In mathematics, the **prime-counting function** is the function counting the number of prime numbers less than or equal to some real number *x*.^{[1]}^{[2]} It is denoted by *π*(*x*) (unrelated to the number π).

Main article: Prime number theorem |

Of great interest in number theory is the growth rate of the prime-counting function.^{[3]}^{[4]} It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately

where log is the natural logarithm, in the sense that

This statement is the prime number theorem. An equivalent statement is

where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).

In 1899, de la Vallée Poussin proved that
^{[6]}

for some positive constant a. Here,

More precise estimates of *π*(*x*) are now known. For example, in 2002, Kevin Ford proved that^{[7]}

Mossinghoff and Trudgian proved^{[8]} an explicit upper bound for the difference between *π*(*x*) and li(*x*):

For values of *x* that are not unreasonably large, li(*x*) is greater than *π*(*x*). However, *π*(*x*) − li(*x*) is known to change sign infinitely many times. For a discussion of this, see Skewes' number.

For *x* > 1 let *π*_{0}(*x*) = *π*(*x*) − 1/2 when *x* is a prime number, and *π*_{0}(*x*) = *π*(*x*) otherwise. Bernhard Riemann, in his work *On the Number of Primes Less Than a Given Magnitude*, proved that *π*_{0}(*x*) is equal to^{[9]}

where

The Riemann hypothesis suggests that every such non-trivial zero lies along Re(*s*) = 1/2.

The table compares exact values of *π*(*x*) to the two approximations *x* / log *x* and li(*x*). The last column, *x* / *π*(*x*), is the average prime gap below x.

*x**π*(*x*)*π*(*x*) −*x*/log(*x*)li( *x*) −*π*(*x*)*x*/log(*x*)

% errorli( *x*)

% error*x*/*π*(*x*)10 4 0 2 8.22% 42.606% 2.500 10 ^{2}25 3 5 14.06% 18.597% 4.000 10 ^{3}168 23 10 14.85% 5.561% 5.952 10 ^{4}1,229 143 17 12.37% 1.384% 8.137 10 ^{5}9,592 906 38 9.91% 0.393% 10.425 10 ^{6}78,498 6,116 130 8.11% 0.164% 12.739 10 ^{7}664,579 44,158 339 6.87% 0.051% 15.047 10 ^{8}5,761,455 332,774 754 5.94% 0.013% 17.357 10 ^{9}50,847,534 2,592,592 1,701 5.23% 3.34×10 ^{−3}%19.667 10 ^{10}455,052,511 20,758,029 3,104 4.66% 6.82×10 ^{−4}%21.975 10 ^{11}4,118,054,813 169,923,159 11,588 4.21% 2.81×10 ^{−4}%24.283 10 ^{12}37,607,912,018 1,416,705,193 38,263 3.83% 1.02×10 ^{−4}%26.590 10 ^{13}346,065,536,839 11,992,858,452 108,971 3.52% 3.14×10 ^{−5}%28.896 10 ^{14}3,204,941,750,802 102,838,308,636 314,890 3.26% 9.82×10 ^{−6}%31.202 10 ^{15}29,844,570,422,669 891,604,962,452 1,052,619 3.03% 3.52×10 ^{−6}%33.507 10 ^{16}279,238,341,033,925 7,804,289,844,393 3,214,632 2.83% 1.15×10 ^{−6}%35.812 10 ^{17}2,623,557,157,654,233 68,883,734,693,928 7,956,589 2.66% 3.03×10 ^{−7}%38.116 10 ^{18}24,739,954,287,740,860 612,483,070,893,536 21,949,555 2.51% 8.87×10 ^{−8}%40.420 10 ^{19}234,057,667,276,344,607 5,481,624,169,369,961 99,877,775 2.36% 4.26×10 ^{−8}%42.725 10 ^{20}2,220,819,602,560,918,840 49,347,193,044,659,702 222,744,644 2.24% 1.01×10 ^{−8}%45.028 10 ^{21}21,127,269,486,018,731,928 446,579,871,578,168,707 597,394,254 2.13% 2.82×10 ^{−9}%47.332 10 ^{22}201,467,286,689,315,906,290 4,060,704,006,019,620,994 1,932,355,208 2.03% 9.59×10 ^{−10}%49.636 10 ^{23}1,925,320,391,606,803,968,923 37,083,513,766,578,631,309 7,250,186,216 1.94% 3.76×10 ^{−10}%51.939 10 ^{24}18,435,599,767,349,200,867,866 339,996,354,713,708,049,069 17,146,907,278 1.86% 9.31×10 ^{−11}%54.243 10 ^{25}176,846,309,399,143,769,411,680 3,128,516,637,843,038,351,228 55,160,980,939 1.78% 3.21×10 ^{−11}%56.546 10 ^{26}1,699,246,750,872,437,141,327,603 28,883,358,936,853,188,823,261 155,891,678,121 1.71% 9.17×10 ^{−12}%58.850 10 ^{27}16,352,460,426,841,680,446,427,399 267,479,615,610,131,274,163,365 508,666,658,006 1.64% 3.11×10 ^{−12}%61.153 10 ^{28}157,589,269,275,973,410,412,739,598 2,484,097,167,669,186,251,622,127 1,427,745,660,374 1.58% 9.05×10 ^{−13}%63.456 10 ^{29}1,520,698,109,714,272,166,094,258,063 23,130,930,737,541,725,917,951,446 4,551,193,622,464 1.53% 2.99×10 ^{−13}%65.759

In the On-Line Encyclopedia of Integer Sequences, the *π*(*x*) column is sequence OEIS: A006880, *π*(*x*) − *x*/log *x* is sequence OEIS: A057835, and li(*x*) − *π*(*x*) is sequence OEIS: A057752.

The value for *π*(10^{24}) was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.^{[11]}
It was later verified unconditionally in a computation by D. J. Platt.^{[12]}
The value for *π*(10^{25}) is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.^{[13]}
The value for *π*(10^{26}) was computed by D. B. Staple.^{[14]} All other prior entries in this table were also verified as part of that work.

The values for 10^{27}, 10^{28}, and 10^{29} were announced by David Baugh and Kim Walisch in 2015,^{[15]} 2020,^{[16]} and 2022,^{[17]} respectively.

A simple way to find *π*(*x*), if *x* is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to *x* and then to count them.

A more elaborate way of finding *π*(*x*) is due to Legendre (using the inclusion–exclusion principle): given *x*, if *p*_{1}, *p*_{2},…, *p _{n}* are distinct prime numbers, then the number of integers less than or equal to

(where ⌊*x*⌋ denotes the floor function). This number is therefore equal to

when the numbers *p*_{1}, *p*_{2},…, *p _{n}* are the prime numbers less than or equal to the square root of

Main article: Meissel–Lehmer algorithm |

In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating *π*(*x*): Let *p*_{1}, *p*_{2},…, *p _{n}* be the first

Given a natural number *m*, if *n* = *π*(^{3}√*m*) and if *μ* = *π*(√*m*) − *n*, then

Using this approach, Meissel computed *π*(*x*), for *x* equal to 5×10^{5}, 10^{6}, 10^{7}, and 10^{8}.

In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real *m* and for natural numbers *n* and *k*, *P _{k}*(

where the sum actually has only finitely many nonzero terms. Let *y* denote an integer such that ^{3}√*m* ≤ *y* ≤ √*m*, and set *n* = *π*(*y*). Then *P*_{1}(*m*,*n*) = *π*(*m*) − *n* and *P _{k}*(

The computation of *P*_{2}(*m*,*n*) can be obtained this way:

where the sum is over prime numbers.

On the other hand, the computation of Φ(*m*,*n*) can be done using the following rules:

Using his method and an IBM 701, Lehmer was able to compute the correct value of *π*(10^{9}) and missed the correct value of *π*(10^{10}) by 1.^{[18]}

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.^{[19]}

Other prime-counting functions are also used because they are more convenient to work with.

Riemann's prime-power counting function is usually denoted as Π_{0}(*x*) or *J*_{0}(*x*). It has jumps of 1/*n* at prime powers *p ^{n}* and it takes a value halfway between the two sides at the discontinuities of

Formally, we may define Π_{0}(*x*) by

where the variable p in each sum ranges over all primes within the specified limits.

We may also write

where Λ is the von Mangoldt function and

The Möbius inversion formula then gives

where *μ*(*n*) is the Möbius function.

Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function Λ, and using the Perron formula we have

The **Chebyshev function** weights primes or prime powers *p ^{n}* by log(

For *x* ≥ 2,^{[20]}

and

Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were the first used to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulae.^{[21]}

We have the following expression for the second Chebyshev function *ψ*:

where

Here *ρ* are the zeros of the Riemann zeta function in the critical strip, where the real part of *ρ* is between zero and one. The formula is valid for values of *x* greater than one, which is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula.

For Π_{0}(*x*) we have a more complicated formula

Again, the formula is valid for *x* > 1, while *ρ* are the nontrivial zeros of the zeta function ordered according to their absolute value. The integral is equal to the series over the trivial zeros:

The first term li(*x*) is the usual logarithmic integral function; the expression li(*x*^{ρ}) in the second term should be considered as Ei(*ρ* log *x*), where Ei is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals.

Thus, Möbius inversion formula gives us^{[10]}

valid for *x* > 1, where

is Riemann's R-function^{[22]} and *μ*(*n*) is the Möbius function. The latter series for it is known as Gram series.^{[23]}^{[24]} Because log *x* < *x* for all *x* > 0, this series converges for all positive *x* by comparison with the series for *e ^{x}*. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as

Folkmar Bornemann proved,^{[25]} when assuming the conjecture that all zeros of the Riemann zeta function are simple,^{[note 1]} that

where *ρ* runs over the non-trivial zeros of the Riemann zeta function and *t* > 0.

The sum over non-trivial zeta zeros in the formula for *π*_{0}(*x*) describes the fluctuations of *π*_{0}(*x*) while the remaining terms give the "smooth" part of prime-counting function,^{[26]} so one can use

as a good estimator of *π*(*x*) for *x* > 1. In fact, since the second term approaches 0 as *x* → ∞, while the amplitude of the "noisy" part is heuristically about √*x*/log *x*, estimating *π*(*x*) by R(*x*) alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function

Here are some useful inequalities for *π*(*x*).

for *x* ≥ 17.

The left inequality holds for *x* ≥ 17 and the right inequality holds for *x* > 1. The constant 1.25506 is 30 log 113/113 to 5 decimal places, as *π*(*x*) log *x*/*x* has its maximum value at *x* = 113.^{[27]}

Pierre Dusart proved in 2010:^{[28]}

and

Here are some inequalities for the *n*th prime, *p*_{n}. The upper bound is due to Rosser (1941),^{[29]} the lower one to Dusart (1999):^{[30]}

The left inequality holds for *n* ≥ 2 and the right inequality holds for *n* ≥ 6.

An approximation for the *n*th prime number is

Ramanujan^{[31]} proved that the inequality

holds for all sufficiently large values of *x*.

In 2010^{[28]} Dusart proved (Proposition 6.6) that, for *n* ≥ 688383,

and (Proposition 6.7) that, for *n* ≥ 3,

More recently, Dusart^{[32]}
has proved (Theorem 5.1) that, for *x* > 1,

and that, for *x* ≥ 88789,

The Riemann hypothesis implies a much tighter bound on the error in the estimate for *π*(*x*), and hence to a more regular distribution of prime numbers,

Specifically,^{[33]}

Dudek (2015) proved that the Riemann hypothesis implies that for all *x* ≥ 2 there is a prime *p* satisfying