## Prime Counting Function

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• According to Prime Number Theorom a) pi_a(n) ~ Li(n) above is the improvement over b) pi_b(n) ~ n/(ln(n)+B) with B = -1.08336 For those who cannot compute the
Message 1 of 3 , Sep 7, 2005
According to Prime Number Theorom

a) pi_a(n) ~ Li(n)

above is the improvement over

b) pi_b(n) ~ n/(ln(n)+B) with B = -1.08336

For those who cannot compute the logarithmic integral Li(n) the
following counting functions yields better results than b)

c) pi_c(n) ~ n{1 +0.5x +sqrt[x^2/4 -x-1 -1/(4x^2)]}/(2 x)
where x=ln(n)

let n=10^k below the differences to the actual pi(10^k)

k ; a) ; b) ; c)

3 ; 9 ; 3 ; 9
6 ; 129 ; 43 ; 105
9 ; 1700 ; 69207 ; 312
12 ; 3.8 10^4 ; 6.1 10^7 ; -1.1 10^5
15 ; 1.1 10^6 ; 4.6 10^10 ; -2.5 10^7
18 ; 2.2 10^7 ; 3.5 10^13 ; -2.5 10^8
21 ; 5.9 10^8 ; 2.7 10^16 ; -4.1 10^12
23 ; 7.2 10^9 ; 2.3 10^18 ; -4.3 10^14

pi_a(n) ~ Li(n) is still the best

Has any one seen the approximation pi_c() before

I discovered it by investigating the number of primes
between a^b and a^(b+1)

regards
Anton
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