This is Dataplot data file FREQPRIM.DAT
Frequency of Prime Numbers
Number of observations = 10
Number of variables per line image = 2
Order of variables on a line image--
1. Response variable = number of primes < n
2. Factor = log10(n)
Reference--Dence, T. (1980). The Fortran Cookbook, Tab Books, pp 68-70.
Reference--Steen, L. A. (1978). Mathematics Today, Vintage, pp 44-53.
Reference--Davis, P. J. & Hersh, Reuben (1981). The Mathematical
Experience, Houghton Mifflin, Boston, pp. 209-214.
Note--Dence & Steen differ in value for 10**9: 50847534 vs. 50879478
Note--The prime number freq. function is typically denoted as pi(n)
pi(n) = number of primes < n
Note--Erdos gave a lower bound of log(n)/(2*log(2))
Note--Gauss in his Prime Number Theorem showed: lim pi(n)/(n/log(n)) = 1
and thus Gauss's approx for pi(n) is n/log(n)
Note--Legendre improved the approx. by pi(n) = n/(log(n)-1.08)
Note--Riemann showed that a good approximation for pi(n) is
pi(n) = Li(n) - (1/2)*Li(n**0.5) - (1/3)*Li(n**.3333) - ...
where Li(x) is the logarithmic integral:
Li(x) = integral for 2 to x of 1/log(x)
pi(n) log10(n)
-------------------
4 1
25 2
168 3
1229 4
9592 5
78498 6
664579 7
5761455 8
50847534 9
455052512 10