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Prime Counting Function

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  • Anton
    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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