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.