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Old 2011-07-24, 16:06   #1
wblipp
 
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Default Prime Density Correction Terms?

Is there a known correction term to the density primes being 1/ln(x)? There are well known improvements to the nth prime being n*ln(n), but I think these are mostly improved approximations for the Log Integral (Li). What I'm asking about would be an improved integrand for the Log Integral.

I know this isn't stated in a mathematically rigorous manner, but I think the question is clear enough.
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Old 2011-07-24, 20:23   #2
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Is there a known correction term to the density primes being 1/ln(x)?
I'm pretty sure the answer is no. Better estimates would probably be stronger than RH.
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Old 2011-07-25, 11:43   #3
R.D. Silverman
 
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I'm pretty sure the answer is no. Better estimates would probably be stronger than RH.
Indeed. One needs to get a count of the number of primes in a
short interval. Cramer's conjecture asserts the existence of a prime
in the (short) interval x, x + O(log^2 x), but says nothing
about how many there are.
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Old 2011-07-25, 20:42   #4
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I was on that too and came up with 1/ln(x+1/2*ln(x)*sqrt(x)) (ln being the natural logarithm here, as being used in the OP) which was met with about the same criticism as in this thread.
Sure, I understand why this is a moot point, but there are some numerical and heuristical indications...

Last fiddled with by mart_r on 2011-07-25 at 20:47
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Old 2011-07-30, 01:34   #5
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Originally Posted by wblipp View Post
Is there a known correction term to the density primes being 1/ln(x)?
pi(x) = x/[ln(x) - 1.08] was offered as one "improvement", very long time ago.
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Old 2011-07-30, 10:54   #6
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pi(x) = x/[ln(x) - 1.08] was offered as one "improvement", very long time ago.
http://oeis.org/A193257
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Old 2011-07-30, 16:37   #7
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Arkadiusz, I didn't realize you posted here! (Or lurked, as the case may be.)
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Old 2011-09-01, 03:59   #8
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Originally Posted by mart_r View Post
I was on that too and came up with 1/ln(x+1/2*ln(x)*sqrt(x)) (ln being the natural logarithm here, as being used in the OP) which was met with about the same criticism as in this thread.
Sure, I understand why this is a moot point, but there are some numerical and heuristical indications...
This seems like arguing about whether sin x for large x is more like 0 or 1/x: it seems to miss the point that the osculating portion is large compared to the correction.

As an example, between 10^10 and 10^10 + 10^7 the standard 1/log x predicts 434,285 primes, while this predicts 2 fewer. But there are 434,650, so the actual errors are 365 and 367.

IIRC |pi(x) - li(x)| is known to be >> sqrt(x)/log x infinitely often, while under RH it's << sqrt(x) log x. So it's a priori possible that the correction could be meaningful, but I'm not holding my breath.
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Old 2011-09-02, 22:19   #9
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Just this question:
Does the Riemann prime counting formula / RH if true imply that \int (Li(x)-\pi(x)-\frac{\sqrt x}{\log x}) is - most of the time - closer to zero than \int (Li(x)-\pi(x))?
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Old 2011-09-03, 00:09   #10
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William,

I just noticed this question. I'll take a look at my files on this when I get back to work on Tuesday. (If you don't hear from me, remind me.)
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Old 2011-09-03, 02:26   #11
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Quote:
Originally Posted by mart_r View Post
Just this question:
Does the Riemann prime counting formula / RH if true imply that \int (Li(x)-\pi(x)-\frac{\sqrt x}{\log x}) is - most of the time - closer to zero than \int (Li(x)-\pi(x))?
I don't think that's known to be true under the RH.
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