Prime-counting function

The values of Template:Math for the first 60 positive integers

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number Template:Mvar. It is denoted by Template:Math (unrelated to the [[pi|number Template:Pi]]).

A symmetric variant seen sometimes is Template:Math, which is equal to Template:Math if Template:Mvar is exactly a prime number, and equal to Template:Math otherwise. That is, the number of prime numbers less than Template:Mvar, plus half if Template:Mvar equals a prime.

Growth rate

Template:Main Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately $$\frac{x}{\log x}$$ where Template:Math is the natural logarithm, in the sense that $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\pi (x)}{x/\log x}=1.$$ This statement is the prime number theorem. An equivalent statement is $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\pi (x)}{\mathrm{li}(x)}=1$$ where Template:Math 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).

More precise estimates

In 1899, de la Vallée Poussin proved that $$\pi (x)=\mathrm{li}(x)+O\left(x{e}^{-a\sqrt{\log x}}\right)\text{as }x\to \mathrm{\infty }$$ for some positive constant Template:Mvar. Here, Template:Math is the [[big O notation|big Template:Mvar notation]].

More precise estimates of Template:Math are now known. For example, in 2002, Kevin Ford proved that $$\pi (x)=\mathrm{li}(x)+O(x\exp (-0.2098(\log x{)}^{3/5}(\log \log x{)}^{-1/5})).$$

Mossinghoff and Trudgian proved an explicit upper bound for the difference between Template:Math and Template:Math: $$|\pi (x)-\mathrm{li}(x)|\le 0.2593\frac{x}{(\log x{)}^{3/4}}\exp (-\sqrt{\frac{\log x}{6.315}})\text{for }x\ge 229.$$

For values of Template:Mvar that are not unreasonably large, Template:Math is greater than Template:Math. However, Template:Math is known to change sign infinitely many times. For a discussion of this, see Skewes' number.

Exact form

For Template:Math let Template:Math when Template:Mvar is a prime number, and Template:Math otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that Template:Math is equal to

Riemann's explicit formula using the first 200 non-trivial zeros of the zeta function

$${\pi }_0(x)=\mathrm{R}(x)-\sum _{\rho }\mathrm{R}(x^{\rho }),$$ where $$\mathrm{R}(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}\mathrm{li}\left(x^{1/n}\right),$$ Template:Math is the Möbius function, Template:Math is the logarithmic integral function, Template:Mvar indexes every zero of the Riemann zeta function, and Template:Math is not evaluated with a branch cut but instead considered as Template:Math where Template:Math is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros Template:Mvar of the Riemann zeta function, then Template:Math may be approximated by $${\pi }_0(x)\approx \mathrm{R}(x)-\sum _{\rho }\mathrm{R}\left(x^{\rho }\right)-\frac{1}{\log x}+\frac{1}{\pi }\arctan \frac{\pi }{\log x}.$$

The Riemann hypothesis suggests that every such non-trivial zero lies along Template:Math.

Table of Template:Math, Template:Math, and Template:Math

The table shows how the three functions Template:Math, Template:Math, and Template:Math compared at powers of 10. See also, and

Template:Mvar	Template:Math	Template:Math	Template:Math	Template:Math	Template:Math  % error
10	4	0	2	2.500	−8.57%
102	25	3	5	4.000	+13.14%
103	168	23	10	5.952	+13.83%
104	1,229	143	17	8.137	+11.66%
105	9,592	906	38	10.425	+9.45%
106	78,498	6,116	130	12.739	+7.79%
107	664,579	44,158	339	15.047	+6.64%
108	5,761,455	332,774	754	17.357	+5.78%
109	50,847,534	2,592,592	1,701	19.667	+5.10%
1010	455,052,511	20,758,029	3,104	21.975	+4.56%
1011	4,118,054,813	169,923,159	11,588	24.283	+4.13%
1012	37,607,912,018	1,416,705,193	38,263	26.590	+3.77%
1013	346,065,536,839	11,992,858,452	108,971	28.896	+3.47%
1014	3,204,941,750,802	102,838,308,636	314,890	31.202	+3.21%
1015	29,844,570,422,669	891,604,962,452	1,052,619	33.507	+2.99%
1016	279,238,341,033,925	7,804,289,844,393	3,214,632	35.812	+2.79%
1017	2,623,557,157,654,233	68,883,734,693,928	7,956,589	38.116	+2.63%
1018	24,739,954,287,740,860	612,483,070,893,536	21,949,555	40.420	+2.48%
1019	234,057,667,276,344,607	5,481,624,169,369,961	99,877,775	42.725	+2.34%
1020	2,220,819,602,560,918,840	49,347,193,044,659,702	222,744,644	45.028	+2.22%
1021	21,127,269,486,018,731,928	446,579,871,578,168,707	597,394,254	47.332	+2.11%
1022	201,467,286,689,315,906,290	4,060,704,006,019,620,994	1,932,355,208	49.636	+2.02%
1023	1,925,320,391,606,803,968,923	37,083,513,766,578,631,309	7,250,186,216	51.939	+1.93%
1024	18,435,599,767,349,200,867,866	339,996,354,713,708,049,069	17,146,907,278	54.243	+1.84%
1025	176,846,309,399,143,769,411,680	3,128,516,637,843,038,351,228	55,160,980,939	56.546	+1.77%
1026	1,699,246,750,872,437,141,327,603	28,883,358,936,853,188,823,261	155,891,678,121	58.850	+1.70%
1027	16,352,460,426,841,680,446,427,399	267,479,615,610,131,274,163,365	508,666,658,006	61.153	+1.64%
1028	157,589,269,275,973,410,412,739,598	2,484,097,167,669,186,251,622,127	1,427,745,660,374	63.456	+1.58%
1029	1,520,698,109,714,272,166,094,258,063	23,130,930,737,541,725,917,951,446	4,551,193,622,464	65.759	+1.52%

Graph showing ratio of the prime-counting function Template:Math to two of its approximations, Template:Math and Template:Math. As Template:Mvar increases (note Template:Mvar-axis is logarithmic), both ratios tend towards 1. The ratio for Template:Math converges from above very slowly, while the ratio for Template:Math converges more quickly from below.

In the On-Line Encyclopedia of Integer Sequences, the Template:Math column is sequence Template:OEIS2C, Template:Math is sequence Template:OEIS2C, and Template:Math is sequence Template:OEIS2C.

The value for Template:Math was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis. It was later verified unconditionally in a computation by D. J. Platt. The value for Template:Math is by the same four authors. The value for Template:Math was computed by D. B. Staple. All other prior entries in this table were also verified as part of that work.

The values for 10²⁷, 10²⁸, and 10²⁹ were announced by David Baugh and Kim Walisch in 2015, 2020, and 2022, respectively.

Algorithms for evaluating Template:Math

A simple way to find Template:Math, if Template:Mvar is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to Template:Mvar and then to count them.

A more elaborate way of finding Template:Math is due to Legendre (using the inclusion–exclusion principle): given Template:Mvar, if Template:Math are distinct prime numbers, then the number of integers less than or equal to Template:Mvar which are divisible by no Template:Mvar is

$\lfloor x\rfloor -\sum _i\lfloor \frac{x}{p_i}\rfloor +\sum _{i<j}\lfloor \frac{x}{p_i{p}_j}\rfloor -\sum _{i<j<k}\lfloor \frac{x}{p_i{p}_j{p}_k}\rfloor +\cdots$

(where Template:Math denotes the floor function). This number is therefore equal to

$\pi (x)-\pi \left(\sqrt{x}\right)+1$

when the numbers Template:Math are the prime numbers less than or equal to the square root of Template:Mvar.

The Meissel–Lehmer algorithm

In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating Template:Math: Let Template:Math be the first Template:Mvar primes and denote by Template:Math the number of natural numbers not greater than Template:Mvar which are divisible by none of the Template:Mvar for any Template:Math. Then

$\mathrm{\Phi }(m,n)=\mathrm{\Phi }(m,n-1)-\mathrm{\Phi }(\frac{m}{p_n},n-1).$

Given a natural number Template:Mvar, if Template:Math and if Template:Math, then

$\pi (m)=\mathrm{\Phi }(m,n)+n(\mu +1)+\frac{{\mu }^2-\mu }{2}-1-\sum _{k=1}^{\mu }\pi \left(\frac{m}{p_{n+k}}\right).$

Using this approach, Meissel computed Template:Math, for Template:Mvar equal to Template:Val, 10⁶, 10⁷, and 10⁸.

In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real Template:Mvar and for natural numbers Template:Mvar and Template:Mvar, Template:Math as the number of numbers not greater than Template:Mvar with exactly Template:Mvar prime factors, all greater than Template:Mvar. Furthermore, set Template:Math. Then

$\mathrm{\Phi }(m,n)=\sum _{k=0}^{+\mathrm{\infty }}{P}_k(m,n)$

where the sum actually has only finitely many nonzero terms. Let Template:Mvar denote an integer such that Template:Math, and set Template:Math. Then Template:Math and Template:Math when Template:Math. Therefore,

$\pi (m)=\mathrm{\Phi }(m,n)+n-1-P_2(m,n)$

The computation of Template:Math can be obtained this way:

$P_2(m,n)=\sum _{y<p\le \sqrt{m}}(\pi \left(\frac{m}{p}\right)-\pi (p)+1)$

where the sum is over prime numbers.

On the other hand, the computation of Template:Math can be done using the following rules:

1. $\mathrm{\Phi }(m,0)=\lfloor m\rfloor$
2. $\mathrm{\Phi }(m,b)=\mathrm{\Phi }(m,b-1)-\mathrm{\Phi }(\frac{m}{p_b},b-1)$

Using his method and an IBM 701, Lehmer was able to compute the correct value of Template:Math and missed the correct value of Template:Math by 1.

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.

Other prime-counting functions

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

Riemann's prime-power counting function

Riemann's prime-power counting function is usually denoted as Template:Math or Template:Math. It has jumps of Template:Math at prime powers Template:Mvar and it takes a value halfway between the two sides at the discontinuities of Template:Math. That added detail is used because the function may then be defined by an inverse Mellin transform.

Formally, we may define Template:Math by

${\mathrm{\Pi }}_0(x)=\frac{1}{2}(\sum _{p^n<x}\frac{1}{n}+\sum _{p^n\le x}\frac{1}{n})$

where the variable Template:Mvar in each sum ranges over all primes within the specified limits.

We may also write

${\mathrm{\Pi }}_0(x)=\sum _{n=2}^x\frac{\mathrm{\Lambda }(n)}{\log n}-\frac{\mathrm{\Lambda }(x)}{2\log x}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\pi }_0\left(x^{1/n}\right)$

where Template:Math is the von Mangoldt function and

${\pi }_0(x)=\underset{\epsilon \to 0}{\lim }\frac{\pi (x-\epsilon )+\pi (x+\epsilon )}{2}.$

The Möbius inversion formula then gives

${\pi }_0(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}{\mathrm{\Pi }}_0\left(x^{1/n}\right),$

where Template:Math is the Möbius function.

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

$\log \zeta (s)=s\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathrm{\Pi }}_0(x)x^{-s-1}\mathrm{d}x$

Chebyshev's function

The Chebyshev function weights primes or prime powers Template:Mvar by Template:Math:

$\begin{array}{cc}\vartheta (x)& =\sum _{p\le x}\log p\\ \psi (x)& =\sum _{p^n\le x}\log p=\sum _{n=1}^{\mathrm{\infty }}\vartheta \left(x^{1/n}\right)=\sum _{n\le x}\mathrm{\Lambda }(n).\end{array}$

For Template:Math,

$\vartheta (x)=\pi (x)\log x-\underset{2}{\overset{x}{\int }}\frac{\pi (t)}{t}\mathrm{d}t$

and

$\pi (x)=\frac{\vartheta (x)}{\log x}+\underset{2}{\overset{x}{\int }}\frac{\vartheta (t)}{t{\log }^2(t)}\mathrm{d}t.$

Formulas for prime-counting functions

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.

We have the following expression for the second Chebyshev function Template:Mvar:

${\psi }_0(x)=x-\sum _{\rho }\frac{x^{\rho }}{\rho }-\log 2\pi -\frac{1}{2}\log (1-x^{-2}),$

where

${\psi }_0(x)=\underset{\epsilon \to 0}{\lim }\frac{\psi (x-\epsilon )+\psi (x+\epsilon )}{2}.$

Here Template:Mvar are the zeros of the Riemann zeta function in the critical strip, where the real part of Template:Mvar is between zero and one. The formula is valid for values of Template:Mvar 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 Template:Math we have a more complicated formula

${\mathrm{\Pi }}_0(x)=\mathrm{li}(x)-\sum _{\rho }\mathrm{li}\left(x^{\rho }\right)-\log 2+\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{\mathrm{d}t}{t(t^2-1)\log t}.$

Again, the formula is valid for Template:Math, while Template:Mvar are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term Template:Math is the usual logarithmic integral function; the expression Template:Math in the second term should be considered as Template:Math, where Template:Math is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals. The final integral is equal to the series over the trivial zeros:

$\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{\mathrm{d}t}{t(t^2-1)\log t}=\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{1}{t\log t}\left(\sum _m{t}^{-2m}\right)\mathrm{d}t=\sum _m\underset{x}{\overset{\mathrm{\infty }}{\int }}\frac{t^{-2m}}{t\log t}\mathrm{d}t\stackrel{(u=t^{-2m})}{=}-\sum _m\mathrm{li}\left(x^{-2m}\right)$

Thus, Möbius inversion formula gives us

${\pi }_0(x)=\mathrm{R}(x)-\sum _{\rho }\mathrm{R}\left(x^{\rho }\right)-\sum _m\mathrm{R}\left(x^{-2m}\right)$

valid for Template:Math, where

$\mathrm{R}(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}\mathrm{li}\left(x^{1/n}\right)=1+\sum _{k=1}^{\mathrm{\infty }}\frac{{(\log x)}^k}{k!k\zeta (k+1)}$

is Riemann's R-function and Template:Math is the Möbius function. The latter series for it is known as Gram series. Because Template:Math for all Template:Math, this series converges for all positive Template:Mvar by comparison with the series for Template:Mvar. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as Template:Math and not Template:Math.

Folkmar Bornemann proved, when assuming the conjecture that all zeros of the Riemann zeta function are simple, that

$\mathrm{R}\left(e^{-2\pi t}\right)=\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k-1}{t}^{-2k-1}}{(2k+1)\zeta (2k+1)}+\frac{1}{2}\sum _{\rho }\frac{t^{-\rho }}{\rho \cos \frac{\pi \rho }{2}{\zeta }^{\prime }(\rho )}$

where Template:Mvar runs over the non-trivial zeros of the Riemann zeta function and Template:Math.

The sum over non-trivial zeta zeros in the formula for Template:Math describes the fluctuations of Template:Math while the remaining terms give the "smooth" part of prime-counting function, so one can use

$\mathrm{R}(x)-\sum _{m=1}^{\mathrm{\infty }}\mathrm{R}\left(x^{-2m}\right)$

as a good estimator of Template:Math for Template:Math. In fact, since the second term approaches 0 as Template:Math, while the amplitude of the "noisy" part is heuristically about Template:Math, estimating Template:Math by Template:Math alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function

$({\pi }_0(x)-\mathrm{R}(x))\frac{\log x}{\sqrt{x}}.$

Inequalities

Ramanujan proved that the inequality

$\pi (x{)}^2<\frac{ex}{\log x}\pi \left(\frac{x}{e}\right)$

holds for all sufficiently large values of Template:Mvar.

Here are some useful inequalities for Template:Math.

$\frac{x}{\log x}<\pi (x)<1.25506\frac{x}{\log x}\text{for }x\ge 17.$

The left inequality holds for Template:Math and the right inequality holds for Template:Math. The constant 1.25506 is Template:Math to 5 decimal places, as Template:Math has its maximum value at Template:Math.

Pierre Dusart proved in 2010:

$\frac{x}{\log x-1}<\pi (x)<\frac{x}{\log x-1.1}\text{for }x\ge 5393\text{ and }x\ge 60184,\text{ respectively.}$

More recently, Dusart has proved (Theorem 5.1) that

$\frac{x}{\log x}(1+\frac{1}{\log x}+\frac{2}{{\log }^{2}x})\le \pi (x)\le \frac{x}{\log x}(1+\frac{1}{\log x}+\frac{2}{{\log }^{2}x}+\frac{7.59}{{\log }^{3}x}),$

for Template:Math and Template:Math, respectively.

Going in the other direction, an approximation for the Template:Mvarth prime, Template:Mvar, is

$p_n=n(\log n+\log \log n-1+\frac{\log \log n-2}{\log n}+O\left(\frac{(\log \log n{)}^2}{(\log n{)}^2}\right)).$

Here are some inequalities for the Template:Mvarth prime. The lower bound is due to Dusart (1999) and the upper bound to Rosser (1941).

$n(\log n+\log \log n-1)<p_n<n(\log n+\log \log n)\text{for }n\ge 6.$

The left inequality holds for Template:Math and the right inequality holds for Template:Math. A variant form sometimes seen substitutes $\log n+\log \log n=\log (n\log n).$ An even simpler lower bound is

$n\log n<p_n,$

which holds for all Template:Math, but the lower bound above is tighter for Template:Math.

In 2010 Dusart proved (Propositions 6.7 and 6.6) that

$n(\log n+\log \log n-1+\frac{\log \log n-2.1}{\log n})\le p_n\le n(\log n+\log \log n-1+\frac{\log \log n-2}{\log n}),$

for Template:Math and Template:Math, respectively.

In 2024, Axler further tightened this (equations 1.12 and 1.13) using bounds of the form

$f(n,g(w))=n(\log n+\log \log n-1+\frac{\log \log n-2}{\log n}-\frac{g(\log \log n)}{2{\log }^{2}n})$

proving that

$f(n,w^2-6w+11.321)\le p_n\le f(n,w^2-6w)$

for Template:Math and Template:Math, respectively. The lower bound may also be simplified to Template:Math without altering its validity. The upper bound may be tightened to Template:Math if Template:Math.

There are additional bounds of varying complexity.

The Riemann hypothesis

The Riemann hypothesis implies a much tighter bound on the error in the estimate for Template:Math, and hence to a more regular distribution of prime numbers,

$\pi (x)=\mathrm{li}(x)+O(\sqrt{x}\log x).$

Specifically,

$|\pi (x)-\mathrm{li}(x)|<\frac{\sqrt{x}}{8\pi }\log x,\text{for all }x\ge 2657.$

Template:Harvtxt proved that the Riemann hypothesis implies that for all Template:Math there is a prime Template:Mvar satisfying

$x-\frac{4}{\pi }\sqrt{x}\log x<p\le x.$

See also

• Bertrand's postulate
• Oppermann's conjecture
• Ramanujan prime

External links

• Chris Caldwell, The Nth Prime Page at The Prime Pages.
• Tomás Oliveira e Silva, Tables of prime-counting functions.
• Template:Citation