 mersenneforum.org > Math Connection between Li(x)-Pi(x) and x-theta(x)
 Register FAQ Search Today's Posts Mark Forums Read 2011-01-25, 20:04 #1 mart_r   Dec 2008 you know...around... 22×163 Posts Connection between Li(x)-Pi(x) and x-theta(x) I'm still not quite satisfied with what I find when looking for this one... Once upon a time, I think it was about two years ago, I tried to find a way to get the value of x-(x) from known values of Li(x)-(x). (x), some of you may recognize, is Chebyshev's sum of log(p) for all primes p<=x. I found x-(x) = log(x)*[Li(x)-(x)]+1- [[log(y+1)-log(y)]*[Li(y)-(y)]] To simplify the summation term, I split log(y+1)-log(y) into , and thus only the most significant terms have to be added, since the remaining terms converge for appropriately large x (to give a ballpark figure, y>108 to get at least five correct decimal digits): x-(x) ~ log(x)*[Li(x)-(x)]+1.279143- Yet I don't exactly know how to proceed with the summation term. Any comments/suggestions? Precisely, the question is "What accuracy is feasible without having to compute every single value of the sum?" (And ultimately "When will (x) surpass x for the first time?")   2011-01-26, 13:52   #2
R.D. Silverman

Nov 2003

22·5·373 Posts Quote:
 Originally Posted by mart_r I'm still not quite satisfied with what I find when looking for this one... Once upon a time, I think it was about two years ago, I tried to find a way to get the value of x-(x) from known values of Li(x)-(x). (x), some of you may recognize, is Chebyshev's sum of log(p) for all primes p<=x. I found x-(x) = log(x)*[Li(x)-(x)]+1- [[log(y+1)-log(y)]*[Li(y)-(y)]] To simplify the summation term, I split log(y+1)-log(y) into , and thus only the most significant terms have to be added, since the remaining terms converge for appropriately large x (to give a ballpark figure, y>108 to get at least five correct decimal digits): x-(x) ~ log(x)*[Li(x)-(x)]+1.279143- Yet I don't exactly know how to proceed with the summation term. Any comments/suggestions? Precisely, the question is "What accuracy is feasible without having to compute every single value of the sum?" (And ultimately "When will (x) surpass x for the first time?")
Have you tried a Stieltje's integral approximation (i.e. Euler-MacLauren
summation)???   2011-01-26, 16:57   #3
mart_r

Dec 2008
you know...around...

65210 Posts Quote:
 Originally Posted by R.D. Silverman Have you tried a Stieltje's integral approximation (i.e. Euler-MacLauren summation)???
No, but I'll see what I can do. Will probably take a few days to catch up on those integrals.   2011-02-27, 21:14   #4
mart_r

Dec 2008
you know...around...

22·163 Posts Quote:
 Originally Posted by R.D. Silverman Have you tried a Stieltje's integral approximation (i.e. Euler-MacLauren summation)???
I'm afraid these don't help much. Li(x)-Pi(x) is an erratic function, and about the only approximation I don't have too many troubles working with is sqrt(x)/log(x), as extracted from Riemanns approach to Pi(x). I would want to sum the exact terms up to a given number (maybe z = 109 or 1012) and then use a rough approx value ~ for the gap in-between (and if x is appropriately large, I could go directly for , which I know is an underestimate for the sum, but grows asymptotically).
Now I wondered if there's a way to get any kind of error bound.   2011-03-18, 16:52 #5 mart_r   Dec 2008 you know...around... 22·163 Posts May I ask one concise question: is there a fuction f(x) such that , i.e. ? Darn it, that can't be quite right. May I ask another question: does someone at least understand this equation? (Silverman: No. This is gibberish. You compare infinity to some function whose n isn't defined. me: I know, I just have extreme difficulty to get things that I want to say into precise mathematical depictions.) Last fiddled with by mart_r on 2011-03-18 at 17:04   2011-03-18, 17:25   #6
fivemack
(loop (#_fork))

Feb 2006
Cambridge, England

24×3×7×19 Posts Quote:
 Originally Posted by mart_r May I ask one concise question: is there a fuction f(x) such that , i.e. ? Darn it, that can't be quite right.
Do you really want the infinities there? At the moment the left-hand side is only defined for x<-1.

And should the right-hand integral also be from 2 to infinity?

Plugging things into Wolfram Alpha,   2011-03-18, 18:00 #7 mart_r   Dec 2008 you know...around... 22×163 Posts What I meant is that for a given value of x the equation approaches some constant if n goes to infinity. To give some numerical examples: for x=0, this would be f(0)=1, since the prime counting functions Li(x) and x/(log(x)-1) both are asymptotic to the number of primes up to x. If I'm not mistaken, f(1/2) should be somewhere near 2.24. Last fiddled with by mart_r on 2011-03-18 at 18:01   2011-03-18, 19:57 #8 CRGreathouse   Aug 2006 10111011000012 Posts I still can't quite understand what function you're getting at.    2011-03-18, 22:34   #9
mart_r

Dec 2008
you know...around...

12148 Posts Quote:
 Originally Posted by CRGreathouse I still can't quite understand what function you're getting at. Does it make the situation any better if I write
~ ?

Well... okay, I'll try it again tomorrow. I see I need to get rid of comparing two infinities. Again, I highly recommend the well-known example:
Quote:
 Originally Posted by mart_r For x=0, this would be f(0)=1, since Li(m) ~ m/(log(m)-1)
~

[Quote Jake Long] Aw maan! [\Quote]

Last fiddled with by mart_r on 2011-03-18 at 22:40   2011-03-19, 18:51 #10 fivemack (loop (#_fork))   Feb 2006 Cambridge, England 638410 Posts So you want f(n) = I'm not sure that this is not equal to 1 for all n. Last fiddled with by fivemack on 2011-03-19 at 18:52   2011-03-19, 20:18 #11 mart_r   Dec 2008 you know...around... 65210 Posts Well, firstly I notice that one of these integrals wasn't necessary at all: That looks more like it. And now I see that I may have made things too complicated to begin with. f(x) is, judging by the values I get with MathCad, nothing more than 1/(x+1). Last fiddled with by mart_r on 2011-03-19 at 20:19  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post danaj Computer Science & Computational Number Theory 9 2018-03-31 14:59 davieddy Math 2 2011-08-02 09:50 Dougy Math 2 2009-01-05 05:09 rogue Math 5 2007-03-16 10:59 michael PrimeNet 5 2004-01-30 20:46

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