## Meissel's Formula

A modification of Legendre's Formula for the Prime Counting Function . It starts with

 (1)

where is the Floor Function, is the number of Integers with , and is the number of Integers with . Identities satisfied by the s include
 (2)

for and
 (3)

Meissel's formula is
 (4)

where
 (5) (6)

Taking the derivation one step further yields Lehmer's Formula.

Riesel, H. Meissel's Formula.'' Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, p. 12, 1994.