PrimePi

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

π(x) is a nonanalytical function which is defined over the reals ℝ.

Symmetries and periodicities

Symmetry

No symmetry

Periodicity

No periodicity

Series representations

Other series representations

Integral representations

On the real axis

Of the direct function

Identities

Functional identities

Integration

Definite integration

For the direct function itself

Representations through equivalent functions

With related functions

Inequalities

Zeros

Theorems

The prime number theorem

π(x)∝(x/(log(x)))/;  ( x->∞).

History

– Euclid proved that π(n)->∞  as n->∞ 
– L. Euler proved that π(n)/n->0  as n->∞
– P.G. Lejeune–Dirichlet (1838)
– P. L. Chebychev (1849, 1851)
– J. Hadamard and Ch. Vallée de la Poisson (1896) independently proved the prime number theorem

Copyright