on the distribution of prime numbers

8/11/2019 On the Distribution of Prime Numbers

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Jh tea Kfstrfcutfjh jn Prfma Humcars

^fjla, [email protected]

Dprfl 31, 304?

FH[SJKQG[FJH

_a prasaht savardl gjhtahtfjhs wfte tea Prfma Humcar [eajram (PH[) dhkjffar dltarhdtfva axpldhdtfjhs. [ea fflrst sagtfjh jn tefs pdpar kadls wfte teaprjcdcflfty d humcar fs prfma. _a lfhb tefs prjcdcflfty tj tea ffijjr jn teasqudra rjjt jn tea humcar. _a teah fllustrdta ejw fhgjrpjrdtfho tea ffijjr jntea squdra rjjt jn tea humcar oahardtas d stap nuhgtfjh tedt mjra dggurdtalydpprjxfmdtas(x) tedh x

lh(x) .

[ea sagjhk sagtfjh jn tefs pdpar kadls wfte gjhvaroahgas. _a kamjhstrdtafhgradsak dggurdgy tj(x) cy fkahtfnyfho tea jptfmdl gjagfaht fh jur nuhgtfjh._a teah trdhspjsa tea gjhvaroahga droumaht tj tea Ljo cdsa suge tedt dtfhfflhfty, x

lh(x) fs vdlfk, cut hjt njr dhy fflhfta humcar.

Nfhdlly, wa ljjb dt tea efstjrfgdl tradtmaht jn gjhvaroahgas dhk tea sucstf-tutfjh jn lfmft gjhkftfjhs karfvak nrjm Aulars prjkugt njrmuld.

4 Prjcdcflfty d Humcar fs Prfma

[ea prjcdcflfty d humcar fs prfma fs aqudl tj tea sfza jn fts squdra rjjt, sfhgajha ndgtjr must rasfka wftefh tea rdhoa R0,

x]. Fn twj humcars edva tea sdma

wejla humcar pdrt jn teafr squdra rjjt, teay edva aqudl prjcdcflftfas jn cafhoprfma.

Njr axdmpla, 404 dhk 446 edva tea sdma prjcdcflfty jn cafho prfma. Cjtehumcars edva 40 wejla humcars fh teafr squdra rjjts, 40.0;66 dhk 40.60?4raspagtfvaly. [eus tea prjcdcflfty d humcar fs prfma jvar tea rdhoa nrjm 400tj 434 fs uhfnjrm (njr dll humcars ahkfho fh 4, 7, ?, dhk 6). PH[ dssarts tedttea prjcdcflfty d humcar fs prfma fs aqufvdlaht tj 4

lh(x) . [efs sfmply gdhhjt ca,

dhk tefs kaglfhfho prjcdcflfty jvar tea rdhoa rdtear tedh tea teajratfgdl uhfnjrm

ladks tj d systamfg uhkarastfmdta jn prfmas (aspagfdlly weah rdhoas oat ldroarvfd tea kfstdhga catwaah squdras jn ldroar humcars) ds hjtfgak wfte x

lh(x) .

4


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4.4 Qhfnjrm Prjcdcflfty Dssumptfjh

_a parnjrm twj tasts tj kamjhstrdta tea uhfnjrm prjcdcflfty dssumptfjh us-fho p dhk tea fhtarvdl madh, ftarvdl. _eah tastfho tea uhfnjrm prjcdcflftydssumptfjh, wa njuhk tea rdtfj jn prfma humcars dcjva tea madh nrjm twjwejla humcars squdrak tj tea prfma humcars caljw tefs madh dsymptjtfg tj 4._a jhly tastak savardl euhkrak tejusdhk, cut tefs gjulk ca dh drad jn nurtearrasadrge tj nurtear cjlstar tefs dssumptfjh prasahtak fh Nfoura 4 caljw.

8 raqufra(prdgmd)2raqufra(humcars)2raqufra(HHW)

8 rdtfj=humarfg()

8 madh.pjfht=humarfg()

8 njr(f fh 494000){

+ madh.pjfhtRf]=madh(g(fX3,(f+4)X3))

+ rdtfjRf]=sum(Prfmas(fX3,(f+4)X3)a+01 5a+01 4a+0>

4

3

7

;

Gumuldtfva Dcjva Madh / Caljw Madh

madh.pjfht

rdtfj

^fawfho tefs uhfnjrmfty dssumptfjh nrjm d kfffaraht parspagtfva, tea squdrarjjt jn d prfma humcar eds dh aqudl prjcdcflfty jn ahkfho fh d .31 jr .6, sfhgatea rdhoa fs rasat wfte avary parnagt squdra. Fh ndgt, weah tdbfho tea kagfmdl jntea prfma humcar njr tea fflrst >>;,1?6 prfmas (x= 40, 000, 000), wa njuhk tedttea kagfmdl fs uhfnjrmly kfstrfcutak. Fn x

lh(x) wara d vfdcla dpprjxfmdtfjh, teah

3


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teara sejulk axfst mjra .04 prfmas tedh .66 prfmas, dggjmpdhfak cy d hjh-lfhadrgumuldtfva jutput. Ejwavar, tefs fs hjt tea gdsa ds sejwh fh Nfoura 3.

8 efst(sqrt(Prfmas(4a?))-nljjr(sqrt(Prfmas(4a?))),cradbs = 66,

+ mdfh="Wqudra Sjjt jn Prfma Lass _ejla Humcar")

Wqudra Sjjt jn Prfma Lass _ejla Humcar

sqrt(Prfmas(4a+0?)) nljjr(sqrt(Prfmas(4a+0?)))

Nraquahgy

0.0 0.3 0.; 0.> 0.5 4.0

0

4000

3000

7000

;000

1000

>000

?000

Ds tea humcar oats ldroar, tefs uhfnjrmfty ejlks njr dhy aqudl sfzak kagfmdlrdhoa catwaah tea parnagt squdras. Njr axdmpla, tea humcar jn prfmas wftesqudra rjjts ahkfho fh R.04, .31] = R.31, .1] = R.1, .?1] = R.?1, .00].

3 Gjhvaroahgas

_a wfll kamjhstrdta njr adge x, dh jptfmum ljo cdsa axfsts suge tedt (x) =x

ljoJP[FMQMCdsa (x)dhk tefs jptfmum ljodrftemfg cdsa fs ftsaln d gjhvaroaht nuhg-

tfjh wfte lfmft a. [ea rdta jn gjhvaroahga jn Aqudtfjh (4), jntah ofvah ds teakafflhftfjh jna, fs pratty qufgb.

lfmx

4 +

4

x

(4+x)

=a (4)

Qsfho Aqudtfjh (4) ds tea ljo cdsa ealps sjmawedt varsus xlh(x) , cut hjt tj

7


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tea dggurdgy kasfrak. _a gdh sljw tefs gjhvaroahga kjwh nurtear cy fhgjrpj-rdtfho dkkftfjhdl ljos tjx ds sejwh fh Aqudtfjh (3).

lfmx

4 +

4

ljo40(x)

(4+ljo40(x))=a (3)

Nfoura 7 caljw sejws d gjmpdrfsjh jn tea twj gjhvaroahgas9

8 aq.4=humarfg()

8 aq.3=humarfg()

8 njr(f fh saq(4,47,4)){

+ aq.4Rf]=(4+(4/(4*40Xf)))X(4+(4*40Xf))

+ aq.3Rf]=(4+(4/ljo(4*40Xf,40)))X(4+ljo(4*40Xf,40))

+ }

8 pljt(aq.4,typa = l ,ylfm = g(mfh(aq.4),mdx(aq.3)),

+ mdfh = "Gjhvaroahga tj a",yldc = ,xldc = 4aXx )

8 lfhas(aq.3,gjl= clua )8 dclfha(e=axp(4),lwk=4,gjl= rak ,lty=3)

8

3 ; > 5 40 43

3.

5

7.

0

7.

3

7.

;

7.

>

7.

5

;.

0

Gjhvaroahga tj a

4aXx

3.4 Fhgjrpjrdtfhox fhtj (x)Aqudtfjh (7) raprasahts jur astfmdta jn (x) cy fhgjrpjrdtfhox tj gjm-pahsdta njr tea prjcdcflfty d humcar fs prfma vfd fts wejla humcar squdra rjjt

;


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dljho wfte tea sljwar gjhvaroahga tj a.

^ N(x) =

x3

lh

x3

4+ 4ljo40(x)

(4+ljo40(x))

(7)

x3 fs tea ffijjr jn tea squdra rjjt jn tea humcar tj ca astfmdtak, squdrak.[efs mdy saam gjuhtarfhtuftfva dt fflrst oldhga, cut ft ragjhgflas d prjcdcflftyfssua wfte tea prfma humcar teajram (PH[) ds kfsgussak fh Wagtfjh 4.

Eara fs tea S-nuhgtfjh njr Aqudtfjh(7), eahganjrte ^N9

8 ^fjla.Nuhgtfjh


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+ ^N.cdsaRf]=uhfrjjt(nuhgtfjh(x)

+ ^fjla.Nuhgtfjh(4*40Xf)-((4*40Xf)/ljo((4*40Xf),cdsa=x)),

+ ljwar = 3, uppar = ;, tjl = 4a-6)$rjjt+ lf.x.cdsaRf]=uhfrjjt(nuhgtfjh(x)

+ (lf(4*40Xf)-lf(3))-((4*40Xf)/ljo((4*40Xf),cdsa=x)),

+ ljwar = 3, uppar = ;, tjl = 4a-6)$rjjt

+ }

8 jptfmum.cdsas=gcfhk(jptfmum.cdsa,^N.cdsa,lf.x.cdsa)

8 jptfmum.cdsasRgjmplata.gdsas(jptfmum.cdsas),]

jptfmum.cdsa ^N.cdsa lf.x.cdsa

R4,] 7.4>33?5 7.50456; 7.54>065

R3,] 7.464175 7.464175 7.75>014

R7,] 7.404?04 7.445560 7.4;5041

R;,] 7.04?4?3 7.04;?;4 7.0366>6

R1,] 3.61?674 3.615151 3.6>745?R>,] 3.64550> 3.64??31 3.630766

R?,] 3.560437 3.556;77 3.560131

Aqudtfjh (4) fs tjj ndst jn d gjhvaroahga tj a, dhk wjulk ndr jutpdga teajptfmum jr Aqudtfjh (7) (eahganjrte ^N(x)). Ejwavar, wa gdh saa tedt avahtea ^N(x) astfmdta haaks tj ca ljwarak jhga wa oat fhtj tea radlm jn ldroahumcars, prasumdcly dt dhjtear kagradsfho rdta gjhvarofho tj a. [eara fs dlsjhj d prfjrf radsjh wey tefs ljo cdsa astfmdta fsht ftsaln nuhgtfjhdl tjwdrks dgjhvaroahga tja wfte d kagradsfho cdsa (


7/16

(x). [ea fhffiaxfcflfty fhearaht fh fts humardtjr ndgtjrfdls ladks tj fhgramahtstedt gdhhjt raplfgdta tea jptfmum cdsa, avah wfte dvardofho tagehfquas. _a

gdh saa tea gjhvaroahga tj tea jptfmum njr Lf(x), cut tea lfmft fs a, hjt dkyhdmfgdlly fhkugak jptfmum ljo cdsa vdlua. [eus, ft fs gjhsfstaht wfte teahjtfjh tedtLf(x) wfll fh turh edva ljwar vdluas tedh tea jptfmum ljo cdsa dhkuhkarastfmdta (x) njr ldroar vdluas jnx. Ft fs hjt gjhiagturak tedt dh fhfflhftadmjuht jn grjssas axfst, ds tedt wjulk raqufra d vary kyhdmfg sefnt fh tea fmplfakljo cdsa gjhvaroahga nuhgtfjh uhkarlyfho Lf(x) jr jn(x). Nurtearmjra, weahdvardofho Lf(x) tj mdhfpuldta tea affagtfva ljo cdsa rdta, sljwfho d kfvaroahgakjas hjt hagassdrfly oudrdhtaa d syhgerjhfzdtfjh jr gjhvaroahga tj tea jptfmumljo cdsa rdta (nrjm dcjva jr caljw).

3.7 Fhtarsagtfjh jn (x) dhk ^N(x)

Fn ^N(x) wara d vdlfk astfmdta jn(x), teah (x) sejulk grjss ^N(x) jhga fh

adge fhtarvdl nrjm uhkarhadte ds ^N(x) fs d stap nuhgtfjh2 jvarastfmdtfho (x)caljw tea madh jn tea fhtarvdl catwaah twj squdras, dhk uhkarastfmdtfho (x)dcjva tea madh. [ea stap gjmas nrjm tea haxt wejla humcar fh tea squdrarjjt jnx. [efs squdra rjjt stap nuhgtfjh uhkjuctakly eds sjma raldtfjh tj teaSfamdhh Eypjteasfs ds tea grftfgdl strfp, s = 4/3 fs gjhsfstaht wfte tea radlpdrt jn tea lfha cafho d squdra rjjt fh tea zatd nuhgtfjh (s).

3.7.4 ^N(x) Astfmdta ^fsudlfzdtfjh

8 h=4000

8 ^N.astfmdta=humarfg()

8 pf.dgtudl=humarfg()

8 x.lhx=humarfg()

8 lf.x=humarfg()8 njr(f fh 49h){

+ ^N.astfmdtaRf]=^fjla.Nuhgtfjh(f)

+ pf.dgtudlRf]=lahote(Prfmas(4,f)RPrfmas(4,f)


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0 300 ;00 >00 500 4000

0

10

400

410

Fhkax

^N.astfmdta

^N

Pf(x)

x/lh(x)

lf(x)

3.7.3 ^N(x) Arrjrs

Dhk wa gdh gjmpdra tea arrjrs jn cjte matejks9

8 ^N.arrjr=^N.astfmdta-pf.dgtudl

8 x.lhx.arrjr=x.lhx-pf.dgtudl

8 lf.x.arrjr=lf.x-pf.dgtudl

8 pljt(^N.arrjr,typa = l ,gjl= purpla ,lwk=7,ylfm = g(mfh(x.lhx.arrjr),

+ mdx(g(lf.x.arrjr,^N.arrjr))))

8 lfhas(x.lhx.arrjr,gjl= lfmaoraah ,lwk=7)

8 lfhas(lf.x.arrjr,gjl= clua ,lwk=7)

8 dclfha(e=0)

8 laoahk( cjttjmlant ,laoahk=g("^N arrjrs","x/lh(x) arrjrs","lf(x) arrjrs"),

+ nfll = g( purpla , lfmaoraah , clua ),

+ cty=

h

)

5


9/16

0 300 ;00 >00 500 4000

30

41

40

1

0

1

40

Fhkax

^N.arrjr

^N arrjrs

x/lh(x) arrjrs

lf(x) arrjrs

7 Mfkpjfhts jn ^N(x)

_a gdh fmprjva teasa arrjrs cy rakugfho tea ^N(x) stap nuhgtfjh tj d gjhtfhujuslfha. CV fsjldtfho tea mfkpfjhts jn^N(x), wa gdh teah fflt team dhk fhtarpjldtaastfmdtas.4

8 mfkpjfht.kn=gcfhk((49h),^N.astfmdta)

8 mfkpjfhts=dooraodta(x = mfkpjfht.knR,4], cy =lfst(mfkpjfht.knR,3]), NQH = "madh")

8 pljt(^N.astfmdta,gjl= purpla ,typa= l ,lwk=7,cty= h )

8 pjfhts(mfkpjfhts$x,mfkpjfhts$Orjup.4,pge=46)

4Gjhhagtfho dll pjfhts lfhadrly dhk fhtarpjldtfho dll fhtaoar vdluas wftefh R4, x] wfll lfbalyca ndstar tedh HHW.rao.

6


10/16

0 300 ;00 >00 500 4000

0

10

400

410

Fhkax

^N.astfmdta

Hjw wa dkk cdgb (x) dhk fsjldta jur ^N(x) mfkpjfhts.

8 pljt(pf.dgtudl,gjl= rak ,typa = l ,lwk=3)

8 pjfhts(mfkpjfhts$x,mfkpjfhts$Orjup.4,pge=46)

8 laoahk(

tjplant

,laoahk=g("^N mfkpjfhts","Pf(x)"),nfll = g(

cldgb

,

rak

),+ cty= h )

40


11/16

0 300 ;00 >00 500 4000

0

10

400

410

Fhkax

pf.dgtudl

^N mfkpjfhtsPf(x)

Eara fs tea hjhlfhadr raorassfjh fflttfho jur ^N(x) mfkpjfhts.3

8 HHW.rao(mfkpjfhts$x,mfkpjfhts$Orjup.4)$S3

3Wurprfsfholy, teasa mfkpjfhts wjulk hjt suggumc tj dh axpjhahtfdl fflt...eahga tea hjhlfh-

adr raorassfjh.

44


12/16

0 300 ;00 >00 500 4000

0

10

400

410

HHW Jrkar = 1

U (Waomahts = 45)

V

S3= 0.6666

_fte (x) jvarldyak fh oraah tefs tfma, wa gdh cdraly saa jur fflttak ^N(x)mfkpjfhts raorassfjh.

8 HHW.rao(mfkpjfhts$x,mfkpjfhts$Orjup.4,jrkar=

mdx

)$S3

8 lfhas(pf.dgtudl,gjl=

oraah

,lwk=7)

43


13/16

0 300 ;00 >00 500 4000

0

10

400

410

HHW Jrkar = mdx

U (Waomahts = 36)

V

S3= 4

Wj lats gjmputa jur arrjrs nrjm jur ^N(x) mfkpjfhts raorassfjh tj (x) dhksaa fn wa fmprjvak jur astfmdta jvar tea hdfva stap ^N(x). _a haak tj oahardtajur fhtarpjldtak astfmdtas usfho tea pjfht.ast pdrdmatar fh HHW.rao.

8 HHW.^N.arrjrs=HHW.rao(mfkpjfhts$x,mfkpjfhts$Orjup.4,pjfht.ast = g(49h),pljt = N,+ jrkar= mdx )$Pjfht.ast

8 pljt(^N.arrjr,typa = l ,gjl= purpla ,lwk=7,

+ ylfm = g(mfh(g(^N.arrjr,lf.x.arrjr)),

+ mdx(g(^N.arrjr,lf.x.arrjr))))

8 lfhas(HHW.^N.arrjrs-pf.dgtudl,gjl= cldgb ,lwk=7)


8 laoahk( tjplant ,laoahk=g("^N","^N mfkpjfhts","lf(x)"),

+ nfll = g( purpla , cldgb , clua ),

+ cty=

h

,ejrfz = N)

8 dclfha(e=0)

47


14/16

0 300 ;00 >00 500 4000

;

3

0

3

;

>

5

40

Fhkax

^N.arrjr

^N^N mfkpjfhts

lf(x)

7.4 Fhgradsak x

_a wfll fhgradsa x tj 40,000 tj saa fn tea fflt stfll ejlks wall, dhk fn jur ^N(U)

mfkpjfht arrjr tarms dra stfll wall caedvak.

8 h=40000

8 ^N.astfmdta=humarfg()

8 pf.dgtudl=humarfg()

8 x.lhx=humarfg()

8 lf.x=humarfg()

8 njr(f fh 49h){

+ ^N.astfmdtaRf]=^fjla.Nuhgtfjh(f)

+ pf.dgtudlRf]=lahote(Prfmas(4,f)RPrfmas(4,f)


15/16

+ cty= h )

8 ^N.arrjr=^N.astfmdta-pf.dgtudl

8 x.lhx.arrjr=x.lhx-pf.dgtudl8 lf.x.arrjr=lf.x-pf.dgtudl

8 mfkpjfht.kn=gcfhk((49h),^N.astfmdta)

8 mfkpjfhts=dooraodta(x = mfkpjfht.knR,4], cy = lfst(mfkpjfht.knR,3]), NQH = "madh")

0 3000 ;000 >000 5000 40000

0

300

;00

>00

500

4000

4300

Fhkax

^N.astfmdta

^N

Pf(x)

x/lh(x)

lf(x)

Nft ljjbs ojjk, dra tea arrjrs wall caedvak:

8 HHW.^N.arrjrs=HHW.rao(mfkpjfhts$x,mfkpjfhts$Orjup.4,pjfht.ast = g(49h),pljt = N,

+ jrkar= mdx )$Pjfht.ast

8 pljt(^N.arrjr,typa = l ,gjl= purpla ,lwk=7,

+ ylfm = g(mfh(^N.arrjr),

+ mdx(g(lf.x.arrjr,^N.arrjr))))

8 lfhas(HHW.^N.arrjrs-pf.dgtudl,gjl= cldgb ,lwk=7)


8 dclfha(e=0)

8 laoahk(

cjttjmlant

,laoahk=g("^N arrjrs","^N mfkpjfht arrjrs","lf(x) arrjrs"),+ nfll = g( purpla , cldgb , clua ),

+ cty= h ,ejrfz = N)

41


16/16

0 3000 ;000 >000 5000 40000

40

0

40

30

Fhkax

^N.a

rrjr

^N arrjrs^N mfkpjfht arrjrs

lf(x) arrjrs

[eay dra.

Lfmfts & Gjmmahts

D fflhdl tejuoet jh prfmas wefge tdbas us cdgb dll tea wdy tj Aulars prjkugtnjrmuld9

h=4

4

hs =

p=prfma

4

4 ps

Njr dhy fflhfta humcar h, teara fs hj aqudlfty. [efs fs dlsj radlfzak nrjm teandgt tedta fs dh frrdtfjhdl humcar fh tea hdturdl ljo cdsa. [earanjra, wedtavarhumcar jha stjps dt ft fh utflfzfho a, dh uhkarastfmdta jn(x) fs oudrdhtaak.

Qhnjrtuhdtaly, lfmfts wara usak tj parnjrm dhdlysfs wfte cadutfnul dggjmpd-hyfho wjrb sfhga Odusss fhftfdl prfma gjhiagtura dhk Aulars sucsaquaht prjjn.Ejwavar,tea stjry jn tea prfma humcars fs dt fts eadrt d gjhvaroahga stjry2 dhktj usa lfmfts fs tj sfmply tall d kfffaraht stjry.

[efs wjrb wjulk ca oradtly dkvdhgak cy d mjra gdpdcla ldhoudoa tedh S, cutwa ejpa tea prjvfkak gjka fs radkdcla ahjuoe tj ca pjrtak tj jtear ldhoudoas.

_a ljjb tj axtahk tea dhdlysfs tj muge ldroar humcars wfte kyhdmfg ljocdsas dhk ultfmdtaly gjmpdra tj Sfamdhhs astfmdta.

4>

on the distribution of prime numbers

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