# PrimePi

PrimePi[x]

gives the number of primes less than or equal to x.

# Details and Options • PrimePi is also known as prime counting function.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• counts the prime numbers less than or equal to x.
• has the asymptotic expansion as .
• • The following option can be given:
•  Method Automatic method to use ProgressReporting False whether to report the progress of the computation
• Possible settings for Method include:
•  "DelegliseRivat" use the Deléglise–Rivat algorithm "Legendre" use the Legendre formula "Lehmer" use the Lehmer formula "LMO" use the Lagarias–Miller–Odlyzko algorithm "Meissel" use the Meissel formula "Sieve" use the sieve of Erastosthenes

# Examples

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## Basic Examples(3)

Compute the number of primes up to 15:

Plot the prime counting function:

Find the leading asymptotic term for PrimePi at Infinity:

## Scope(10)

### Numerical Manipulation(5)

PrimePi works over integers:

Rational numbers:

Real numbers:

PrimePi works over large numbers:

### Symbolic Manipulation(5)

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:

## Options(6)

### Method(5)

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 DelégliseRivat 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 LagariasMillerOdlyzko (LMO) to be used for counting primes:

Compare the timing:

### ProgressReporting(1)

By default, PrimePi does not report progress:

With the setting , PrimePi shows the progress of the computation: ## Applications(22)

### Basic Applications(7)

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(7)

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 HardyWright formula:

Compute PrimePi using Accumulate:

### Number Theory(8)

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 HardyLittlewood 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 :

## Properties & Relations(5)

The largest domain of definitions of PrimePi:

PrimePi is asymptotically equivalent to as :

PrimePi is the inverse of Prime:

Use PrimeQ to count prime numbers:

Mathematical function entity:

An integral representation of PrimePi:

## Possible Issues(1)

Evaluation time increases exponentially:

## Neat Examples(3)

Ulam spiral colored based on the difference in PrimePi values:

Generate a path based on the PrimePi function:

Construct polyhedra using directed graphs generated by primes less than :