Logön fonäti pada: "Primanumaleset"

In [[numateor]], '''primanumaleset''' (PNL) bepenon kondöti zaik, lasümptotik primanumas.

Spikölo no kuratiko, primanumaleset steton, das, if väloy ma fäd numi nilü num gretik ''N'', mög, das num pevälöl binon primanum, leigon zao ko 1/ln&nbsp;''N'', kö el ln&nbsp;''N'' malon logariti natik ela ''N''. Samo, nilü ''N'' = 10.000, num za bal se zöls binon primanum; nilü ''N'' = 1.000.000.000, te num bal se teldegbals binon primanum.

Me vöds votik, primanums nesuvöfikons maä betikoy numis ai gretikumis, e primanumaleset bepenon kuratiko modi nesuvöfikama at.


== Leset ==
[[Ragiv:PrimeNumberTheorem.png|thumb|300px|Leigod bevü els π(''x'') (blöv), ''x'' / ln ''x'' (grün) e Li(''x'') (red).]]

Büolasumobsöd, das el π(''x'') binon numamasekät primanumas, o.b. sekät, kel givon numi primanumas läs ka, u leigölas ko, el ''x'', if el ''x'' binon num jenöfik seimik. As sam, π(10)&nbsp;=&nbsp;4 bi dabinons primanums fol (2, 3, 5, 7) läs ka, ü leigöls ko, el 10. Primanumaleset lesagon täno, das miedot ''müedota'' sekätas tel: π(''x'') e ''x'' / ln (''x'') du el ''x'' nilikon nenfine binon 1. Gebölo penamamodi hiela Landau, sek at kanon papenön as:

:<math>\pi(x)\sim\frac{x}{\ln x}</math>.

Küpälolös, das atos ''no'' sinifon, das miedot ''näedota'' sekätas tel, du el ''x'' nilikon nenfine, binon ser.

Stabü taibs hielas Anton Felkel e Jurij Vega, leset at päniludon fa hiel Adrien-Marie Legendre ün 1796, e päblöfon nensekidiko fa hiel Jacques Hadamard e hiel Charles de la Vallée-Poussin ün 1896. Blöfam matematik gebon metodis dileta komplitik, pato sekäti „zeta“ hiela Riemann.

== Numamasekät primanumas stabü lintegral logaritik ==
Hiel [[Carl Friedrich Gauss]] äniludom, das nilikam nog gudikum ele π(''x'') pagivon dub lintegralasekät logaritik Li(''x''), pamiedetöl as:

:<math> \mbox{Li}(x) = \int_2^x \frac1{\ln t} \,\mbox{d}t = \mbox{li}(x) - \mbox{li}(2).  </math>

Lintegral at vo tikodükon, das „densit“ primanumas zü ''t'' muton binön 1/ln&nbsp;''t''. Sekät at tefon logariti medü stäänükam asümptotik:

:<math> \mbox{Li}(x) = \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k} 
= \frac{x}{\ln x} + \frac{x}{(\ln x)^2} + \frac{2x}{(\ln x)^3} + \cdots </math>

Klu primanumaleset kanon papenön i π(''x'') ~ Li(''x''). Frut fomama at binon notod smalikum pöka. Ibo klülos se blöfam hielas Hadamard e de la Vallée Poussin, das:

:<math> \pi(x)={\rm Li} (x) + O \left(x \mathrm{e}^{-a\sqrt{\ln x}}\right) \quad\mbox{as } x \to \infty</math>

pro num positik semik ''a'', kö ''O''(...) binon penamamod hiela Landau. Atos pegudükumon ad:

:<math>\pi(x)={\rm Li} (x) + O \left(x \, \exp \left( -\frac{A(\ln x)^{3/5}}{(\ln \ln x)^{1/5}} \right) \right). </math>

Kodü tef vü sekät „zeta“ ela Riemann ed el π(''x''), nilud ela Riemann labon veüti gretik pro numateor: if pablüfonöv, givonöv täxeti gudikum pöka pö primanumaleset, ka ut, kel anu gebidon. Kuratikumo, hiel [[Helge von Koch]] äjonom ün 1901, das, if e te if nilud ela Riemann veraton, pök in tef löpo pemäniotöl kanon pagudükumön ad:

:<math> \pi(x) = {\rm Li} (x) + O\left(\sqrt x \ln x\right). </math>

Num: ''a'' in el ''O'' gretik pätäxeton ün 1976 fa hiel Lowell Schoenfeld (if lasumoy niludi ela Riemann) as: 

:<math>|\pi(x)-{\rm Li}(x)|<\frac{\sqrt x\,\ln x}{8\pi}</math>

pro els ''x'' valik ≥ 2657. Ätüvom i miedoti sümik pro numamasekät primanumas hiela Chebyshev ψ:

:<math>|\psi(x)-x|<\frac{\sqrt x\,\ln^2 x}{8\pi}</math>

pro els ''x'' valik ≥ 73.2.

Lintegral logaritik Li(''x'') binon gretikum ka π(''x'') pro els ''x'' „smalik“. Ün 1914 ye hiel J. E. Littlewood äblöfom, das atos no ai veraton. Völad balid ela ''x'', pö kel π(''x'') pluons tefü el Li(''x'') binon zao ''x'' = 10<sup>316</sup> (l. yegedi dö num hiela Skewes pro pats pluik).

== Säk dö „dibät“ ==
In laf balid tumyela 20id, matematans anik äsenälons, das dabinon nivodaleod metodas matematik, e das primanumaleset binon leset „dibätik“, kela blöfam flagons dileti komplitik. Metods gebü te sekäts jenöfik päcedons nefägikis. Hiel G. H. Hardy äbinon liman famik grupa at.

Kred at boso äsmalikumon sekü blöfam primanumaleseta stabü leset hiela Norbert Wiener dö leset hiela Tauber, do ämögos ad vüdön säkädi ägevölo lesete hiela Wiener „dibäti“ sümik ad ut dileta komplitik. Hiels Paul Erdős e Atle Selberg ye ätuvoms blöfami „balugik“ primanumaleseta, kel gebon te metodis numateora. Vobot elas Selberg e Erdős ibo äseilükon lesagis dö „dibät“ äjonölo, das metods „balugik“ (ön jenet at, metods yümätavik) äbinons fägikum, kas spetoy. Poso pub „sibametodas“ äjonon, das metods at älabons rouli fümik in teor primanumas.

Avigad ''et al.'' 2005 keninükon fomami blöfama at pefümodöl medü nünömaprograms in „teorodiblöfian Isabelle“.

== Primanumaleset pro progedasökods kalkulavik ==
El <math>\pi_{n,a}(x)</math> malonöd numi primanumas in progedasökod kalkulavik ''a'', ''a'' + ''n'', ''a'' + 2''n'', ''a'' + 3''n'', … läs ka ''x''. Hiels [[Johann Peter Gustav Lejeune Dirichlet|Dirichlet]] e Legendre äniludoms, ed el Vallée Poussin äblöfom, das, if ''a'' e ''n'' binons keprimanums, tän:

:<math>
\pi_{n,a}(x) \sim \frac{1}{\phi(n)}\mathrm{Li}(x),
</math>
kö el φ(·) binon sekät totienik ela [[Euler]]. Me vöds votik, primanums padilons leigöfiko bevü klads bliböl [''a''] modulo ''n'', binölo gcd(''a'', ''n'') = 1.

== Mieds numamasekäta primanumas ==
Primanumaleset binon sek ''lasümptotik''. Kludo no kanon pagebön ad ''miedükön'' eli π(''x'').

Mieds anik pro el π(''x'') sevädons, as sam:

<math> \frac{x}{\ln x} < \pi(x) < 1.25506 \, \frac{x}{\ln x}</math>

Neleig balid veraton pro els ''x'' ≥ 17, e telid pro el ''x'' > 1.

Meid frutik votik binon:

<math> \frac {x}{\ln x + 2} < \pi(x) < \frac {x}{\ln x - 4} \quad\mbox{for } x \ge 55
</math>

== Nilikams primanume ''n''id ==
As sek primanumaleseta, kanoy getön notodoti lasümptotik pro primanum ''n''id, papenöl as ''p''<sub>''n''</sub>:
:<math>p_n \sim n \ln n.</math>
Nilikam gudikum binon
:<math> p_n = n \ln n +  n \ln \ln n + \frac{n}{\ln n} \big( \ln \ln n - \ln n- 2 \big) 
+ O\left( \frac {n (\ln \ln n)^2} {(\ln n)^2}\right). </math>

[[Leset di Rosser]] lesagon, das ''p''<sub>''n''</sub> binon gretikum ka ''n''&nbsp;ln&nbsp;''n''. Atos kanon gudükumön medü pär sököl miedas:
:<math> n \ln n + n\ln\ln n - n  < p_n <  n \ln n + n \ln \ln n \quad\mbox{for } n \ge 6. </math>
Neleig nedetik pätüvon fa hiel Pierre Dusart (1999) e lonöfon pro ''n'' ≥ 2.

== Taib elas π(''x''), ''x'' / ln ''x'', e Li(''x'') ==
Is pajonon taib, in kel paleigodons sekäts kil: π(''x''), ''x'' / ln ''x'' e Li(''x'').
:{| class="wikitable" style="text-align: right"
! ''x'' 
! π(''x'')
! π(''x'') − ''x'' / ln ''x''
! π(''x'') / (''x'' / ln ''x'')
! Li(''x'') − π(''x'')
! ''x'' / π(''x'')
|-
| 10
| 4
| −0.3
| 0.921
| 2.2
| 2.500
|-
| 10<sup>2
| 25
| 3.3
| 1.151
| 5.1
| 4.000
|-
| 10<sup>3</sup>
| 168
| 23
| 1.161
| 10
| 5.952
|-
| 10<sup>4</sup>
| 1,229
| 143
| 1.132
| 17
| 8.137
|-
| 10<sup>5</sup>
| 9,592
| 906
| 1.104
| 38
| 10.425
|-
| 10<sup>6</sup>
| 78,498
| 6,116
| 1.084
| 130
| 12.740
|-
| 10<sup>7</sup>
| 664,579
| 44,158
| 1.071
| 339
| 15.047
|-
| 10<sup>8</sup>
| 5,761,455
| 332,774
| 1.061
| 754
| 17.357
|-
| 10<sup>9</sup>
| 50,847,534
| 2,592,592
| 1.054
| 1,701
| 19.667
|-
| 10<sup>10</sup>
| 455,052,511
| 20,758,029
| 1.048
| 3,104
| 21.975
|-
| 10<sup>11</sup>
| 4,118,054,813
| 169,923,159
| 1.043
| 11,588
| 24.283
|-
| 10<sup>12</sup>
| 37,607,912,018
| 1,416,705,193
| 1.039
| 38,263
| 26.590
|-
| 10<sup>13</sup>
| 346,065,536,839
| 11,992,858,452
| 1.034
| 108,971
| 28.896
|-
| 10<sup>14</sup>
| 3,204,941,750,802
| 102,838,308,636
| 1.033
| 314,890
| 31.202
|-
| 10<sup>15</sup>
| 29,844,570,422,669
| 891,604,962,452
| 1.031
| 1,052,619
| 33.507
|-
| 10<sup>16</sup>
| 279,238,341,033,925
| 7,804,289,844,393
| 1.029
| 3,214,632
| 35.812
|-
| 10<sup>17</sup>
| 2,623,557,157,654,233
| 68,883,734,693,281
| 1.027
| 7,956,589
| 38.116
|-
| 10<sup>18</sup>
| 24,739,954,287,740,860
| 612,483,070,893,536
| 1.025
| 21,949,555
| 40.420
|-
| 10<sup>19</sup>
| 234,057,667,276,344,607
| 5,481,624,169,369,960
| 1.024
| 99,877,775
| 42.725
|-
| 10<sup>20</sup>
| 2,220,819,602,560,918,840
| 49,347,193,044,659,701
| 1.023
| 222,744,644
| 45.028
|-
| 10<sup>21</sup>
| 21,127,269,486,018,731,928
| 446,579,871,578,168,707
| 1.022
| 597,394,254
| 47.332
|-
| 10<sup>22</sup>
| 201,467,286,689,315,906,290
| 4,060,704,006,019,620,994
| 1.021
| 1,932,355,208
| 49.636
|-
| 10<sup>23</sup>
| 1,925,320,391,606,818,006,727
| 37,083,513,766,592,669,113
| 1.020
| 7,236,148,412
| 51.939
|}

== Süm ko polünoms no diletoviks love fel finilabik ==
Dabinon sümod primanumaleseta, kel bepenon „dilami“ polünomas no diletoviks love fel finilabik; fom ona sümon vemo ad jenet primanumaleset rigik.

Kuratiko spikölo, el ''F'' = GF(''q'') binonöd fel finilabik labü binets ''q'', pegivölo el ''q'' semik fümik, ed el ''N''<sub>''n''</sub> binonöd num polünomas no diletovik labü koäf balid = 1 love ''F'', kela grad = ''n''. Atos sinifon, das sukoy polünomis labü koäfs pevälöl se ''F'', kels no kanons papenön as naedam polünomas grada smalikum. Is polünoms at labons rouli primanumas, ibä polünoms votik valik labü koäf balid = 1 pabumons me naedams onas. Kanoy tän blöfön, das

:<math>N_n \sim \frac{q^n}{n}</math>

If plaädoy eli ''x'' dub ''q''<sup>''n''</sup>, tän flan detik leiga at vedon:

:<math>\frac{x}{\log_q x}</math>

kelos kleilükon sümi. Bi dabinons ebo ''q''<sup>''n''</sup> polünomas grada ''n'' labü koäf balid = 1 (keninükamü diletoviks), atos kanon pavotanotodön ön mod soik: if väloy polünomi grada ''n'' labü koäf balid = 1 ma fäd, tän mög, das binon no diletovik, binon za 1/''n''.

Kanoy igo blöfön sümodi niluda ela Riemann:

:<math>N_n = \frac{q^n}n + O\left(\frac{q^{n/2}}{n}\right).</math>

Blöfams lesetas at binons balugikum, ka jenets klatädik. Tefons blöfädi yümätavik brefik, ma kel binet alik stäänükama grada ''n'' ela ''F'' binon vul polünoma no diletovik semik, kela grad ''d'' müedon ''n''; vulis at ön mods difik tel änumölo, kanoy fümedön, das

:<math>q^n = \sum_{d\mid n} d N_d</math>

kö saedam lonöfon love müedians valik ''d'' ela ''n''. Güükam hiela Möbius givon:

:<math>N_n = \frac1n \sum_{d\mid n} \mu(n/d) q^d</math>

kö el μ(k) binon dinod ela Möbius. (Leig at ya päsevon fa hiel Gauss.) Cifadil komon ven ''d'' = ''n'', e no fikulos ad miedükön dilis votik. „Nilud ela Riemann“ sekidon de jenöfot, das müedian legik gretikün ela ''n'' no kanon pluon tefü ''n''/2.

== Reidolös i yegedis (Linglänapükik): ==
* [[:en:Abstract analytic number theory|Numateor diletik nedabinotik]] ad getön nünis dö valemükams leseta.
* [[:en:Landau prime ideal theorem|Leset ela Landau tefü primadials]] ad logön valemükami in jäfüd numafelas lalgebrik.
* [[:en:Prime gap|Primanumanäedots]].

== Literat ==
* Avigad, Jeremy, Kevin Donnelly, David Gray, e Paul Raff. 2005. ''[http://arxiv.org/abs/cs.AI/0509025 A formally verified proof of the prime number theorem],'' e-print cs. AI/0509025 in el [[ArXiv]].
* Bach, E., e J. Shallit. 1996. ''Algorithmic Number Theory'', toum: 1. Dabüköp ela MIT. ISBN 0-262-02405-5. (L. pad: 233, diläd: 8.8.)
* Dusart, P. 1999. [http://www.ams.org/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf "The ''k''th prime is greater than ''k(ln k + ln ln k-1)'' for ''k''>=2".] Mathematics of Computation, nüm: 68, pads: 411-415.
* Granville, A. 1995. [http://www.dms.umontreal.ca/~andrew/Postscript/cramer Harald Cramér and the distribution of prime numbers], ''Scandinavian Actuarial Journal,'' toum:&nbsp;1, pads: 12-28.
* Hardy, G. H., e J.E. Littlewood. 1916. Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes. ''Acta Mathematica'', nüm: 41, pads: 119-196.
* Knuth, D. ''[[The Art of Computer Programming]]'', volume 2, section 4.5.4, exercise 36, ISBN 0-201-89684-2.
* Schoenfeld, L. 1976. ''Sharper bounds for the Chebyshev functions θ(''x'') and ψ(''x''), II'', Mathematics of Computation, toum: 30, nüm: 134, pads: 337-360.
* Soprounov, I. 1998. [http://www.math.umass.edu/~isoprou/pdf/primes.pdf A Short Proof of the Prime Number Theorem for Arithmetic Progressions].

== Yüms plödik ==
In Linglänapük:
* [http://www.scs.uiuc.edu/~mainzv/exhibitmath/exhibit/felkel.htm Taib Primanumas fa hiel Anton Felkel.]
* [http://mathworld.wolfram.com/PrimeFormulas.html Prime formulas] and [http://mathworld.wolfram.com/PrimeNumberTheorem.html Primanumaleset] in bevüresodatopäd: MathWorld.
* [http://planetmath.org/?op=getobj&from=objects&id=199 Primanumaleset] in bevüresodatopäd: PlanetMath.
* [http://primes.utm.edu/howmany.shtml Primanums liomödotik dabinons-li?] e [http://primes.utm.edu/notes/gaps.html Fagot bevü primanums] fa hiel Chris Caldwell se Nivera di Tennessee.
* [http://www.ieeta.pt/~tos/primes.html Taibs numamasekätas primanumas] hiela Tomás Oliveira e Silva.

[[Klad:Matemat]]

{{Yeged gudik}}



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