Prime-counting function

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

$${\displaystyle {\frac {x}{\log(x)}}}$$

where log is the natural logarithm, in the sense that

$${\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\log(x)}}=1.}$$

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

$${\displaystyle \lim _{x\rightarrow \infty }\pi (x)/\operatorname {li} (x)=1}$$

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).[5]

More precise estimates

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

$${\displaystyle \pi (x)=\operatorname {li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }$$

for some positive constant a. Here, O(...) is the big O notation.

More precise estimates of ${\displaystyle \pi (x)\!}$ are now known. For example, in 2002, Kevin Ford proved that[7]

$${\displaystyle \pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\log x)^{\frac {3}{5}}(\log \log x)^{-{\frac {1}{5}}}\right)\right).}$$

Mossinghoff and Trudgian proved[8] an explicit upper bound for the difference between ${\displaystyle \pi (x)}$ and ${\displaystyle \operatorname {li} (x)}$:

$${\displaystyle {\big |}\pi (x)-\operatorname {li} (x){\big |}\leq 0.2593{\frac {x}{(\log x)^{3/4}}}\exp \left(-{\sqrt {\frac {\log x}{6.315}}}\right)}$$

for ${\displaystyle x\geq 229}$.

For values of ${\displaystyle x}$ that are not unreasonably large, ${\displaystyle \operatorname {li} (x)}$ is greater than ${\displaystyle \pi (x)}$. However, ${\displaystyle \pi (x)-\operatorname {li} (x)}$ is known to change sign infinitely many times. For a discussion of this, see Skewes' number.

Exact form

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

$${\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho }),}$$

where

$${\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n}),}$$

μ(n) is the Möbius function, li(x) is the logarithmic integral function, ρ indexes every zero of the Riemann zeta function, and li(x^ρ/n) is not evaluated with a branch cut but instead considered as Ei(ρ/n log x) where Ei(x) is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then ${\displaystyle \pi _{0}(x)}$ may be approximated by[10]

$${\displaystyle \pi _{0}(x)\approx \operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-{\frac {1}{\log {x}}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log {x}}}.}$$

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

The table shows how the three functions π(x), x / log x and li(x) compare at powers of 10. See also,[3][11] and[12]

x	π(x)	π(x) − x / log x	li(x) − π(x)	x / π(x)	x / log x  % 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 π(x) to two of its approximations, x/log x and Li(x). As x increases (note x axis is logarithmic), both ratios tend towards 1. The ratio for x/log x converges from above very slowly, while the ratio for Li(x) converges more quickly from below.

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(10²⁴) was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.[13] It was later verified unconditionally in a computation by D. J. Platt.[14] The value for π(10²⁵) is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.[15] The value for π(10²⁶) was computed by D. B. Staple.[16] All other prior entries in this table were also verified as part of that work.

The value for 10²⁷ was announced in 2015 by David Baugh and Kim Walisch.[17]

The value for 10²⁸ was announced in 2020 by David Baugh and Kim Walisch.[18]

The value for 10²⁹ was announced in 2022 by David Baugh and Kim Walisch.[19]

A simple way to find ${\displaystyle \pi (x)}$, if ${\displaystyle x}$ is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to ${\displaystyle x}$ and then to count them.

A more elaborate way of finding ${\displaystyle \pi (x)}$ is due to Legendre (using the inclusion–exclusion principle): given ${\displaystyle x}$, if ${\displaystyle p_{1},p_{2},\ldots ,p_{n}}$ are distinct prime numbers, then the number of integers less than or equal to ${\displaystyle x}$ which are divisible by no ${\displaystyle p_{i}}$ is

${\displaystyle \lfloor x\rfloor -\sum _{i}\left\lfloor {\frac {x}{p_{i}}}\right\rfloor +\sum _{i<j}\left\lfloor {\frac {x}{p_{i}p_{j}}}\right\rfloor -\sum _{i<j<k}\left\lfloor {\frac {x}{p_{i}p_{j}p_{k}}}\right\rfloor +\cdots }$

(where ${\displaystyle \lfloor {x}\rfloor }$ denotes the floor function). This number is therefore equal to

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

when the numbers ${\displaystyle p_{1},p_{2},\ldots ,p_{n}}$ are the prime numbers less than or equal to the square root of ${\displaystyle x}$.

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

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 formulas.[23]

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

${\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log 2\pi -{\frac {1}{2}}\log \left(1-x^{-2}\right),}$

where

${\displaystyle \psi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}$

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 ${\displaystyle \Pi _{0}(x)}$ we have a more complicated formula

${\displaystyle \Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\log 2+\int _{x}^{\infty }{\frac {dt}{t\left(t^{2}-1\right)\log t}}.}$

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

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:

${\displaystyle \int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}=\int _{x}^{\infty }{\frac {1}{t\log t}}\left(\sum _{m}t^{-2m}\right)\,\mathrm {d} t=\sum _{m}\int _{x}^{\infty }{\frac {t^{-2m}}{t\log t}}\,\mathrm {d} t\,\,{\overset {(u=t^{-2m})}{=}}-\sum _{m}\operatorname {li} (x^{-2m})}$

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]

${\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-\sum _{m}\operatorname {R} (x^{-2m})}$

valid for x > 1, where

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

is Riemann's R-function[24] and μ(n) is the Möbius function. The latter series for it is known as Gram series.[25][26] Because ${\displaystyle \log(x)<x}$ for all ${\displaystyle x>0}$, this series converges for all positive x by comparison with the series for ${\displaystyle e^{x}}$. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as ${\displaystyle \rho \log x}$ and not ${\displaystyle \log x^{\rho }}$.

Folkmar Bornemann proved,[27] when assuming the conjecture that all zeros of the Riemann zeta function are simple,[note 1] that

${\displaystyle \operatorname {R} (e^{-2\pi t})={\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta (2k+1)}}+{\frac {1}{2}}\sum _{\rho }{\frac {t^{-\rho }}{\rho \cos(\pi \rho /2)\zeta '(\rho )}}}$

where ${\displaystyle \rho }$ runs over the non-trivial zeros of the Riemann zeta function and ${\displaystyle t>0}$.

The sum over non-trivial zeta zeros in the formula for ${\displaystyle \pi _{0}(x)}$ describes the fluctuations of ${\displaystyle \pi _{0}(x),}$ while the remaining terms give the "smooth" part of prime-counting function,[28] so one can use

${\displaystyle \operatorname {R} (x)-\sum _{m=1}^{\infty }\operatorname {R} (x^{-2m})}$

as a good estimator of ${\displaystyle \pi (x)}$ for x > 1. In fact, since the second term approaches 0 as ${\displaystyle x\to \infty }$, while the amplitude of the "noisy" part is heuristically about ${\displaystyle {\sqrt {x}}/\log x,}$ estimating ${\displaystyle \pi (x)}$ by ${\displaystyle \operatorname {R} (x)}$ alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function

${\displaystyle {\bigl (}\pi _{0}(x)-\operatorname {R} (x){\bigr )}{\frac {\log x}{\sqrt {x}}}.}$

Here are some useful inequalities for π(x).

${\displaystyle {\frac {x}{\log x}}<\pi (x)<1.25506{\frac {x}{\log x}}}$

for x ≥ 17.

The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is ${\textstyle {\frac {30\log 113}{113}}}$ to 5 decimal places, as ${\textstyle {\frac {\pi (x)\log x}{x}}}$ has its maximum value at x = 113.[29]

Pierre Dusart proved in 2010:

${\displaystyle {\frac {x}{\log x-1}}<\pi (x)}$ for ${\displaystyle x\geq 5393}$, and

${\displaystyle \pi (x)<{\frac {x}{\log x-1.1}}}$ for ${\displaystyle x\geq 60184}$.[30]

Here are some inequalities for the nth prime, p_n. The upper bound is due to Rosser (1941),[31] the lower one to Dusart (1999):[32]

${\displaystyle n(\log(n\log n)-1)<p_{n}<n{\log(n\log n)}}$ for n ≥ 6.

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

An approximation for the nth prime number is

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

Ramanujan[33] proved that the inequality

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

holds for all sufficiently large values of ${\displaystyle x}$.

In [30] Dusart proved (Proposition 6.6) that, for ${\displaystyle n\geq 688383}$,

${\displaystyle p_{n}\leq n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}\right),}$

and (Proposition 6.7) that, for ${\displaystyle n\geq 3}$,

${\displaystyle p_{n}\geq n\left(\log n+\log \log n-1+{\frac {\log \log n-2.1}{\log n}}\right).}$

More recently, Dusart[34] has proved (Theorem 5.1) that, for ${\displaystyle x>1}$,

${\displaystyle \pi (x)\leq {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}+{\frac {7.59}{\log ^{3}x}}\right)}$ ,

and that, for ${\displaystyle x\geq 88789}$,

${\displaystyle \pi (x)>{\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}\right).}$

The Riemann hypothesis implies a much tighter bound on the error in the estimate for ${\displaystyle \pi (x)}$, and hence to a more regular distribution of prime numbers,

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

Specifically,[35]

${\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {\sqrt {x}}{8\pi }}\,\log {x},\qquad {\text{for all }}x\geq 2657.}$

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

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