20110724, 16:06  #1 
"William"
May 2003
New Haven
2^{2}×593 Posts 
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. 
20110724, 20:23  #2 
Aug 2006
5,981 Posts 

20110725, 11:43  #3  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×3×311 Posts 
Quote:
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. 

20110725, 20:42  #4 
Dec 2008
you know...around...
3^{3}×29 Posts 
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 20110725 at 20:47 
20110730, 01:34  #5 
Jun 2011
Henlopen Acres, Delaware
7×19 Posts 

20110730, 10:54  #6  
Dec 2009
3^{3} Posts 
Quote:


20110730, 16:37  #7 
Aug 2006
5,981 Posts 
Arkadiusz, I didn't realize you posted here! (Or lurked, as the case may be.)

20110901, 03:59  #8  
Aug 2006
5,981 Posts 
Quote:
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. 

20110902, 22:19  #9 
Dec 2008
you know...around...
3^{3}×29 Posts 
Just this question:
Does the Riemann prime counting formula / RH if true imply that is  most of the time  closer to zero than ? 
20110903, 00:09  #10 
May 2003
7·13·17 Posts 
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.) 
20110903, 02:26  #11 
Aug 2006
5,981 Posts 

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