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) |
and | |  | |
|  | (3) |
Meissel's formula is
 | |  | |
|  | (4) |
where
 |  |  | (5) |
 | |  | (6) |
Taking the derivation one step further yields Lehmer's Formula.See also Legendre's Formula, Lehmer's Formula, Prime Counting Function
References
Riesel, H. ``Meissel's Formula.'' Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, p. 12, 1994.
- Leg
- Legendre Addition Theorem
- Legendre Differential Equation
- Legendre Duplication Formula
- Legendre Function of the First Kind
- Legendre Function of the Second Kind
- Legendre-Gauss Quadrature
- Legendre-Jacobi Elliptic Integral
- Legendre Polynomial
- Legendre Polynomial of the Second Kind
- Legendre Quadrature
- Legendre Relation
- Legendre's Chi-Function
- Legendre's Constant
- Legendre Series
- Legendre's Factorization Method
- Legendre's Formula
- Legendre's Quadratic Reciprocity Law
- Legendre Sum
- Legendre Symbol
- Legendre Transformation
- Lehmer Method
- Lehmer Number
- Lehmer-Schur Method
- Lehmer's Constant