Prime-counting function
Template:Short description Template:Redirect Template:Log(x) Template:Duplication
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.<ref name="Bach">Template:Cite book</ref><ref name="mathworld_pcf">{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PrimeCountingFunction%7CPrimeCountingFunction.html}} |title = Prime Counting Function |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> 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 rateEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of great interest in number theory is the growth rate of the prime-counting function.<ref name="Caldwell">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="Dickson">Template:Cite book</ref> It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately <math display=block> \frac{x}{\log x} </math> where Template:Math is the natural logarithm, in the sense that <math display=block>\lim_{x\rightarrow\infty} \frac{\pi(x)}{x/\log x}=1. </math> This statement is the prime number theorem. An equivalent statement is <math display=block>\lim_{x\rightarrow\infty}\frac{\pi(x)}{\operatorname{li}(x)}=1</math> 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).<ref name="Ireland">Template:Cite book</ref>
More precise estimatesEdit
In 1899, de la Vallée Poussin proved that <ref>See also Theorem 23 of Template:Cite book</ref> <math display=block>\pi(x) = \operatorname{li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty</math> 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<ref name="Ford">Template:Cite journal</ref> <math display=block>\pi(x) = \operatorname{li} (x) + O \left(x \exp \left( -0.2098(\log x)^{3/5} (\log \log x)^{-1/5} \right) \right).</math>
Mossinghoff and Trudgian proved<ref>Template:Cite journal</ref> an explicit upper bound for the difference between Template:Math and Template:Math: <math display=block>\bigl| \pi(x) - \operatorname{li}(x) \bigr| \le 0.2593 \frac{x}{(\log x)^{3/4}} \exp \left( -\sqrt{ \frac{\log x}{6.315} } \right) \quad \text{for } x \ge 229.</math>
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 formEdit
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<ref>{{#invoke:citation/CS1|citation |CitationClass=web
}}</ref>
<math display="block">\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^\rho),</math> where <math display=block>\operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \operatorname{li}\left(x^{1/n}\right),</math> 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<ref name="RieselGohl">Template:Cite journal</ref> <math display=block>\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}} .</math>
The Riemann hypothesis suggests that every such non-trivial zero lies along Template:Math.
Table of Template:Math, Template:Math, and Template:MathEdit
The table shows how the three functions Template:Math, Template:Math, and Template:Math compared at powers of 10. See also,<ref name="Caldwell" /><ref name="Silva">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and<ref name="Gourdon">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Template:Mvar Template:Math Template:Math Template:Math Template:Math Template:Math
% error10 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%
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.<ref name="Franke">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> It was later verified unconditionally in a computation by D. J. Platt.<ref name="Platt2012">Template:Cite journal</ref> The value for Template:Math is by the same four authors.<ref name="Buethe">{{#invoke:citation/CS1|citation |CitationClass=web }} Includes 600,000 value of Template:Math for Template:Math</ref> The value for Template:Math was computed by D. B. Staple.<ref name="Staple">Template:Cite thesis</ref> 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,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 2020,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and 2022,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> respectively.
Algorithms for evaluating Template:MathEdit
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
- <math>\lfloor x\rfloor - \sum_{i}\left\lfloor\frac{x}{p_i}\right\rfloor + \sum_{i<j} \left\lfloor\frac{x}{p_ip_j}\right\rfloor - \sum_{i<j<k}\left\lfloor\frac{x}{p_ip_jp_k}\right\rfloor + \cdots</math>
(where Template:Math denotes the floor function). This number is therefore equal to
- <math>\pi(x)-\pi\left(\sqrt{x}\right)+1</math>
when the numbers Template:Math are the prime numbers less than or equal to the square root of Template:Mvar.
The Meissel–Lehmer algorithmEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
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
- <math>\Phi(m,n)=\Phi(m,n-1)-\Phi\left(\frac m {p_n},n-1\right).</math>
Given a natural number Template:Mvar, if Template:Math and if Template:Math, then
- <math>\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) .</math>
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
- <math>\Phi(m,n) = \sum_{k=0}^{+\infty} P_k(m,n)</math>
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,
- <math>\pi(m) = \Phi(m,n) + n - 1 - P_2(m,n)</math>
The computation of Template:Math can be obtained this way:
- <math>P_2(m,n) = \sum_{y < p \le \sqrt{m} } \left( \pi \left( \frac m p \right) - \pi(p) + 1\right)</math>
where the sum is over prime numbers.
On the other hand, the computation of Template:Math can be done using the following rules:
- <math>\Phi(m,0) = \lfloor m\rfloor</math>
- <math>\Phi(m,b) = \Phi(m,b-1) - \Phi\left(\frac m{p_b},b-1\right)</math>
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.<ref name="lehmer">Template:Cite journal</ref>
Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.<ref name="pix_comp">Template:Cite journal</ref>
Other prime-counting functionsEdit
Other prime-counting functions are also used because they are more convenient to work with.
Riemann's prime-power counting functionEdit
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
- <math>\Pi_0(x) = \frac{1}{2} \left( \sum_{p^n < x} \frac{1}{n} + \sum_{p^n \le x} \frac{1}{n} \right)\ </math>
where the variable Template:Mvar in each sum ranges over all primes within the specified limits.
We may also write
- <math>\ \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)</math>
where Template:Math is the von Mangoldt function and
- <math>\pi_0(x) = \lim_{\varepsilon \to 0} \frac{\pi(x-\varepsilon) + \pi(x+\varepsilon)}{2}.</math>
The Möbius inversion formula then gives
- <math>\pi_0(x) = \sum_{n=1}^\infty \frac{\mu(n)}{n}\ \Pi_0\left(x^{1/n}\right),</math>
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
- <math>\log \zeta(s) = s \int_0^\infty \Pi_0(x) x^{-s-1}\, \mathrm{d}x</math>
Chebyshev's functionEdit
The Chebyshev function weights primes or prime powers Template:Mvar by Template:Math:
- <math>\begin{align}
\vartheta(x) &= \sum_{p\le x} \log p \\ \psi(x)&=\sum_{p^n \le x} \log p = \sum_{n=1}^\infty \vartheta \left( x^{1/n} \right) = \sum_{n \le x}\Lambda(n) . \end{align}</math>
For Template:Math,<ref>Template:Cite book</ref>
- <math>\vartheta(x) = \pi(x)\log x - \int_2^x \frac{\pi(t)}{t}\, \mathrm{d}t </math>
and
- <math>\pi(x)=\frac{\vartheta(x)}{\log x} + \int_2^x \frac{\vartheta(t)}{t\log^{2}(t)} \mathrm{d} t .</math>
Formulas for prime-counting functionsEdit
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.<ref name="Titchmarsh">Template:Cite book</ref>
We have the following expression for the second Chebyshev function Template:Mvar:
- <math>\psi_0(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \log 2\pi - \frac{1}{2} \log\left(1-x^{-2}\right),</math>
where
- <math>\psi_0(x) = \lim_{\varepsilon \to 0} \frac{\psi(x - \varepsilon) + \psi(x + \varepsilon)}{2}.</math>
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
- <math>\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}.</math>
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:
- <math>\int_x^\infty \frac{\mathrm dt}{t \left(t^2 - 1\right) \log t}=\int_x^\infty \frac{1}{t\log t}
\left(\sum_{m}t^{-2m}\right)\,\mathrm dt=\sum_{m}\int_x^\infty \frac{t^{-2m}}{t\log t} \,\mathrm dt \,\,\overset{\left(u=t^{-2m}\right)}{=}-\sum_{m} \operatorname{li}\left(x^{-2m}\right) </math>
Thus, Möbius inversion formula gives us<ref name="RieselGohl" />
- <math>\pi_0(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}\left(x^\rho\right) - \sum_{m} \operatorname{R}\left(x^{-2m}\right)</math>
valid for Template:Math, where
- <math>\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)}</math>
is Riemann's R-function<ref name="mathworld_r">{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:RiemannPrimeCountingFunction%7CRiemannPrimeCountingFunction.html}} |title = Riemann Prime Counting Function |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> and Template:Math is the Möbius function. The latter series for it is known as Gram series.<ref name="Riesel94">Template:Cite book</ref><ref name="mathworld_gram">{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:GramSeries%7CGramSeries.html}} |title = Gram Series |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> 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,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> when assuming the conjecture that all zeros of the Riemann zeta function are simple,<ref group="note">Montgomery showed that (assuming the Riemann hypothesis) at least two thirds of all zeros are simple.</ref> that
- <math>\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)}+\frac12\sum_{\rho}\frac{t^{-\rho}}{\rho\cos\frac{\pi\rho}{2}\zeta'(\rho)}</math>
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,<ref name="Watkins">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> so one can use
- <math>\operatorname{R}(x) - \sum_{m=1}^\infty \operatorname{R}\left(x^{-2m}\right)</math>
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
- <math>\bigl( \pi_0(x) - \operatorname{R}(x)\bigr) \frac{\log x}{\sqrt x}.</math>
InequalitiesEdit
Ramanujan<ref>Template:Cite book</ref> proved that the inequality
- <math>\pi(x)^2 < \frac{ex}{\log x} \pi\left( \frac{x}{e} \right)</math>
holds for all sufficiently large values of Template:Mvar.
Here are some useful inequalities for Template:Math.
- <math> \frac x {\log x} < \pi(x) < 1.25506 \frac x {\log x} \quad \text{for }x \ge 17.</math>
The left inequality holds for Template:Math and the right inequality holds for Template:Math. The constant {{#expr:30*ln(113)/113 round 5}} is Template:Math to 5 decimal places, as Template:Math has its maximum value at Template:Math.<ref name=Rosser1962>Template:Cite journal</ref>
Pierre Dusart proved in 2010:<ref name = "Dusart2010">Template:Cite arXiv</ref>
- <math> \frac {x} {\log x - 1} < \pi(x) < \frac {x} {\log x - 1.1}\quad \text{for }x \ge 5393 \text{ and }x \ge 60184,\text{ respectively.}</math>
More recently, Dusart has proved<ref>Template:Cite journal</ref> (Theorem 5.1) that
- <math>\frac{x}{\log x} \left( 1 + \frac{1}{\log x} + \frac{2}{\log^2 x} \right) \le \pi(x) \le \frac{x}{\log x} \left( 1 + \frac{1}{\log x} + \frac{2}{\log^2 x} + \frac{7.59}{\log^3 x} \right),</math>
for Template:Math and Template:Math, respectively.
Going in the other direction, an approximation for the Template:Mvarth prime, Template:Mvar, is
- <math>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).</math>
Here are some inequalities for the Template:Mvarth prime. The lower bound is due to Dusart (1999)<ref>Template:Cite journal</ref> and the upper bound to Rosser (1941).<ref>Template:Cite journal</ref>
- <math> n (\log n + \log\log n - 1) < p_n < n (\log n + \log\log n)\quad \text{for } n \ge 6.</math>
The left inequality holds for Template:Math and the right inequality holds for Template:Math. A variant form sometimes seen substitutes <math>\log n +\log\log n = \log(n \log n).</math> An even simpler lower bound is<ref name=Rosser62>Template:Cite journal</ref>
- <math>n \log n < p_n,</math>
which holds for all Template:Math, but the lower bound above is tighter for Template:Math.
In 2010 Dusart proved<ref name = "Dusart2010" /> (Propositions 6.7 and 6.6) that
- <math> n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2.1}{\log n} \right) \le p_n \le n \left( \log n + \log \log n - 1 + \frac{\log \log n - 2}{\log n} \right),</math>
for Template:Math and Template:Math, respectively.
In 2024, Axler<ref>Template:Cite journal</ref> further tightened this (equations 1.12 and 1.13) using bounds of the form
- <math> 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)</math>
proving that
- <math> f(n, w^2 - 6w + 11.321) \le p_n \le f(n, w^2 - 6w)</math>
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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
The Riemann hypothesisEdit
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,
- <math>\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log{x}).</math>
Specifically,<ref>Template:Cite journal</ref>
- <math>|\pi(x) - \operatorname{li}(x)| < \frac{\sqrt{x}}{8\pi} \, \log{x}, \quad \text{for all } x \ge 2657. </math>
Template:Harvtxt proved that the Riemann hypothesis implies that for all Template:Math there is a prime Template:Mvar satisfying
- <math>x - \frac{4}{\pi} \sqrt{x} \log x < p \leq x.</math>
See alsoEdit
ReferencesEdit
NotesEdit
External linksEdit
- Chris Caldwell, The Nth Prime Page at The Prime Pages.
- Tomás Oliveira e Silva, Tables of prime-counting functions.
- Template:Citation