On the Error between my estimate for pi(Pn^2) and Pn^2/ln(Pn^2)


I’m rather excited about a mini-breakthrough, although I can’t yet make a definitive claim about its significance.

I’ve previously provided an estimate for the number of primes less than Pn^2 as:

PQ(Pn^2) = Pn^2 x (product for i=1 to n [(Pi – 1)/Pi] ~ pi(Pn^2)

I’ve shown in previous posts the results of numerical experimentation that show this estimate is higher than pi(Pn^2), with the difference increasing with increasing Pn^2, but with the rate of increase appearing to “slow down” with increasing Pn^2. I wondered if, with increasing Pn^2, the ratio between my estimate and the estimate from the Prime Number Theorem, being pi(Pn^2) ~ Pn^2/ln(Pn^2), might converge upon a fixed real number. This appears to be the case.

I suggest that in the limit as n –> infinity:

PQ(Pn^2)/[Pn^2/ln(Pn^2)] = 1.1229…

And by numerical experimentation using Pn^2 values to 4 x 10^11, we find this ratio reaching 1.12282714621192….

This is exciting to me because this error term is the same as mentioned by Prof. Tao here: http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/#comment-119197

where he talks about adjusting something by a factor of 1.1229, or 2e^gamma, where gamma is the Euler-Mascheroni constant, and in this same article he makes the point that “This discrepancy reflects the difficulty in cutting off the product in primes (4) at the right place (for instance, the sieve of Eratosthenes suggests that one might want to cut off at sqrt(x) instead)…”

To me, the fact that this constant appears when comparing my naive estimate with x/log(x) suggests I’m on a fruitful track. The further fact that comparing my estimate for the number of primes with that for the number of twin primes leads to the twin prime constant suggests I’m on a fruitful track with respect to the twin primes.

I will try to re-state everything in a clear, concise form leading to these results in a new post in the coming days.

About petequinn

I'm a Canadian geotechnical engineer specializing in the study of landslides. I started this page to discuss some mathematical topics that interest me, initially this involved mostly prime numbers, but more recently I've diverted focus back to a number of topics of interest in geotechnique, geographic information systems and risk. I completed undergraduate training in engineering physics at Royal Military College (Kingston, Ontario), did a masters degree in civil (geotechnical) engineering at University of British Columbia (Vancouver), and doctorate in geological engineering at Queen's University (Kingston). I was a military engineer for several years at the beginning of my career, and did design and construction work across Canada and abroad. I've worked a few years for the federal government managing large environmental clean up projects in Canada's arctic, and I've worked across Canada, on both coasts and in the middle, as a consulting geotechnical engineer. My work has taken me everywhere in Canada's north, to most major Canadian cities and many small Canadian towns, and to Alaska, Chile, Bermuda, the Caribbean, Germany, Norway, Sweden, Bosnia, and Croatia. My main "hobby" is competitive distance running, which I may write about in future.
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