PrimePi

PrimePi[x]

gives the number of primes TemplateBox[{x}, PrimePi] 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.
  • TemplateBox[{x}, PrimePi] counts the prime numbers less than or equal to x.
  • TemplateBox[{x}, PrimePi] has asymptotic expansion as .
  • The following option can be given:
  • MethodAutomaticmethod to use
  • Possible settings for Method include:
  • "DelegliseRivat"use the DelégliseRivat algorithm
    "Legendre"use the Legendre formula
    "Lehmer"use the Lehmer formula
    "LMO"use the LagariasMillerOdlyzko algorithm
    "Meissel"use the Meissel formula
    "PrimeSieve"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:

PrimePi threads over lists:

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  (5)

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:

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:

TemplateBox[{x}, PrimePi] is bounded below by and is bounded above by for greater than 3:

is within of TemplateBox[{x}, PrimePi] for :

TemplateBox[{{TemplateBox[{x}, PrimePi], -, TemplateBox[{x}, LogIntegral]}}, Abs]<(sqrt(x) log(x))/(8 pi) 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 ^(th) Ramanujan prime is the smallest number Rn such that π(x) - π(x/2) n for all x Rn:

Plot the sequence of Ramanujan primes:

Compare the count of Ramanujan primes to PrimePi:

Calculate the primorial up to the ^(th) 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 TemplateBox[{x}, PrimePi]=TemplateBox[{x}, EulerPhi]:

Properties & Relations  (5)

The largest domain of definitions of PrimePi:

PrimePi is asymptotically equivalent to as :

And to LogIntegral:

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 :

Introduced in 1991
 (2.0)
 |
Updated in 2020
 (12.1)