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Prime-counting function

From Wikipedia, the free encyclopedia
Function representing the number of primes less than or equal to a given number
"Π(x)" redirects here. For the variant of the gamma function, seeGamma function § Pi function.
This article uses technical mathematical notation for logarithms. All instances oflog(x) without a subscript base should be interpreted as anatural logarithm, also commonly written asln(x) orloge(x).
This articleduplicates the scope of other articles, specificallyPrime number theorem. Pleasediscuss this issue and help introduce asummary style to the article.(December 2024)
The values ofπ(n) for the first 60 positive integers

Inmathematics, theprime-counting function is thefunction counting the number ofprime numbers less than or equal to somereal numberx.[1][2] It is denoted byπ(x) (unrelated to thenumberπ).

A symmetric variant seen sometimes isπ0(x), which is equal toπ(x) −1⁄2 ifx is exactly a prime number, and equal toπ(x) otherwise. That is, the number of prime numbers less thanx, plus half ifx equals a prime.

Growth rate

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Main article:Prime number theorem

Of great interest innumber theory is thegrowth rate of the prime-counting function.[3][4] It wasconjectured in the end of the 18th century byGauss and byLegendre to be approximatelyxlog⁡x{\displaystyle {\frac {x}{\log x}}}wherelog is thenatural logarithm, in the sense thatlimx→∞π(x)x/log⁡x=1.{\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{x/\log x}}=1.}This statement is theprime number theorem. An equivalent statement islimx→∞π(x)li⁡(x)=1{\displaystyle \lim _{x\rightarrow \infty }{\frac {\pi (x)}{\operatorname {li} (x)}}=1}whereli is thelogarithmic integral function. The prime number theorem was first proved in 1896 byJacques Hadamard and byCharles de la Vallée Poussin independently, using properties of theRiemann zeta function introduced byRiemann in 1859. Proofs of the prime number theorem not using the zeta function orcomplex analysis were found around 1948 byAtle Selberg and byPaul Erdős (for the most part independently).[5]

More precise estimates

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In 1899,de la Vallée Poussin proved that[6]π(x)=li⁡(x)+O(xe−alog⁡x)as x→∞{\displaystyle \pi (x)=\operatorname {li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty }for some positive constanta. Here,O(...) is thebigO notation.

More precise estimates ofπ(x) are now known. For example, in 2002,Kevin Ford proved that[7]π(x)=li⁡(x)+O(xexp⁡(−0.2098(log⁡x)3/5(log⁡log⁡x)−1/5)).{\displaystyle \pi (x)=\operatorname {li} (x)+O\left(x\exp \left(-0.2098(\log x)^{3/5}(\log \log x)^{-1/5}\right)\right).}

Mossinghoff andTrudgian proved[8] an explicit upper bound for the difference betweenπ(x) andli(x):|π(x)−li⁡(x)|≤0.2593x(log⁡x)3/4exp⁡(−log⁡x6.315)for x≥229.{\displaystyle {\bigl |}\pi (x)-\operatorname {li} (x){\bigr |}\leq 0.2593{\frac {x}{(\log x)^{3/4}}}\exp \left(-{\sqrt {\frac {\log x}{6.315}}}\right)\quad {\text{for }}x\geq 229.}

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

Exact form

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Forx > 1 letπ0(x) =π(x) −⁠1/2⁠ whenx is a prime number, andπ0(x) =π(x) otherwise.Bernhard Riemann, in his workOn the Number of Primes Less Than a Given Magnitude, proved thatπ0(x) is equal to[9]

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

π0(x)=R⁡(x)−∑ρR⁡(xρ),{\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho }),}whereR⁡(x)=∑n=1∞μ(n)nli⁡(x1/n),{\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right),}μ(n) is theMöbius function,li(x) is thelogarithmic integral function,ρ indexes every zero of the Riemann zeta function, andli(x⁠ρ/n⁠) is not evaluated with abranch cut but instead considered asEi(⁠ρ/n⁠ logx) whereEi(x) is theexponential integral. If the trivial zeros are collected and the sum is takenonly over the non-trivial zerosρ of the Riemann zeta function, thenπ0(x) may be approximated by[10]π0(x)≈R⁡(x)−∑ρR⁡(xρ)−1log⁡x+1πarctan⁡πlog⁡x.{\displaystyle \pi _{0}(x)\approx \operatorname {R} (x)-\sum _{\rho }\operatorname {R} \left(x^{\rho }\right)-{\frac {1}{\log x}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log x}}.}

TheRiemann hypothesis suggests that every such non-trivial zero lies alongRe(s) =⁠1/2⁠.

Table ofπ(x),⁠x/logx⁠, andli(x)

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The table shows how the three functionsπ(x),⁠x/logx⁠, andli(x) compared at powers of 10. See also,[3][11] and[12]

xπ(x)π(x) −⁠x/logx⁠li(x) −π(x)⁠x/π(x)⁠⁠x/logx⁠
 % error
104022.500−8.57%
10225354.000+13.14%
10316823105.952+13.83%
1041,229143178.137+11.66%
1059,5929063810.425+9.45%
10678,4986,11613012.739+7.79%
107664,57944,15833915.047+6.64%
1085,761,455332,77475417.357+5.78%
10950,847,5342,592,5921,70119.667+5.10%
1010455,052,51120,758,0293,10421.975+4.56%
10114,118,054,813169,923,15911,58824.283+4.13%
101237,607,912,0181,416,705,19338,26326.590+3.77%
1013346,065,536,83911,992,858,452108,97128.896+3.47%
10143,204,941,750,802102,838,308,636314,89031.202+3.21%
101529,844,570,422,669891,604,962,4521,052,61933.507+2.99%
1016279,238,341,033,9257,804,289,844,3933,214,63235.812+2.79%
10172,623,557,157,654,23368,883,734,693,9287,956,58938.116+2.63%
101824,739,954,287,740,860612,483,070,893,53621,949,55540.420+2.48%
1019234,057,667,276,344,6075,481,624,169,369,96199,877,77542.725+2.34%
10202,220,819,602,560,918,84049,347,193,044,659,702222,744,64445.028+2.22%
102121,127,269,486,018,731,928446,579,871,578,168,707597,394,25447.332+2.11%
1022201,467,286,689,315,906,2904,060,704,006,019,620,9941,932,355,20849.636+2.02%
10231,925,320,391,606,803,968,92337,083,513,766,578,631,3097,250,186,21651.939+1.93%
102418,435,599,767,349,200,867,866339,996,354,713,708,049,06917,146,907,27854.243+1.84%
1025176,846,309,399,143,769,411,6803,128,516,637,843,038,351,22855,160,980,93956.546+1.77%
10261,699,246,750,872,437,141,327,60328,883,358,936,853,188,823,261155,891,678,12158.850+1.70%
102716,352,460,426,841,680,446,427,399267,479,615,610,131,274,163,365508,666,658,00661.153+1.64%
1028157,589,269,275,973,410,412,739,5982,484,097,167,669,186,251,622,1271,427,745,660,37463.456+1.58%
10291,520,698,109,714,272,166,094,258,06323,130,930,737,541,725,917,951,4464,551,193,622,46465.759+1.52%
Graph showing ratio of the prime-counting functionπ(x) to two of its approximations,⁠x/logx⁠ andLi(x). Asx increases (notex-axis is logarithmic), both ratios tend towards 1. The ratio for⁠x/logx⁠ converges from above very slowly, while the ratio forLi(x) converges more quickly from below.

In theOn-Line Encyclopedia of Integer Sequences, theπ(x) column is sequenceOEIS: A006880,π(x) −⁠x/logx⁠ is sequenceOEIS: A057835, andli(x) −π(x) is sequenceOEIS: A057752.

The value forπ(1024) was originally computed by J. Buethe,J. Franke, A. Jost, and T. Kleinjung assuming theRiemann hypothesis.[13]It was later verified unconditionally in a computation by D. J. Platt.[14]The value forπ(1025) is by the same four authors.[15] The value forπ(1026) was computed by D. B. Staple.[16] 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,[17] 2020,[18] and 2022,[19] respectively.

Algorithms for evaluatingπ(x)

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A simple way to findπ(x), ifx is not too large, is to use thesieve of Eratosthenes to produce the primes less than or equal tox and then to count them.

A more elaborate way of findingπ(x) is due toLegendre (using theinclusion–exclusion principle): givenx, ifp1,p2,…,pn are distinct prime numbers, then the number of integers less than or equal tox which are divisible by nopi is

⌊x⌋−∑i⌊xpi⌋+∑i<j⌊xpipj⌋−∑i<j<k⌊xpipjpk⌋+⋯{\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⌊x⌋ denotes thefloor function). This number is therefore equal to

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

when the numbersp1,p2,…,pn are the prime numbers less than or equal to the square root ofx.

The Meissel–Lehmer algorithm

[edit]
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 evaluatingπ(x): Letp1,p2,…,pn be the firstn primes and denote byΦ(m,n) the number of natural numbers not greater thanm which are divisible by none of thepi for anyi ≤n. Then

Φ(m,n)=Φ(m,n−1)−Φ(mpn,n−1).{\displaystyle \Phi (m,n)=\Phi (m,n-1)-\Phi \left({\frac {m}{p_{n}}},n-1\right).}

Given a natural numberm, ifn =π(3√m) and ifμ =π(√m) −n, then

π(m)=Φ(m,n)+n(μ+1)+μ2−μ2−1−∑k=1μπ(mpn+k).{\displaystyle \pi (m)=\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 computedπ(x), forx equal to5×105, 106, 107, and 108.

In 1959,Derrick Henry Lehmer extended and simplified Meissel's method. Define, for realm and for natural numbersn andk,Pk(m,n) as the number of numbers not greater thanm with exactlyk prime factors, all greater thanpn. Furthermore, setP0(m,n) = 1. Then

Φ(m,n)=∑k=0+∞Pk(m,n){\displaystyle \Phi (m,n)=\sum _{k=0}^{+\infty }P_{k}(m,n)}

where the sum actually has only finitely many nonzero terms. Lety denote an integer such that3√m ≤y ≤√m, and setn =π(y). ThenP1(m,n) =π(m) −n andPk(m,n) = 0 whenk ≥ 3. Therefore,

π(m)=Φ(m,n)+n−1−P2(m,n){\displaystyle \pi (m)=\Phi (m,n)+n-1-P_{2}(m,n)}

The computation ofP2(m,n) can be obtained this way:

P2(m,n)=∑y<p≤m(π(mp)−π(p)+1){\displaystyle P_{2}(m,n)=\sum _{y<p\leq {\sqrt {m}}}\left(\pi \left({\frac {m}{p}}\right)-\pi (p)+1\right)}

where the sum is over prime numbers.

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

  1. Φ(m,0)=⌊m⌋{\displaystyle \Phi (m,0)=\lfloor m\rfloor }
  2. Φ(m,b)=Φ(m,b−1)−Φ(mpb,b−1){\displaystyle \Phi (m,b)=\Phi (m,b-1)-\Phi \left({\frac {m}{p_{b}}},b-1\right)}

Using his method and anIBM 701, Lehmer was able to compute the correct value ofπ(109) and missed the correct value ofπ(1010) by 1.[20]

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.[21]

Other prime-counting functions

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

Riemann's prime-power counting function

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Riemann's prime-power counting function is usually denoted asΠ0(x) orJ0(x). It has jumps of⁠1/n⁠ at prime powerspn and it takes a value halfway between the two sides at the discontinuities ofπ(x). That added detail is used because the function may then be defined by an inverseMellin transform.

Formally, we may defineΠ0(x) by

Π0(x)=12(∑pn<x1n+∑pn≤x1n) {\displaystyle \Pi _{0}(x)={\frac {1}{2}}\left(\sum _{p^{n}<x}{\frac {1}{n}}+\sum _{p^{n}\leq x}{\frac {1}{n}}\right)\ }

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

We may also write

 Π0(x)=∑n=2xΛ(n)log⁡n−Λ(x)2log⁡x=∑n=1∞1nπ0(x1/n){\displaystyle \ \Pi _{0}(x)=\sum _{n=2}^{x}{\frac {\Lambda (n)}{\log n}}-{\frac {\Lambda (x)}{2\log x}}=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}\left(x^{1/n}\right)}

whereΛ is thevon Mangoldt function and

π0(x)=limε→0π(x−ε)+π(x+ε)2.{\displaystyle \pi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\pi (x-\varepsilon )+\pi (x+\varepsilon )}{2}}.}

TheMöbius inversion formula then gives

π0(x)=∑n=1∞μ(n)n Π0(x1/n),{\displaystyle \pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\ \Pi _{0}\left(x^{1/n}\right),}

whereμ(n) is theMöbius function.

Knowing the relationship between the logarithm of theRiemann zeta function and thevon Mangoldt functionΛ, and using thePerron formula we have

log⁡ζ(s)=s∫0∞Π0(x)x−s−1dx{\displaystyle \log \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s-1}\,\mathrm {d} x}

Chebyshev's function

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TheChebyshev function weights primes or prime powerspn bylogp:

ϑ(x)=∑p≤xlog⁡pψ(x)=∑pn≤xlog⁡p=∑n=1∞ϑ(x1/n)=∑n≤xΛ(n).{\displaystyle {\begin{aligned}\vartheta (x)&=\sum _{p\leq x}\log p\\\psi (x)&=\sum _{p^{n}\leq x}\log p=\sum _{n=1}^{\infty }\vartheta \left(x^{1/n}\right)=\sum _{n\leq x}\Lambda (n).\end{aligned}}}

Forx ≥ 2,[22]

ϑ(x)=π(x)log⁡x−∫2xπ(t)tdt{\displaystyle \vartheta (x)=\pi (x)\log x-\int _{2}^{x}{\frac {\pi (t)}{t}}\,\mathrm {d} t}

and

π(x)=ϑ(x)log⁡x+∫2xϑ(t)tlog2⁡(t)dt.{\displaystyle \pi (x)={\frac {\vartheta (x)}{\log x}}+\int _{2}^{x}{\frac {\vartheta (t)}{t\log ^{2}(t)}}\mathrm {d} t.}

Formulas for prime-counting functions

[edit]

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 theprime number theorem. They stem from the work of Riemann andvon Mangoldt, and are generally known asexplicit formulae.[23]

We have the following expression for the secondChebyshev functionψ:

ψ0(x)=x−∑ρxρρ−log⁡2π−12log⁡(1−x−2),{\displaystyle \psi _{0}(x)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\log 2\pi -{\frac {1}{2}}\log \left(1-x^{-2}\right),}

where

ψ0(x)=limε→0ψ(x−ε)+ψ(x+ε)2.{\displaystyle \psi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\psi (x-\varepsilon )+\psi (x+\varepsilon )}{2}}.}

Hereρ are the zeros of the Riemann zeta function in the critical strip, where the real part ofρ is between zero and one. The formula is valid for values ofx 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 lastsubtrahend in the formula.

ForΠ0(x) we have a more complicated formula

Π0(x)=li⁡(x)−∑ρli⁡(xρ)−log⁡2+∫x∞dtt(t2−1)log⁡t.{\displaystyle \Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} \left(x^{\rho }\right)-\log 2+\int _{x}^{\infty }{\frac {\mathrm {d} t}{t\left(t^{2}-1\right)\log t}}.}

Again, the formula is valid forx > 1, whileρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The first termli(x) is the usuallogarithmic integral function; the expressionli(xρ) in the second term should be considered asEi(ρ logx), whereEi is theanalytic continuation of theexponential 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:

∫x∞dtt(t2−1)log⁡t=∫x∞1tlog⁡t(∑mt−2m)dt=∑m∫x∞t−2mtlog⁡tdt=(u=t−2m)−∑mli⁡(x−2m){\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 {\left(u=t^{-2m}\right)}{=}}-\sum _{m}\operatorname {li} \left(x^{-2m}\right)}

Thus,Möbius inversion formula gives us[10]

π0(x)=R⁡(x)−∑ρR⁡(xρ)−∑mR⁡(x−2m){\displaystyle \pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} \left(x^{\rho }\right)-\sum _{m}\operatorname {R} \left(x^{-2m}\right)}

valid forx > 1, where

R⁡(x)=∑n=1∞μ(n)nli⁡(x1/n)=1+∑k=1∞(log⁡x)kk!kζ(k+1){\displaystyle \operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} \left(x^{1/n}\right)=1+\sum _{k=1}^{\infty }{\frac {\left(\log x\right)^{k}}{k!k\zeta (k+1)}}}

is Riemann's R-function[24] andμ(n) is theMöbius function. The latter series for it is known asGram series.[25][26] Becauselogx <x for allx > 0, this series converges for all positivex by comparison with the series forex. The logarithm in the Gram series of the sum over the non-trivial zero contribution should be evaluated asρ logx and notlogxρ.

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

R⁡(e−2πt)=1π∑k=1∞(−1)k−1t−2k−1(2k+1)ζ(2k+1)+12∑ρt−ρρcos⁡πρ2ζ′(ρ){\displaystyle \operatorname {R} \left(e^{-2\pi t}\right)={\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 {\frac {\pi \rho }{2}}\zeta '(\rho )}}}

whereρ runs over the non-trivial zeros of the Riemann zeta function andt > 0.

The sum over non-trivial zeta zeros in the formula forπ0(x) describes the fluctuations ofπ0(x) while the remaining terms give the "smooth" part of prime-counting function,[28] so one can use

R⁡(x)−∑m=1∞R⁡(x−2m){\displaystyle \operatorname {R} (x)-\sum _{m=1}^{\infty }\operatorname {R} \left(x^{-2m}\right)}

as a good estimator ofπ(x) forx > 1. In fact, since the second term approaches 0 asx → ∞, while the amplitude of the "noisy" part is heuristically about⁠√x/logx⁠, estimatingπ(x) byR(x) alone is just as good, and fluctuations of thedistribution of primes may be clearly represented with the function

(π0(x)−R⁡(x))log⁡xx.{\displaystyle {\bigl (}\pi _{0}(x)-\operatorname {R} (x){\bigr )}{\frac {\log x}{\sqrt {x}}}.}

Inequalities

[edit]

Ramanujan[29] proved that the inequality

π(x)2<exlog⁡xπ(xe){\displaystyle \pi (x)^{2}<{\frac {ex}{\log x}}\pi \left({\frac {x}{e}}\right)}

holds for all sufficiently large values ofx.

Here are some useful inequalities forπ(x).

xlog⁡x<π(x)<1.25506xlog⁡xfor x≥17.{\displaystyle {\frac {x}{\log x}}<\pi (x)<1.25506{\frac {x}{\log x}}\quad {\text{for }}x\geq 17.}

The left inequality holds forx ≥ 17 and the right inequality holds forx > 1. The constant 1.25506 is30⁠log 113/113⁠ to 5 decimal places, asπ(x)⁠logx/x⁠ has its maximum value atx =p30 = 113.[30]

Pierre Dusart proved in 2010:[31]

xlog⁡x−1<π(x)<xlog⁡x−1.1for x≥5393 and x≥60184, respectively.{\displaystyle {\frac {x}{\log x-1}}<\pi (x)<{\frac {x}{\log x-1.1}}\quad {\text{for }}x\geq 5393{\text{ and }}x\geq 60184,{\text{ respectively.}}}

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

xlog⁡x(1+1log⁡x+2log2⁡x)≤π(x)≤xlog⁡x(1+1log⁡x+2log2⁡x+7.59log3⁡x),{\displaystyle {\frac {x}{\log x}}\left(1+{\frac {1}{\log x}}+{\frac {2}{\log ^{2}x}}\right)\leq \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),}

forx ≥ 88789 andx > 1, respectively.

Going in the other direction, an approximation for thenth prime,pn, is

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

Here are some inequalities for thenth prime. The lower bound is due to Dusart (1999)[33] and the upper bound to Rosser (1941).[34]

n(log⁡n+log⁡log⁡n−1)<pn<n(log⁡n+log⁡log⁡n)for n≥6.{\displaystyle n(\log n+\log \log n-1)<p_{n}<n(\log n+\log \log n)\quad {\text{for }}n\geq 6.}

The left inequality holds forn ≥ 2 and the right inequality holds forn ≥ 6. A variant form sometimes seen substituteslog⁡n+log⁡log⁡n=log⁡(nlog⁡n).{\displaystyle \log n+\log \log n=\log(n\log n).} An even simpler lower bound is[35]

nlog⁡n<pn,{\displaystyle n\log n<p_{n},}

which holds for alln ≥ 1, but the lower bound above is tighter forn >ee ≈15.154.

In 2010 Dusart proved[31] (Propositions 6.7 and 6.6) that

n(log⁡n+log⁡log⁡n−1+log⁡log⁡n−2.1log⁡n)≤pn≤n(log⁡n+log⁡log⁡n−1+log⁡log⁡n−2log⁡n),{\displaystyle n\left(\log n+\log \log n-1+{\frac {\log \log n-2.1}{\log n}}\right)\leq p_{n}\leq n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}\right),}

forn ≥ 3 andn ≥ 688383, respectively.

In 2024, Axler[36] 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+log⁡log⁡n−2log⁡n−g(log⁡log⁡n)2log2⁡n){\displaystyle f(n,g(w))=n\left(\log n+\log \log n-1+{\frac {\log \log n-2}{\log n}}-{\frac {g(\log \log n)}{2\log ^{2}n}}\right)}

proving that

f(n,w2−6w+11.321)≤pn≤f(n,w2−6w){\displaystyle f(n,w^{2}-6w+11.321)\leq p_{n}\leq f(n,w^{2}-6w)}

forn ≥ 2 andn ≥ 3468, respectively.The lower bound may also be simplified tof(n,w2) without altering its validity. The upper bound may be tightened tof(n,w2 − 6w + 10.667) ifn ≥ 46254381.

There are additional bounds of varying complexity.[37][38][39]

The Riemann hypothesis

[edit]

TheRiemann hypothesis implies a much tighter bound on the error in the estimate forπ(x), and hence to a more regular distribution of prime numbers,

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

Specifically,[40]

|π(x)−li⁡(x)|<x8πlog⁡x,for all x≥2657.{\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {\sqrt {x}}{8\pi }}\,\log {x},\quad {\text{for all }}x\geq 2657.}

Dudek (2015) proved that the Riemann hypothesis implies that for allx ≥ 2 there is a primep satisfying

x−4πxlog⁡x<p≤x.{\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x.}

See also

[edit]

References

[edit]
  1. ^Bach, Eric; Shallit, Jeffrey (1996).Algorithmic Number Theory. MIT Press. volume 1 page 234 section 8.8.ISBN 0-262-02405-5.
  2. ^Weisstein, Eric W."Prime Counting Function".MathWorld.
  3. ^ab"How many primes are there?". Chris K. Caldwell. Archived fromthe original on 2012-10-15. Retrieved2008-12-02.
  4. ^Dickson, Leonard Eugene (2005).History of the Theory of Numbers, Vol. I: Divisibility and Primality. Dover Publications.ISBN 0-486-44232-2.
  5. ^Ireland, Kenneth; Rosen, Michael (1998).A Classical Introduction to Modern Number Theory (Second ed.). Springer.ISBN 0-387-97329-X.
  6. ^See also Theorem 23 ofA. E. Ingham (2000).The Distribution of Prime Numbers. Cambridge University Press.ISBN 0-521-39789-8.
  7. ^Kevin Ford (November 2002)."Vinogradov's Integral and Bounds for the Riemann Zeta Function"(PDF).Proc. London Math. Soc.85 (3):565–633.arXiv:1910.08209.doi:10.1112/S0024611502013655.S2CID 121144007.
  8. ^Mossinghoff, Michael J.;Trudgian, Timothy S. (2015). "Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function".J. Number Theory.157:329–349.arXiv:1410.3926.doi:10.1016/J.JNT.2015.05.010.S2CID 117968965.
  9. ^Hutama, Daniel (2017)."Implementation of Riemann's Explicit Formula for Rational and Gaussian Primes in Sage"(PDF).Institut des sciences mathématiques.
  10. ^abRiesel, Hans; Göhl, Gunnar (1970)."Some calculations related to Riemann's prime number formula"(PDF).Mathematics of Computation.24 (112). American Mathematical Society:969–983.doi:10.2307/2004630.ISSN 0025-5718.JSTOR 2004630.MR 0277489.
  11. ^"Tables of values ofπ(x) and ofπ2(x)". Tomás Oliveira e Silva. Retrieved2024-03-31.
  12. ^"A table of values ofπ(x)". Xavier Gourdon, Pascal Sebah, Patrick Demichel. Retrieved2008-09-14.
  13. ^Franke, Jens (2010-07-29)."Conditional Calculation of π(1024)". Chris K. Caldwell. Retrieved2024-03-30.
  14. ^Platt, David J. (May 2015) [March 2012]."Computingπ(x) Analytically".Mathematics of Computation.84 (293):1521–1535.arXiv:1203.5712.doi:10.1090/S0025-5718-2014-02884-6.
  15. ^"Analytic Computation of the prime-counting Function". J. Buethe. 27 May 2014. Retrieved2015-09-01. Includes 600,000 value ofπ(x) for1014 ≤x ≤ 1.6×1018
  16. ^Staple, Douglas (19 August 2015).The combinatorial algorithm for computing π(x) (Thesis). Dalhousie University. Retrieved2015-09-01.
  17. ^Walisch, Kim (September 6, 2015)."New confirmed π(1027) prime counting function record".Mersenne Forum.
  18. ^Baugh, David (August 30, 2020)."New prime counting function record, pi(10^28)".Mersenne Forum.
  19. ^Walisch, Kim (March 4, 2022)."New prime counting function record: PrimePi(10^29)".Mersenne Forum.
  20. ^Lehmer, Derrick Henry (1 April 1958)."On the exact number of primes less than a given limit".Illinois J. Math.3 (3):381–388. Retrieved1 February 2017.
  21. ^Deléglise, Marc; Rivat, Joel (January 1996)."Computingπ(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method"(PDF).Mathematics of Computation.65 (213):235–245.doi:10.1090/S0025-5718-96-00674-6.
  22. ^Apostol, Tom M. (2010).Introduction to Analytic Number Theory. Springer.ISBN 978-1441928054.
  23. ^Titchmarsh, E.C. (1960).The Theory of Functions, 2nd ed. Oxford University Press.
  24. ^Weisstein, Eric W."Riemann Prime Counting Function".MathWorld.
  25. ^Riesel, Hans (1994).Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Vol. 126 (2nd ed.). Birkhäuser. pp. 50–51.ISBN 0-8176-3743-5.
  26. ^Weisstein, Eric W."Gram Series".MathWorld.
  27. ^Bornemann, Folkmar."Solution of a Problem Posed by Jörg Waldvogel"(PDF).
  28. ^"The encoding of the prime distribution by the zeta zeros". Matthew Watkins. Retrieved2008-09-14.
  29. ^Berndt, Bruce C. (2012-12-06).Ramanujan's Notebooks, Part IV. Springer Science & Business Media. pp. 112–113.ISBN 9781461269328.
  30. ^Rosser, J. Barkley; Schoenfeld, Lowell (1962)."Approximate formulas for some functions of prime numbers".Illinois J. Math.6:64–94.doi:10.1215/ijm/1255631807.ISSN 0019-2082.Zbl 0122.05001.
  31. ^abDusart, Pierre (2 Feb 2010). "Estimates of Some Functions Over Primes without R.H.".arXiv:1002.0442v1 [math.NT].
  32. ^Dusart, Pierre (January 2018). "Explicit estimates of some functions over primes".Ramanujan Journal.45 (1):225–234.doi:10.1007/s11139-016-9839-4.S2CID 125120533.
  33. ^Dusart, Pierre (January 1999)."Thekth prime is greater thank(lnk + ln lnk − 1) fork ≥ 2"(PDF).Mathematics of Computation.68 (225):411–415.Bibcode:1999MaCom..68..411D.doi:10.1090/S0025-5718-99-01037-6.
  34. ^Rosser, Barkley (January 1941). "Explicit bounds for some functions of prime numbers".American Journal of Mathematics.63 (1):211–232.doi:10.2307/2371291.JSTOR 2371291.
  35. ^Rosser, J. Barkley;Schoenfeld, Lowell (March 1962). "Approximate formulas for some functions of prime numbers".Illinois Journal of Mathematics.6 (1):64–94.doi:10.1215/ijm/1255631807.
  36. ^Axler, Christian (2019) [23 Mar 2017]."New estimates for thenth prime number".Journal of Integer Sequences.19 (4) 2.arXiv:1706.03651.
  37. ^"Bounds forn-th prime".Mathematics StackExchange. 31 December 2015.
  38. ^Axler, Christian (2018) [23 Mar 2017]."New Estimates for Some Functions Defined Over Primes"(PDF).Integers.18 A52.arXiv:1703.08032.doi:10.5281/zenodo.10677755.
  39. ^Axler, Christian (2024) [11 Mar 2022]."Effective Estimates for Some Functions Defined over Primes"(PDF).Integers.24 A34.arXiv:2203.05917.doi:10.5281/zenodo.10677755.
  40. ^Schoenfeld, Lowell (1976). "Sharper bounds for the Chebyshev functionsθ(x) andψ(x). II".Mathematics of Computation.30 (134). American Mathematical Society:337–360.doi:10.2307/2005976.ISSN 0025-5718.JSTOR 2005976.MR 0457374.

Notes

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  1. ^Montgomery showed that (assuming the Riemann hypothesis) at least two thirds of all zeros are simple.

External links

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