# Counting Primes¶

AUTHORS:

• R. Andrew Ohana (2009): initial version of efficient prime_pi
• William Stein (2009): fix plot method
• R. Andrew Ohana (2011): complete rewrite, ~5x speedup

EXAMPLES:

sage: z = sage.functions.prime_pi.PrimePi()
prime_pi
True

class sage.functions.prime_pi.PrimePi

The prime counting function, which counts the number of primes less than or equal to a given value.

INPUT:

• x - a real number
• prime_bound - (default 0) a real number < 2^32, prime_pi will make sure to use all the primes up to prime_bound (although, possibly more) in computing prime_pi, this can potentially speedup the time of computation, at a cost to memory usage.

OUTPUT:

integer – the number of primes $$\leq$$ x

EXAMPLES:

These examples test common inputs:

sage: prime_pi(7)
4
sage: prime_pi(100)
25
sage: prime_pi(1000)
168
sage: prime_pi(100000)
9592
sage: prime_pi(500509)
41581


These examples test a variety of odd inputs:

sage: prime_pi(3.5)
2
sage: prime_pi(sqrt(2357))
15
sage: prime_pi(mod(30957, 9750979))
Traceback (most recent call last):
...
TypeError: cannot coerce arguments: positive characteristic not allowed in symbolic computations


We test non-trivial prime_bound values:

sage: prime_pi(100000, 10000)
9592
sage: prime_pi(500509, 50051)
41581


The following test is to verify that trac ticket #4670 has been essentially resolved:

sage: prime_pi(10^10)
455052511


The prime_pi function also has a special plotting method, so it plots quickly and perfectly as a step function:

sage: P = plot(prime_pi, 50, 100)


NOTES:

Uses a recursive implementation, using the optimizations described in [Oha2011].

AUTHOR:

• R. Andrew Ohana (2011)
plot(xmin=0, xmax=100, vertical_lines=True, **kwds)

Draw a plot of the prime counting function from xmin to xmax. All additional arguments are passed on to the line command.

WARNING: we draw the plot of prime_pi as a stairstep function with explicitly drawn vertical lines where the function jumps. Technically there should not be any vertical lines, but they make the graph look much better, so we include them. Use the option vertical_lines=False to turn these off.

EXAMPLES:

sage: plot(prime_pi, 1, 100)
Graphics object consisting of 1 graphics primitive
sage: prime_pi.plot(-2, sqrt(2501), thickness=2, vertical_lines=False)
Graphics object consisting of 16 graphics primitives

sage.functions.prime_pi.legendre_phi(x, a)

Legendre’s formula, also known as the partial sieve function, is a useful combinatorial function for computing the prime counting function (the prime_pi method in Sage). It counts the number of positive integers $$\leq$$ x that are not divisible by the first a primes.

INPUT:

• x – a real number
• a – a non-negative integer

OUTPUT:

integer – the number of positive integers $$\leq$$ x that are not divisible by the first a primes

EXAMPLES:

sage: legendre_phi(100, 0)
100
sage: legendre_phi(29375, 1)
14688
sage: legendre_phi(91753, 5973)
2893
sage: legendre_phi(7.5, 2)
3
sage: legendre_phi(str(-2^100), 92372)
0
sage: legendre_phi(4215701455, 6450023226)
1


NOTES:

Uses a recursive implementation, using the optimizations described in [Oha2011].

AUTHOR:

• R. Andrew Ohana (2011)
sage.functions.prime_pi.partial_sieve_function(x, a)

Legendre’s formula, also known as the partial sieve function, is a useful combinatorial function for computing the prime counting function (the prime_pi method in Sage). It counts the number of positive integers $$\leq$$ x that are not divisible by the first a primes.

INPUT:

• x – a real number
• a – a non-negative integer

OUTPUT:

integer – the number of positive integers $$\leq$$ x that are not divisible by the first a primes

EXAMPLES:

sage: legendre_phi(100, 0)
100
sage: legendre_phi(29375, 1)
14688
sage: legendre_phi(91753, 5973)
2893
sage: legendre_phi(7.5, 2)
3
sage: legendre_phi(str(-2^100), 92372)
0
sage: legendre_phi(4215701455, 6450023226)
1


NOTES:

Uses a recursive implementation, using the optimizations described in [Oha2011].

AUTHOR:

• R. Andrew Ohana (2011)