Prime Density Correction Terms?

wblipp · 2011-07-24, 16:06

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.

CRGreathouse · 2011-07-24, 20:23

Quote:

Originally Posted by wblipp

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.

R.D. Silverman · 2011-07-25, 11:43

Quote:

Originally Posted by CRGreathouse

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.

mart_r · 2011-07-25, 20:42

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...

LiquidNitrogen · 2011-07-30, 01:34

Quote:

Originally Posted by wblipp

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.

Arkadiusz · 2011-07-30, 10:54

Quote:

Originally Posted by LiquidNitrogen

pi(x) = x/[ln(x) - 1.08] was offered as one "improvement", very long time ago.

http://oeis.org/A193257

CRGreathouse · 2011-07-30, 16:37

Arkadiusz, I didn't realize you posted here! (Or lurked, as the case may be.)

CRGreathouse · 2011-09-01, 03:59

Quote:

Originally Posted by mart_r

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.

mart_r · 2011-09-02, 22:19

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 $\int (Li(x)-\pi(x))$ ?

Zeta-Flux · 2011-09-03, 00:09

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.)

CRGreathouse · 2011-09-03, 02:26

Quote:

Originally Posted by mart_r

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.

2011-07-24, 16:06	#1
wblipp "William" May 2003 Near Grandkid 2374₁₀ 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.

2011-07-25, 20:42	#4
mart_r Dec 2008 you know...around... 853 Posts	I was on that too and came up with 1/ln(x+1/2ln(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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2011-07-30, 16:37	#7
CRGreathouse Aug 2006 5,987 Posts	Arkadiusz, I didn't realize you posted here! (Or lurked, as the case may be.)

2011-09-02, 22:19	#9
mart_r Dec 2008 you know...around... 853 Posts	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 $\int (Li(x)-\pi(x))$ ?

2011-09-03, 00:09	#10
Zeta-Flux May 2003 3013₈ 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.)