I used the values of PI(N) from the table in the Wikipedia article "Prime Number Theorem" to calculate A, B & C.
I did a simple data regression analysis of N / PI(N) vs LOG(N) & LOG(N)^2. If the function was truly a straight line; A would be 0.000000000000 ... !
My results suggest that up...
PI(N) = N /{A * LOG(N)^2 +B * LOG(N) + C}. Note: LOG(N) is the common log.
This formula works for N up to 10^23. The accuracy depends on the number of digits
after the decimal point in the coefficients A, B & C. I used a Lotus123 spreadsheet to
calculate them. My calculated values are...