
The
function approximates the prime counting function. In 1808 Legendre found that for a = 1.08366 the formula gives an even better approximation for the prime counting. In 2003 Roberto Neumann found a variant on this function that approximates the prime counting even better, with the Neumann prime counting approximation:
where the values of the square roots are rounded before using them in the summation. A comparison of these approximations can be found at www.ceagle.com. 