20110724, 16:06  #1 
"William"
May 2003
Near Grandkid
2374_{10} 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
5,987 Posts 

20110725, 11:43  #3  
"Bob Silverman"
Nov 2003
North of Boston
1110101010100_{2} 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...
853 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
85_{16} Posts 

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


20110730, 16:37  #7 
Aug 2006
5,987 Posts 
Arkadiusz, I didn't realize you posted here! (Or lurked, as the case may be.)

20110901, 03:59  #8  
Aug 2006
1763_{16} 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...
853 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
3013_{8} 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
5,987 Posts 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
@ Supermods: Error Correction please  Raman  Forum Feedback  72  20130622 07:24 
Asymptotic density of kalmost primes  CRGreathouse  Math  1  20100822 23:47 
Prime kvalue density rating and factorizations  gd_barnes  No Prime Left Behind  25  20090730 12:06 
Density of Mersenne divisors  Random Poster  Math  1  20081215 01:14 
prime density vs. sieving vs. prp time  jasong  Math  18  20060331 03:14 