Prime-counting function explained

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

Growth rate

See main article: Prime number theorem. Of great interest in number theory is the growth rate of the prime-counting function.^{[2] [3]} It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately$$\backslash frac$$where is the natural logarithm, in the sense that$$\backslash lim\_\; \backslash frac=1.$$This statement is the prime number theorem. An equivalent statement is$$\backslash lim\_\backslash frac=1$$where 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).^[4]

More precise estimates

In 1899, de la Vallée Poussin proved that^[5] $$\backslash pi(x)\; =\; \backslash operatorname\; (x)\; +\; O\; \backslash left(x\; e^\backslash right)\; \backslash quad\backslash text\; x\; \backslash to\; \backslash infty$$for some positive constant . Here, is the big notation.

More precise estimates of are now known. For example, in 2002, Kevin Ford proved that^[6] $$\backslash pi(x)\; =\; \backslash operatorname\; (x)\; +\; O\; \backslash left(x\; \backslash exp\; \backslash left(-0.2098(\backslash log\; x)^\; (\backslash log\; \backslash log\; x)^\; \backslash right)\; \backslash right).$$

Mossinghoff and Trudgian proved^[7] an explicit upper bound for the difference between and :$$\backslash bigl|\; \backslash pi(x)\; -\; \backslash operatorname(x)\; \backslash bigr|\; \backslash le\; 0.2593\; \backslash frac\; \backslash exp\; \backslash left(-\backslash sqrt\; \backslash right)\; \backslash quad\; \backslash text\; x\; \backslash ge\; 229.$$

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

Exact form

For let when is a prime number, and otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that is equal to^[8] $$\backslash pi\_0(x)\; =\; \backslash operatorname(x)\; -\; \backslash sum\_\backslash operatorname(x^\backslash rho),$$where$$\backslash operatorname(x)\; =\; \backslash sum\_^\; \backslash frac\; \backslash operatorname\backslash left(x^\backslash right),$$ is the Möbius function, is the logarithmic integral function, indexes every zero of the Riemann zeta function, and is not evaluated with a branch cut but instead considered as where is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros of the Riemann zeta function, then may be approximated by^[9] $$\backslash pi\_0(x)\; \backslash approx\; \backslash operatorname(x)\; -\; \backslash sum\_\backslash operatorname\backslash left(x^\backslash rho\backslash right)\; -\; \backslash frac\; +\; \backslash frac\; \backslash arctan\; .$$

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

Table of,, and

The table shows how the three functions,, and compared at powers of 10. See also,^[10] and^[11]

					% error
10	4	0	2	2.500	−8.57%
10^2	25	3	5	4.000	+13.14%
10^3	168	23	10	5.952	+13.83%
10^4	1,229	143	17	8.137	+11.66%
10^5	9,592	906	38	10.425	+9.45%
10^6	78,498	6,116	130	12.739	+7.79%
10^7	664,579	44,158	339	15.047	+6.64%
10^8	5,761,455	332,774	754	17.357	+5.78%
10^9	50,847,534	2,592,592	1,701	19.667	+5.10%
10^10	455,052,511	20,758,029	3,104	21.975	+4.56%
10^11	4,118,054,813	169,923,159	11,588	24.283	+4.13%
10^12	37,607,912,018	1,416,705,193	38,263	26.590	+3.77%
10^13	346,065,536,839	11,992,858,452	108,971	28.896	+3.47%
10^14	3,204,941,750,802	102,838,308,636	314,890	31.202	+3.21%
10^15	29,844,570,422,669	891,604,962,452	1,052,619	33.507	+2.99%
10^16	279,238,341,033,925	7,804,289,844,393	3,214,632	35.812	+2.79%
10^17	2,623,557,157,654,233	68,883,734,693,928	7,956,589	38.116	+2.63%
10^18	24,739,954,287,740,860	612,483,070,893,536	21,949,555	40.420	+2.48%
10^19	234,057,667,276,344,607	5,481,624,169,369,961	99,877,775	42.725	+2.34%
10^20	2,220,819,602,560,918,840	49,347,193,044,659,702	222,744,644	45.028	+2.22%
10^21	21,127,269,486,018,731,928	446,579,871,578,168,707	597,394,254	47.332	+2.11%
10^22	201,467,286,689,315,906,290	4,060,704,006,019,620,994	1,932,355,208	49.636	+2.02%
10^23	1,925,320,391,606,803,968,923	37,083,513,766,578,631,309	7,250,186,216	51.939	+1.93%
10^24	18,435,599,767,349,200,867,866	339,996,354,713,708,049,069	17,146,907,278	54.243	+1.84%
10^25	176,846,309,399,143,769,411,680	3,128,516,637,843,038,351,228	55,160,980,939	56.546	+1.77%
10^26	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%
10^27	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%
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	63.456	+1.58%
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	65.759	+1.52%

In the On-Line Encyclopedia of Integer Sequences, the column is sequence, is sequence, and is sequence .

The value for was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis.^[12] It was later verified unconditionally in a computation by D. J. Platt.^[13] The value for is due to J. Buethe, J. Franke, A. Jost, and T. Kleinjung.^[14] The value for was computed by D. B. Staple.^[15] 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,^[16] 2020,^[17] and 2022,^[18] respectively.

Algorithms for evaluating

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

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

\lfloor x\rfloor - \sum _{i} \left\lfloor (x)/(p _{i}) \right\rfloor + \sum _{i<j} \left\lfloor (x)/(p _{ip} _{j}) \right\rfloor - \sum _{i<j<k} \left\lfloor (x)/(p _{ip} _{jp} _{k}) \right\rfloor + …

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

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

when the numbers are the prime numbers less than or equal to the square root of .

The Meissel–Lehmer algorithm

See 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 : Let be the first primes and denote by the number of natural numbers not greater than which are divisible by none of the for any . Then

\Phi(m,n) = \Phi(m,n -1 ) - \Phi\left( (m)/(p _{n}) ,n -1 \right).

Given a natural number, if and if, then

\pi(m) = \Phi(m,n) + n(\mu +1 ) + (\mu ^{2-\mu})/(2) - 1 - \sum ^{\mu\pi\left( (m)/(p _{n+k})} _{k =1} \right) .

Using this approach, Meissel computed, for equal to, 10⁶, 10⁷, and 10⁸.

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

\Phi(m,n) = \sum ^{+inf ty} _{k =0} P _{k(m,n)}

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

\pi(m) = \Phi(m,n) + n - 1 - P _{2(m,n)}

The computation of can be obtained this way:

P _{2(m,n)} = \sum _{y} } \left( \pi \left( (m)/(p) \right) - \pi(p) + 1 \right)

where the sum is over prime numbers.

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

\Phi(m, 0 ) = \lfloor m\rfloor \Phi(m,b) = \Phi(m,b -1 ) - \Phi\left( (m{p)/(_{b},b-1\right)})

Using his method and an IBM 701, Lehmer was able to compute the correct value of and missed the correct value of by 1.^[19]

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.^[20]

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 or . It has jumps of at prime powers and it takes a value halfway between the two sides at the discontinuities of . That added detail is used because the function may then be defined by an inverse Mellin transform.

Formally, we may define by

\Pi _{0(x)} = (1)/(2) \left( \sum _{p ^{n} < x} (1)/(n) + \sum _{p ^{n} \le x} (1)/(n) \right) where the variable in each sum ranges over all primes within the specified limits.

We may also write

\Pi _{0(x)} = \sum ^{x} _{n =2} (Λ (n))/(log n) - (Λ (x))/(2log x) = \sum ^{inf ty} _{n =1} (1)/(n) \pi ^{1 /n} _{0 \left(x} \right) where is the von Mangoldt function and\pi _{0(x)} = \lim _{\varepsilon} (\pi(x - \varepsilon) + \pi(x + \varepsilon))/(2) .

The Möbius inversion formula then gives

\pi _{0(x)} = \sum ^{inf ty} _{n =1} (\mu(n))/(n) \Pi ^{1 /n} _{0 \left(x} \right), where is the Möbius function.

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

log \zeta(s) = s \int ^{inf ty} _{0} \Pi _{0(x)} x ^{-s-1} d x

Chebyshev's function

The Chebyshev function weights primes or prime powers by :

\begin{align} \theta(x) & = \sum _{p\le} log p \\ \psi(x)& = \sum _{p ^{n} \le x} log p = \sum ^{inf ty} _{n =1} \theta \left( x ^{1/n} \right) = \sum _{n} Λ (n) . \end{align}

For,^[21]

\vartheta(x) = \pi(x) log x - \int ^{x} _{2} (\pi(t))/(t) d t and\pi(x) = (\vartheta(x))/(log x) + \int ^{x} _{2} (\vartheta(t))/(t log ^{2} (t)) 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.^[22]

We have the following expression for the second Chebyshev function :

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

where

\psi _{0(x)} = \lim _{\varepsilon} (\psi(x - \varepsilon) + \psi(x + \varepsilon))/(2) .

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

\Pi _{0(x)} = \operatorname{li}(x) - \sum _{\rho} \operatorname{li}\left(x ^{\rho\right)} - log 2 + \int ^{inf ty} _{x} (d t)/(t \left(t ^{2} -1 \right) log t) .

Again, the formula is valid for, while 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:

\int ^{inf ty} _{x} (dt)/(t \left(t ^{2} -1 \right) log t) = \int ^{inf ty} _{x} (1)/(t log t) \left(\sum _{m} t ^{-2m} \right) dt=\sum _{m} \int ^{inf ty} _{x} (t ^{-2m})/(t log t) dt \overset{\left(u = t ^{-2m} \right)}{ = } - \sum _{m} \operatorname{li}\left(x ^{-2m} \right)

The first term is the usual logarithmic integral function; the expression in the second term should be considered as, where 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

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

valid for, where

\operatorname{R}(x) = \sum ^{inf ty} _{n =1} (\mu(n))/(n) \operatorname{li}\left(x ^{1/n} \right) = 1 + \sum ^{inf ty} _{k =1} (\left( log x\right) ^{k})/(k! k \zeta(k +1 ))

is Riemann's R-function and is the Möbius function. The latter series for it is known as Gram series.^[23] Because for all, this series converges for all positive by comparison with the series for . The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated as and not .

Folkmar Bornemann proved,^[24] when assuming the conjecture that all zeros of the Riemann zeta function are simple,^[25] that

\operatorname{R}\left(e ^{-2\pi} \right) = (1)/(\pi) \sum ^{inf ty (( -1 ) ^{k-1} t ^{-2k-1})/(( 2 k +1 )\zeta( 2 k +1 ))} _{k =1} + (12 \sum)/(_{\rho}) (t ^{-\rho})/(\rho\cos (\pi\rho)/(\zeta'(\rho) 2)) where runs over the non-trivial zeros of the Riemann zeta function and .

The sum over non-trivial zeta zeros in the formula for describes the fluctuations of while the remaining terms give the "smooth" part of prime-counting function,^[26] so one can use

\operatorname{R}(x) - \sum ^{inf ty} _{m =1} \operatorname{R}\left(x ^{-2m} \right)

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

l ( \pi _{0(x)} - \operatorname{R}(x) r ) (log x)/(\sqrt x) .

Inequalities

Here are some useful inequalities for .

(x)/(log x) < \pi(x) < 1 . 25506 (x)/(log x)

for .

The left inequality holds for and the right inequality holds for . The constant 1.25506 is to 5 decimal places, as has its maximum value at .^[27]

Pierre Dusart proved in 2010:^[28]

(x)/(log x -1) < \pi(x) for x \ge 5393 ,

and

\pi(x) < (x)/(log x -1 . 1) for x \ge 60184 .

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

n ( log (n log n) - 1 ) < p _{n} < n { log (n log n)} for n \ge 6 .

The left inequality holds for and the right inequality holds for .

An approximation for the nth prime number is

p _{n} = n ( log (n log n) - 1 ) + (n ( loglog n -2 ))/(log n) + O\left( (n ( loglog n) ^{2})/(( log n) ^{2}) \right). Ramanujan^[31] proved that the inequality\pi(x) ^{2} < (ex)/(log x) \pi\left( (x)/(e) \right) holds for all sufficiently large values of .

In 2010 Dusart proved (Proposition 6.6) that, for,

p _{n} \le n \left( log n + log log n - 1 + (loglog n -2)/(log n) \right), and (Proposition 6.7) that, for,p _{n} \ge n \left( log n + log log n - 1 + (loglog n -2 . 1)/(log n) \right) .

More recently, Dusart^[32] has proved (Theorem 5.1) that, for,

\pi(x) \le (x)/(log x) \left( 1 + (1)/(log x) + (2)/(log ^{2} x) + (7 . 59)/(log ^{3} x) \right), and that, for,\pi(x) > (x)/(log x) \left( 1 + (1)/(log x) + (2)/(log ^{2} x) \right) .

The Riemann hypothesis

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

\pi(x) = \operatorname{li}(x) + O(\sqrt{x} log {x}). Specifically,^[33] |\pi(x) - \operatorname{li}(x)| < (\sqrt{x)/() } \, \log, \quad \text x \ge 2657.

proved that the Riemann hypothesis implies that for all there is a prime satisfying

x - (4)/(\pi) \sqrt{x} log x < p \leq x.

See also

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

References

Notes

External links

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

Notes and References

1. Book: Bach, Eric . Shallit, Jeffrey . 1996 . Algorithmic Number Theory . MIT Press . 0-262-02405-5 . volume 1 page 234 section 8.8 . true.
2. Web site: Chris K. Caldwell . How many primes are there? . 2008-12-02 . 2012-10-15 . https://web.archive.org/web/20121015002415/http://primes.utm.edu/howmany.shtml . dead .
3. Book: Dickson, Leonard Eugene . L. E. Dickson. 2005 . History of the Theory of Numbers, Vol. I: Divisibility and Primality . Dover Publications . 0-486-44232-2.
4. Book: Ireland, Kenneth . Rosen, Michael . 1998 . A Classical Introduction to Modern Number Theory . Second . Springer . 0-387-97329-X .
5. See also Theorem 23 of Book: A. E. Ingham . Albert Ingham . The Distribution of Prime Numbers . 2000 . Cambridge University Press . 0-521-39789-8.
6. Kevin Ford . Vinogradov's Integral and Bounds for the Riemann Zeta Function . Proc. London Math. Soc. . November 2002 . 85 . 3 . 565–633 . 10.1112/S0024611502013655 . 1910.08209 . 121144007 .
7. Michael J. . Mossinghoff . Timothy S. . Trudgian . Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function . J. Number Theory . 157 . 2015 . 329–349 . 1410.3926 . 10.1016/J.JNT.2015.05.010. 117968965 .
8. Web site: Implementation of Riemann's Explicit Formula for Rational and Gaussian Primes in Sage. Hutama. Daniel. 2017. Institut des sciences mathématiques.
9. Hans Riesel . Riesel . Hans . Göhl . Gunnar . Some calculations related to Riemann's prime number formula . 10.2307/2004630 . 0277489 . 1970 . . 0025-5718 . 24 . 112 . 969–983 . 2004630 . American Mathematical Society .
10. Web site: Tables of values of and of . Tomás Oliveira e Silva . 2024-03-31.
11. Web site: A table of values of . Xavier Gourdon, Pascal Sebah, Patrick Demichel . 2008-09-14.
12. Web site: Conditional Calculation of π(10²⁴) . Chris K. Caldwell . 2024-03-30.
13. Computing Analytically) . 1203.5712. Platt. David J.. math.NT. 2012.
14. Web site: How Many Primes Are There? . J. Buethe . 2015-09-01.
15. The combinatorial algorithm for computing π(x) . 19 August 2015 . Dalhousie University . 2015-09-01. Thesis . Staple . Douglas .
16. Web site: Mersenne Forum. Kim . Walisch. New confirmed π(10²⁷) prime counting function record . September 6, 2015.
17. Web site: Baugh . David . August 30, 2020 . New prime counting function record, pi(10^28) . Mersenne Forum.
18. Web site: Kim . Walisch . March 4, 2022 . New prime counting function record: PrimePi(10^29) . Mersenne Forum.
19. Lehmer . Derrick Henry . 1 April 1958 . On the exact number of primes less than a given limit . Illinois J. Math. . 3 . 3 . 381–388 . 1 February 2017 .
20. Deléglise, Marc . Rivat, Joel . January 1996 . Computing : The Meissel, Lehmer, Lagarias, Miller, Odlyzko method . Mathematics of Computation . 65 . 213 . 235–245 . 10.1090/S0025-5718-96-00674-6 . free .
21. Book: Apostol, Tom M. . 2010 . Introduction to Analytic Number Theory . Springer.
22. Book: Titchmarsh, E.C. . 1960 . The Theory of Functions, 2nd ed. . Oxford University Press.
23. Book: Riesel, Hans . Prime Numbers and Computer Methods for Factorization . 2nd . Hans Riesel . Progress in Mathematics . 126 . Birkhäuser . 1994 . 0-8176-3743-5 . 50–51 .
24. Web site: Bornemann . Folkmar . Solution of a Problem Posed by Jörg Waldvogel .
25. [Hugh Lowell Montgomery|Montgomery]
26. Web site: The encoding of the prime distribution by the zeta zeros . Matthew Watkins . 2008-09-14.
27. J. Barkley Rosser . Rosser . J. Barkley . Schoenfeld . Lowell . Approximate formulas for some functions of prime numbers . Illinois J. Math. . 1962 . 6 . 64–94 . 10.1215/ijm/1255631807 . 0122.05001 . 0019-2082 . free .
28. Dusart . Pierre . Pierre Dusart . 1002.0442v1 . Estimates of Some Functions Over Primes without R.H. . math.NT . 2 Feb 2010 .
29. J. Barkley Rosser. Rosser. Barkley. 1941. Explicit bounds for some functions of prime numbers. 2371291. American Journal of Mathematics. 63. 1. 211–232. 10.2307/2371291.
30. Pierre Dusart. Dusart. Pierre. 1999. The kth prime is greater than k(ln k + ln ln k − 1) for k ≥ 2. Mathematics of Computation. 68. 225. 411–415. 10.1090/S0025-5718-99-01037-6. free.
31. Book: Berndt, Bruce C.. Ramanujan's Notebooks, Part IV. 2012-12-06. 112–113. Springer Science & Business Media. 9781461269328. en.
32. Dusart . Pierre . Pierre Dusart . Explicit estimates of some functions over primes . Ramanujan Journal . 45 . 1 . 225–234 . January 2018 . 10.1007/s11139-016-9839-4. 125120533 .
33. Schoenfeld . Lowell . Lowell Schoenfeld. Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II . 10.2307/2005976 . 0457374 . 1976 . . 0025-5718 . 30 . 134 . 337–360 . 2005976 . American Mathematical Society.