a feeling about what approach will lead to a desired outcome.) I do have some interesting results which may apply to this question but at present I have more questions than answers. A recent prime I posted is a test value for a certain line of inquiry that is associated with this limit process. I'll have more to post if successful. 
20130313, 22:38  #10 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,281 Posts 
I have sieved to p(n)<30e9 (that's up to n<1300005928; it was a toy sieve, and without bit packing I blew up memory already; need to refactor with yafu or A.Kulsha's program?; need to walk in windows after that) and the growth is slower than ln ln n. Of course, it can oscillate on much larger scale. Oliveira e Silva suggests that it converges to 0.05216, which is on my scale (so far) nowhere near in sight...
Last fiddled with by Batalov on 20130313 at 22:45 Reason: sorry, 30e9; with bit packing can sieve easily to 1e12, of course 
20130314, 13:24  #11 
Aug 2002
Buenos Aires, Argentina
31·43 Posts 
This is my attempt at the conditional proof:
We will perform the sum two terms at a time so we will always add positive numbers. On the Riemann hypothesis: (see http://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture ) is convergent so the original sum must be convergent as well. PD: Forget everything I wrote. I understood that the summand was , not as posted by the OP. Last fiddled with by alpertron on 20130314 at 13:38 
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