PrimePi works over integers:

Rational numbers:

Real numbers:

PrimePi works over large numbers:

PrimePi threads over lists:

The traditional mathematical notation for PrimePi:

Find a solution instance of equalities with PrimePi:

Evaluate an integral:

Recognize the PrimePi function:

Find asymptotic relations:

Verify asymptotic relations:

Specify the Legendre method to be used for counting primes [MathWorld]:

Compare the timing:

Specify the Lehmer method to be used for counting primes [MathWorld]:

Compare the timing:

Specify the Deléglise–Rivat method to be used for counting primes:

Compare the timing:

Specify the Meissel method to be used for counting primes [MathWorld]:

Compare the timing:

Specify the Lagarias–Miller–Odlyzko (LMO) to be used for counting primes:

Compare the timing:

By default, PrimePi does not report progress:

With the setting ProgressReporting->True, PrimePi shows the progress of the computation:

Visualize the PrimePi function:

Spiral of PrimePi:

Hexagonal prime spiral:

ContourPlot of the PrimePi function:

Plot the difference between RiemannR and the PrimePi function:

Find all prime numbers less than or equal to 100:

Count the number of primes in an interval:

Approximations to PrimePi:

Plot PrimePi compared with estimates:

is bounded below by and is bounded above by for greater than 3:

is within of for :

for if the Riemann hypothesis is true:

Count the prime numbers using EulerPhi:

Compute PrimePi based on the Hardy–Wright formula:

Compute PrimePi using Accumulate:

Find twin primes up to , i.e. pairs of primes of the form :

Plot the sequence of twin primes and PrimePi:

The Ramanujan prime is the smallest number such that for all :

Plot the sequence of Ramanujan primes:

Compare the count of Ramanujan primes to PrimePi:

Calculate the primorial up to the prime, i.e. a function that multiplies successive primes, similar to the factorial:

Compare the primorial to the factorial up to :

Plot the differences between the factorial and the primorial up to :

Plot the Chebyshev theta function:

Calculate the prime powers up to :

Count all the prime powers up to :

Graph the count of prime powers:

Compare the count of prime powers to PrimePi:

Visualize the second Hardy–Littlewood conjecture, which states that for :

Plot Brocard's conjecture, which states that if and are consecutive primes greater than 2, then between and there are at least four prime numbers:

Find Goldbach partitions, i.e. pairs of primes (, ) such that :

Graph Goldbach's conjecture/comet:

The only 8 solutions of :