|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the '''prime-counting function''' is the [[Function (mathematics)|function]] counting the number of [[prime number]]s less than or equal to some [[real number]] ''x''.<ref name="Bach">{{cite book |first=Eric |last=Bach |coauthors=Shallit, Jeffrey |year=1996 |title=Algorithmic Number Theory |publisher=MIT Press |isbn=0-262-02405-5 |pages=volume 1 page 234 section 8.8 |nopp=true}}</ref><ref name="mathworld_pcf">{{MathWorld |title=Prime Counting Function |urlname=PrimeCountingFunction}}</ref> It is denoted by <math>\scriptstyle\pi(x)</math> (this does not refer to the number [[pi|π]]).
| | Your Tribe is the a great number strong of all as well as have the planet (virtual) at your toes, and simply all that with single a brief on-line on the web that may direct you step by step in how to get regarding cheat code for Discord of Tribes.<br><br> |
|
| |
|
| [[Image:PrimePi.svg|thumb|right|400px|The values of π(''n'') for the first 60 integers]]
| | As being explained in the last Clash of Clans' Tribe Wars overview, anniversary romantic relationship war is breach back up into a couple phases: Alertness Day and Entertainment Day. Anniversary coloration lasts 24 hours and in addition means that you has the potential to accomplish altered things.<br><br>Interweaving social trends form a formidable net in which we're all trapped. When You see, the Tygers of Pan Tang sang 'It's lonely at the top. Everyones trying to do anybody in', these people stole much from clash of clans identify tool no survey. A society without deviate of clans hack campaign no survey is for a society with no knowledge, in that it very good.<br><br>Pay attention to how much money your teenager is simply spending on video console games. These products commonly cheap and there is without question often the option of a buying more add-ons within the game itself. Set monthly and yearly plans available limits on the wide variety of money that should be spent on games. Also, carry conversations with your boys about budgeting.<br><br>Computing games are a wonderful of fun, but they could be very tricky, also. If you have to are put on an absolute game, go on generally web and also seek for cheats. Should you adored this article in addition to you desire to get more info regarding [http://prometeu.net clash of clans hack no survey] [http://Www.adobe.com/cfusion/search/index.cfm?term=&generously+check&loc=en_us&siteSection=home generously check] out the web page. All games have some model of cheat or hacks that can make him or her a lot easier. Only search in very own favorite search engine and even you can certainly discover cheats to get your action better.<br><br>Our world can be driven by supply and will need. We shall look over the Greek-Roman model. Using special care - highlight the role relating to clash of clans hack into tool no survey within the the vast framework the usually this provides.<br><br>Let's take a try interpreting the honest abstracts differently. Hope of it in plan of bulk with jewelry to skip 1 extra. Skipping added moments expenses added money, but also you get a grander deal. Think akin to it as a couple accretion discounts. |
| | |
| ==History==
| |
| Of great interest in [[number theory]] is the [[Asymptotic analysis|growth rate]] of the prime-counting function.<ref name="Caldwell">{{cite web | publisher=Chris K. Caldwell | title=How many primes are there? | url=http://primes.utm.edu/howmany.shtml |accessdate=2008-12-02}}</ref><ref name="Dickson">{{cite book |authorlink=L. E. Dickson| first=Leonard Eugene | last=Dickson | year=2005 | title=[[History of the Theory of Numbers]], Vol. I: Divisibility and Primality | publisher=Dover Publications | isbn=0-486-44232-2}}</ref> It was [[conjecture]]d in the end of the 18th century by [[Carl Friedrich Gauss|Gauss]] and by [[Adrien-Marie Legendre|Legendre]] to be approximately
| |
| | |
| :<math> x/\operatorname{ln}(x)\!</math>
| |
| | |
| in the sense that | |
| | |
| :<math>\lim_{x\rightarrow\infty}\frac{\pi(x)}{x/\operatorname{ln}(x)}=1.\!</math>
| |
| | |
| This statement is the [[prime number theorem]]. An equivalent statement is
| |
| | |
| :<math>\lim_{x\rightarrow\infty}\pi(x) / \operatorname{li}(x)=1\!</math>
| |
| | |
| where ''li'' is the [[logarithmic integral]] function. The prime number theorem was first proved in 1896 by [[Jacques Hadamard]] and by [[Charles Jean de la Vallée-Poussin|Charles de la Vallée Poussin]] independently, using properties of the [[Riemann zeta function]] introduced by [[Bernhard Riemann|Riemann]] in 1859.
| |
| | |
| More precise estimates of <math>\pi(x)\!</math> are now known; for example{{Citation needed|date=March 2012}}
| |
| | |
| :<math>\pi(x) = \operatorname{li}(x) + O\bigl(xe^{-\sqrt{\ln x}/15}\bigr)\!</math>
| |
| | |
| where the ''O'' is [[big O notation]]. For most values of <math>x</math> we are interested in (i.e., when <math>x</math> is not unreasonably large) <math>\operatorname{li}(x)\!</math> is greater than <math>\pi(x)\!</math>, but infinitely often the opposite is true. For a discussion of this, see [[Skewes' number]].
| |
| | |
| 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">{{cite book | first=Kenneth | last=Ireland | coauthors=Rosen, Michael | year=1998 | title=A Classical Introduction to Modern Number Theory | edition=Second | publisher=Springer | isbn=0-387-97329-X }}</ref>
| |
| | |
| ==Table of π(''x''), ''x'' / ln ''x'', and li(''x'')==
| |
| | |
| The table shows how the three functions π(''x''), ''x'' / ln ''x'' and li(''x'') compare at powers of 10. See also,<ref name="Caldwell" /><ref name="Silva">{{cite web |title=Tables of values of pi(x) and of pi2(x) |url=http://www.ieeta.pt/~tos/primes.html |publisher=[[Tomás Oliveira e Silva]] |accessdate=2008-09-14}}</ref><ref name="Kulsha">{{cite web |title=Values of π(x) and Δ(x) for various x's |url=http://www.primefan.ru/stuff/primes/table.html |publisher=Andrey V. Kulsha |accessdate=2008-09-14}}</ref> and.<ref name="Gourdon">{{cite web |title=A table of values of pi(x) |url=http://numbers.computation.free.fr/Constants/Primes/pixtable.html |publisher=Xavier Gourdon, Pascal Sebah, Patrick Demichel |accessdate=2008-09-14}}</ref>
| |
| | |
| :{| class="wikitable" style="text-align: right"
| |
| ! ''x''
| |
| ! π(''x'')
| |
| ! π(''x'') − ''x'' / ln ''x''
| |
| ! li(''x'') − π(''x'')
| |
| ! ''x'' / π(''x'')
| |
| |-
| |
| | 10
| |
| | 4
| |
| | −0.3
| |
| | 2.2
| |
| | 2.500
| |
| |-
| |
| | 10<sup>2
| |
| | 25
| |
| | 3.3
| |
| | 5.1
| |
| | 4.000
| |
| |-
| |
| | 10<sup>3</sup>
| |
| | 168
| |
| | 23
| |
| | 10
| |
| | 5.952
| |
| |-
| |
| | 10<sup>4</sup>
| |
| | 1,229
| |
| | 143
| |
| | 17
| |
| | 8.137
| |
| |-
| |
| | 10<sup>5</sup>
| |
| | 9,592
| |
| | 906
| |
| | 38
| |
| | 10.425
| |
| |-
| |
| | 10<sup>6</sup>
| |
| | 78,498
| |
| | 6,116
| |
| | 130
| |
| | 12.740
| |
| |-
| |
| | 10<sup>7</sup>
| |
| | 664,579
| |
| | 44,158
| |
| | 339
| |
| | 15.047
| |
| |-
| |
| | 10<sup>8</sup>
| |
| | 5,761,455
| |
| | 332,774
| |
| | 754
| |
| | 17.357
| |
| |-
| |
| | 10<sup>9</sup>
| |
| | 50,847,534
| |
| | 2,592,592
| |
| | 1,701
| |
| | 19.667
| |
| |-
| |
| | 10<sup>10</sup>
| |
| | 455,052,511
| |
| | 20,758,029
| |
| | 3,104
| |
| | 21.975
| |
| |-
| |
| | 10<sup>11</sup>
| |
| | 4,118,054,813
| |
| | 169,923,159
| |
| | 11,588
| |
| | 24.283
| |
| |-
| |
| | 10<sup>12</sup>
| |
| | 37,607,912,018
| |
| | 1,416,705,193
| |
| | 38,263
| |
| | 26.590
| |
| |-
| |
| | 10<sup>13</sup>
| |
| | 346,065,536,839
| |
| | 11,992,858,452
| |
| | 108,971
| |
| | 28.896
| |
| |-
| |
| | 10<sup>14</sup>
| |
| | 3,204,941,750,802
| |
| | 102,838,308,636
| |
| | 314,890
| |
| | 31.202
| |
| |-
| |
| | 10<sup>15</sup>
| |
| | 29,844,570,422,669
| |
| | 891,604,962,452
| |
| | 1,052,619
| |
| | 33.507
| |
| |-
| |
| | 10<sup>16</sup>
| |
| | 279,238,341,033,925
| |
| | 7,804,289,844,393
| |
| | 3,214,632
| |
| | 35.812
| |
| |-
| |
| | 10<sup>17</sup>
| |
| | 2,623,557,157,654,233
| |
| | 68,883,734,693,281
| |
| | 7,956,589
| |
| | 38.116
| |
| |-
| |
| | 10<sup>18</sup>
| |
| | 24,739,954,287,740,860
| |
| | 612,483,070,893,536
| |
| | 21,949,555
| |
| | 40.420
| |
| |-
| |
| | 10<sup>19</sup>
| |
| | 234,057,667,276,344,607
| |
| | 5,481,624,169,369,960
| |
| | 99,877,775
| |
| | 42.725
| |
| |-
| |
| | 10<sup>20</sup>
| |
| | 2,220,819,602,560,918,840
| |
| | 49,347,193,044,659,701
| |
| | 222,744,644
| |
| | 45.028
| |
| |-
| |
| | 10<sup>21</sup>
| |
| | 21,127,269,486,018,731,928
| |
| | 446,579,871,578,168,707
| |
| | 597,394,254
| |
| | 47.332
| |
| |-
| |
| | 10<sup>22</sup>
| |
| | 201,467,286,689,315,906,290
| |
| | 4,060,704,006,019,620,994
| |
| | 1,932,355,208
| |
| | 49.636
| |
| |-
| |
| | 10<sup>23</sup>
| |
| | 1,925,320,391,606,803,968,923
| |
| | 37,083,513,766,578,631,309
| |
| | 7,250,186,216
| |
| | 51.939
| |
| |-
| |
| | 10<sup>24</sup>
| |
| | 18,435,599,767,349,200,867,866
| |
| | 339,996,354,713,708,049,069
| |
| | 17,146,907,278
| |
| | 54.243
| |
| |-
| |
| | 10<sup>25</sup>
| |
| | 176,846,309,399,143,769,411,680
| |
| | 3,128,516,637,843,038,351,228
| |
| | 55,160,980,939
| |
| | 56.546
| |
| |}
| |
| [[File:Prime number theorem ratio convergence.svg|thumb|300px|Graph showing ratio of the prime-counting function π(''x'') to two of its approximations, ''x''/ln ''x'' and Li(''x''). As ''x'' increases (note ''x'' axis is logarithmic), both ratios tend towards 1. The ratio for ''x''/ln ''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 {{OEIS2C|id=A006880}}, π(''x'') - ''x'' / ln ''x'' is sequence {{OEIS2C|id=A057835}}, and li(''x'') − π(''x'') is sequence {{OEIS2C|id=A057752}}. The value for π(10<sup>24</sup>) was originally computed by J. Buethe, [[Jens Franke|J. Franke]], A. Jost, and T. Kleinjung assuming the [[Riemann hypothesis]].<ref name="Franke">{{cite web |title=Conditional Calculation of pi(10<sup>24</sup>) |url=http://primes.utm.edu/notes/pi(10%5E24).html |publisher=Chris K. Caldwell |accessdate=2010-08-03}}</ref> It has since been verified unconditionally in a computation by D. J. Platt.<ref name="PlattARXIV2012">{{cite web |title=Computing π(x) Analytically) |url=http://arxiv.org/abs/1203.5712 |accessdate=Jul 25, 2012}}</ref>
| |
| | |
| ==Algorithms for evaluating π(''x'')==
| |
| | |
| A simple way to find <math>\pi(x)</math>, if <math>x</math> is not too large, is to use the [[sieve of Eratosthenes]] to produce the primes less than or equal to <math>x</math> and then to count them.
| |
| | |
| A more elaborate way of finding <math>\pi(x)</math> is due to [[Adrien-Marie Legendre|Legendre]]: given <math>x</math>, if <math>p_1,p_2,\ldots,p_n</math> are distinct prime numbers, then the number of integers less than or equal to <math>x</math> which are divisible by no <math>p_i</math> 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 <math>\lfloor\cdots\rfloor</math> denotes the [[floor function]]). This number is therefore equal to
| |
| | |
| :<math>\pi(x)-\pi\left(\sqrt{x}\right)+1</math>
| |
| | |
| when the numbers <math>p_1, p_2,\ldots,p_n</math> are the prime numbers less than or equal to the square root of <math>x</math>.
| |
| | |
| In a series of articles published between 1870 and 1885, [[Ernst Meissel]] described (and used) a practical combinatorial way of evaluating <math>\pi(x)</math>. Let <math>p_1</math>, <math>p_2,\ldots,p_n</math> be the first <math>n</math> primes and denote by <math>\Phi(m,n)</math> the number of natural numbers not greater than <math>m</math> which are divisible by no <math>p_i</math>. Then
| |
| | |
| :<math>\Phi(m,n)=\Phi(m,n-1)-\Phi\left(\frac{m}{p_n},n-1\right)</math>
| |
| | |
| Given a natural number <math>m</math>, if <math>n=\pi\left(\sqrt[3]{m}\right)</math> and if <math>\mu=\pi\left(\sqrt{m}\right)-n</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 <math>\pi(x)</math>, for <math>x</math> equal to 5×10<sup>5</sup>, 10<sup>6</sup>, 10<sup>7</sup>, and 10<sup>8</sup>.
| |
| | |
| In 1959, [[Derrick Henry Lehmer]] extended and simplified Meissel's method. Define, for real <math>m</math> and for natural numbers <math>n</math>, and <math>k</math>, <math>P_k(m,n)</math> as the number of numbers not greater than ''m'' with exactly ''k'' prime factors, all greater than <math>p_n</math>. Furthermore, set <math>P_0(m,n)=1</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 <math>y</math> denote an integer such that <math>\sqrt[3]{m}\le y\le\sqrt{m}</math>, and set <math>n=\pi(y)</math>. Then <math>P_1(m,n)=\pi(m)-n</math> and <math>P_k(m,n)=0</math> when <math>k</math> ≥ 3. Therefore
| |
| | |
| :<math>\pi(m)=\Phi(m,n)+n-1-P_2(m,n)</math>
| |
| | |
| The computation of <math>P_2(m,n)</math> can be obtained this way:
| |
| | |
| :<math>P_2(m,n)=\sum_{y<p\le\sqrt{m}}\left(\pi\left(\frac mp\right)-\pi(p)+1\right)</math>
| |
| | |
| On the other hand, the computation of <math>\Phi(m,n)</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 <math>\pi\left(10^{10}\right)</math>.
| |
| | |
| Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise and Rivat.<ref name="pix_comp">{{cite web |title=Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko method |url=http://www.ams.org/mcom/1996-65-213/S0025-5718-96-00674-6/S0025-5718-96-00674-6.pdf |publisher=Marc Deléglise and Jöel Rivat, ''Mathematics of Computation'', vol. '''65''', number 33, January 1996, pages 235–245 |accessdate=2008-09-14}}</ref>
| |
| | |
| The Chinese [[mathematician]] Hwang Cheng, in a conference about prime number functions at the [[University of Bordeaux]],<ref name="Cheng">{{Cite journal |author=Hwang H., Cheng |title=Démarches de la Géométrie et des Nombres de l'Université du Bordeaux |publisher=''Prime Magic'' conference |year=2001 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> used the following identities:
| |
| | |
| :<math> e^{(a-1)\Theta}f(x)=f(ax)</math> | |
| | |
| :<math> J(x)=\sum_{n=1}^{\infty}\frac{\pi(x^{1/n})}{n}</math>
| |
| | |
| and setting <math>x=e^t</math>, Laplace-transforming both sides and applying a geometric sum on <math> e^{n\Theta} </math> got the expression
| |
| | |
| :<math> \frac{1}{2{\pi}i}\int_{c-i\infty}^{c+i\infty}g(s)t^{s}\,ds = \pi(t)</math>
| |
|
| |
| :<math> \frac{\ln \zeta(s)}{s}=(1-e^{\Theta(s)})^{-1}g(s)</math>
| |
| | |
| :<math> \Theta(s)=s\frac{d}{ds}</math>
| |
| | |
| ==Other prime-counting functions==
| |
| | |
| Other prime-counting functions are also used because they are more convenient to work with. One is Riemann's prime-counting function, usually denoted as <math>\Pi_0(x)</math> or <math>J_0(x)</math>. This has jumps of ''1/n'' for prime powers ''p''<sup>''n''</sup>, with it taking a value half-way between the two sides at discontinuities. That added detail is because then it may be defined by an inverse [[Mellin transform]]. Formally, we may define <math>\Pi_0(x)</math> by
| |
| | |
| :<math>\Pi_0(x) = \frac12 \bigg(\sum_{p^n < x} \frac1n\ + \sum_{p^n \le x} \frac1n\bigg)</math>
| |
| | |
| where ''p'' is a prime.
| |
| | |
| We may also write
| |
| | |
| :<math>\Pi_0(x) = \sum_2^x \frac{\Lambda(n)}{\ln n} - \frac12 \frac{\Lambda(x)}{\ln x} = \sum_{n=1}^\infty \frac1n \pi_0(x^{1/n})</math>
| |
| | |
| where Λ(''n'') is the [[von Mangoldt function]] and
| |
| | |
| :<math>\pi_0(x) = \lim_{\varepsilon \rightarrow 0}\frac{\pi(x-\varepsilon)+\pi(x+\varepsilon)}2.</math>
| |
| | |
| [[Möbius inversion formula]] then gives
| |
| | |
| :<math>\pi_{0}(x) = \sum_{n=1}^\infty \frac{\mu(n)}n \Pi_0(x^{1/n})</math>
| |
| | |
| Knowing the relationship between log of the [[Riemann zeta function]] and the [[von Mangoldt function]] <math>\Lambda</math>, and using the [[Perron formula]] we have
| |
| | |
| :<math>\ln \zeta(s) = s \int_0^\infty \Pi_0(x) x^{-s-1}\,dx</math>
| |
| | |
| The [[Chebyshev function]] weights primes or prime powers ''p''<sup>''n''</sup> by ln(''p''):
| |
| | |
| :<math>\theta(x)=\sum_{p\le x}\ln p</math>
| |
| :<math>\psi(x) = \sum_{p^n \le x} \ln p = \sum_{n=1}^\infty \theta(x^{1/n}) = \sum_{n\le x}\Lambda(n).</math>
| |
| | |
| Riemann's prime-counting function has the ordinary generating function:
| |
| | |
| :<math>\sum_{n=1}^\infty \Pi_0(n)x^n = \sum_{a=2}^\infty \frac{x^{a}}{1-x} - \frac{1}{2}\sum_{a=2}^\infty
| |
| \sum_{b=2}^\infty \frac{x^{ab}}{1-x} + \frac{1}{3}\sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \frac{x^{abc}}{1
| |
| -x} - \frac{1}{4}\sum_{a=2}^\infty \sum_{b=2}^\infty \sum_{c=2}^\infty \sum_{d=2}^\infty \frac{x^{abcd}}{1-x} +
| |
| \cdots </math>
| |
| | |
| ==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 [[Hans Carl Friedrich von Mangoldt|von Mangoldt]], and are generally known as [[Explicit formulae (L-function)|explicit formula]]s.<ref name="Titchmarsh">{{cite book |first=E.C. |last=Titchmarsh |year=1960 |title=The Theory of Functions, 2nd ed. |publisher=Oxford University Press}}</ref>
| |
| | |
| We have the following expression for ψ:
| |
| | |
| :<math>\psi_0(x) = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac12 \ln(1-x^{-2})</math>
| |
| | |
| where
| |
| | |
| : <math>\psi_0(x) = \lim_{\varepsilon \rightarrow 0}\frac{\psi(x-\varepsilon)+\psi(x+\varepsilon)}2.</math>
| |
| | |
| 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 <math>\scriptstyle\Pi_0(x)</math> we have a more complicated formula
| |
| | |
| :<math>\Pi_0(x) = \operatorname{li}(x) - \sum_{\rho}\operatorname{li}(x^{\rho}) - \ln 2 + \int_x^\infty \frac{dt}{t(t^2-1) \ln t}.</math>
| |
| | |
| Again, the formula is valid for ''x'' > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value, and, again, the latter integral, taken with minus sign, is just the same sum, but over the trivial zeros. The first term li(''x'') is the usual [[logarithmic integral function]]; the expression li(''x''<sup>ρ</sup>) in the second term should be considered as Ei(ρ ln ''x''), where Ei is the [[analytic continuation]] of the [[exponential integral]] function from positive reals to the complex plane with branch cut along the negative reals.
| |
| | |
| Thus, [[Möbius inversion formula]] gives us<ref>{{Cite journal | author1-link=Hans Riesel | last1=Riesel | first1=Hans | last2=Göhl | first2=Gunnar | title=Some calculations related to Riemann's prime number formula | doi=10.2307/2004630 | mr=0277489 | year=1970 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=24 | issue=112 | pages=969–983 | jstor=2004630 | publisher=American Mathematical Society}}</ref>
| |
| | |
| :<math>\pi_{0}(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x}</math>
| |
| | |
| valid for ''x'' > 1, where
| |
| | |
| :<math>\operatorname{R}(x) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \operatorname{li}(x^{1/n}) = 1 + \sum_{k=1}^\infty \frac{(\ln x)^k}{k! k \zeta(k+1)}</math>
| |
| | |
| is so-called Riemann's R-function.<ref name="mathworld_r">{{MathWorld |title=Riemann Prime Counting Function |urlname=RiemannPrimeCountingFunction}}</ref> The latter series for it is known as Gram series <ref name="mathworld_gram">{{MathWorld|title=Gram Series |urlname=GramSeries}}</ref> and converges for all positive ''x''.
| |
| | |
| [[Image:PlotDelta.gif|thumb|right|220px|Δ-function (red line) on log scale]]The sum over non-trivial zeta zeros in the formula for <math>\scriptstyle\pi_0(x)</math> describes the fluctuations of <math>\scriptstyle\pi_0(x)</math>, while the remaining terms give the "smooth" part of prime-counting function,<ref name="Watkins">{{cite web |title=The encoding of the prime distribution by the zeta zeros |url=http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding1.htm |publisher=Matthew Watkins |accessdate=2008-09-14}}</ref> so one can use
| |
| | |
| :<math>\operatorname{R}(x) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x}</math>
| |
| | |
| as the [http://primefan.ru:8014/WWW/stuff/primes/best_estimator.gif best estimator] of <math>\scriptstyle\pi(x)</math> for ''x'' > 1.
| |
| | |
| The amplitude of the "noisy" part is heuristically about <math>\scriptstyle\sqrt x/\ln x</math>, so the fluctuations of the distribution of primes may be clearly represented with the Δ-function:
| |
| | |
| :<math>\Delta(x) = \left( \pi_0(x) - \operatorname{R}(x) + \frac1{\ln x} - \frac1{\pi}\arctan\frac{\pi}{\ln x} \right) \frac{\ln x}{\sqrt x}.</math>
| |
| | |
| An extensive table of the values of Δ(''x'') is available.<ref name="Kulsha" />
| |
| | |
| ==Inequalities==
| |
| Here are some useful inequalities for π(''x'').
| |
| | |
| :<math>
| |
| \frac {x} {\ln x} < \pi(x) < 1.25506 \frac {x} {\ln x}
| |
| \!</math> for x ≥ 17.<ref>{{Cite journal | last = Rosser | first = J. Barkley | last2 = Schoenfeld | first2 = Lowell | title = Approximate formulas for some functions of prime numbers | journal = Illinois J. Math. | year = 1962 | volume = 6 | pages = 64–94 | zbl=0122.05001 | issn=0019-2082 | url = http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ijm/1255631807}}</ref>
| |
| The left inequality holds for x ≥ 17 and the right inequality holds for x > 1.
| |
| | |
| An explanation of the constant 1.25506 is given at {{OEIS|id=A209883}}.
| |
| | |
| Here are some inequalities for the ''n''th prime, ''p''<sub>''n''</sub>.<ref>{{Citation
| |
| | url = http://functions.wolfram.com/NumberTheoryFunctions/Prime/29/0002/
| |
| | title = Inequalites for the n-th prime number at function.wolfram
| |
| | accessdate = March 22, 2013
| |
| }}</ref>
| |
| :<math>
| |
| n (\ln (n \ln n) - 1) < p_n < n {\ln (n \ln n)}
| |
| \!</math> for ''n'' ≥ 6.
| |
| The left inequality holds for n ≥ 1 and the right inequality holds for n ≥ 6.
| |
| | |
| An approximation for the ''n''th prime number is
| |
| :<math> p_n = n (\ln (n \ln n) - 1) + \frac {n (\ln \ln n - 2)} {\ln n} +
| |
| O\left( \frac {n (\ln \ln n)^2} {(\ln n)^2}\right).
| |
| </math>
| |
| | |
| ==The Riemann hypothesis==
| |
| | |
| The [[Riemann hypothesis]] is equivalent to a much tighter bound on the error in the estimate for <math>\pi(x)</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>{{Cite journal | last1=Schoenfeld | first1=Lowell |authorlink=Lowell Schoenfeld| title=Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II | doi=10.2307/2005976 | mr=0457374 | year=1976 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=30 | issue=134 | pages=337–360 | jstor=2005976 | publisher=American Mathematical Society}}</ref>
| |
| :<math>|\pi(x) - \operatorname{li}(x)| < \frac{1}{8\pi} \sqrt{x} \, \log(x), \qquad \text{for all } x \ge 2657. </math>
| |
| | |
| == See also ==
| |
| | |
| * [[Bertrand's postulate]]
| |
| * [[Oppermann's conjecture]]
| |
| * [[Foias constant]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| *Chris Caldwell, [http://primes.utm.edu/nthprime/ ''The Nth Prime Page''] at The [[Prime Pages]].
| |
| *[https://gist.github.com/4385784 Implementation of Legendre, Meissel, and Lehmer methods in C]
| |
| | |
| {{DEFAULTSORT:Prime-Counting Function}}
| |
| [[Category:Analytic number theory]]
| |
| [[Category:Prime numbers]]
| |