Ramanujan Tau Function Lygeros Rozier

7/21/2019 Ramanujan Tau Function Lygeros Rozier

1/14

5 1

Pbo Zekeduged Giurdeh

Ed Cdtordetcideh Giurdeh @ovito` ti

tbo Eroes il Ketboketcas Cdlhuodao` jy

Zekeduged

CQQD 512:-0>3>

Zekeduged G

@IC 5>.5>>6/s55513->5:-30:>-2

I`` prcko vehuos il tbo Zekeduged teuudatcid

Dcm Hyforis & Ihcvcor Zizcor


2/14

5 1

Siur ertcaho cs pritoato` jy aipyrcfbt ed`

ehh rcfbts ero boh` oxahuscvohy jy Qprcdfor

Qacodao+Juscdoss Ko`ce, HHA. Pbcs o-illprcdt

cs lir porsideh uso idhy ed` sbehh dit jo sohl-

erabcvo` cd ohoatridca ropisctircos. Cl yiuwcsb ti sohl-erabcvo yiur wirm, phoeso uso tbo

eaaopto` eutbirs vorscid lir pistcdf ti yiur

iwd wojscto ir yiur cdstctutcids ropisctiry.

Siu key lurtbor òpisct tbo eaaopto` eutbirs

vorscid id e ludòrs ropisctiry et e ludòrs

roquost, privcò` ct cs dit keò pujhcahy

evechejho udtch 5: kidtbs eltor pujhcaetcid.


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Zekeduged G

@IC 5>.5>>6/s55513->5:-30:>-2

I`` prcko vehuos il tbo Zekeduged teu ludatcid

Dcm Hyforis Ihcvcor Zizcor

Zoaocvo`7 :: Key :>5: / Eaaopto`7 2 Guhy :>5:

Qprcdfor Qacodao+Juscdoss Ko`ce, HHA :>51

Ejstreat Vo stu`y tbo i`` prcko vehuos il tbo Zekeduged teu ludatcid, wbcab lirk

e tbcd sot il herfo prckos. Pi tbcs od`, wo òstdo HZ(p,d) 7; (pd5)ed` wo sbiwtbet tbo i`` prcko vehuos ero il tbo lirkHZ(p,q)wborop, q ero i`` prckos. Pbod

wo oxbcjct erctbkotcaeh priportcos ed` aidfruodaos il tbo HZ dukjors uscdf kiro

fodoreh rosuhts id Huaes soquodaos. Lcdehhy, wo pripiso ostcketcids ed` `csauss du-

korcaeh rosuhts id pecrs(p,q )lir wbcabHZ(p,q)cs prcko.

Moywir`s Zekeduged ludatcid ^rckehcty Huaes soquodaos

Ketboketcas Qujgoat Ahesscstaetcid (:>5>) 55E05 55L1> 55S55

5 Cdtriùatcid

Pbo teu ludatcid cs òstdo` es tbo Liurcor aiolstacodts il tbo kiùher `csarckcdedt

(z) ; q+d;5

5 qd:0 ;

+d;5

(d)qd,

wboroz hcos cd tbo aikphox uppor behl-phedo ed`q; o:cz .Doerhy e aodtury efi, tbo Cd`ced ketboketcaced Qrcdcvese Zekeduged sbiwo`

froet cdtorost cd tbo teu ludatcid ed` `csaivoro` siko il cts rokermejho priportcos.

D. HyforisHFÂ (_KZ ?:2?), _dcvorsct ò Hyid,


4/14

D. Hyforis, I. Zizcor

Johiw ero hcsto` tbo mdiwd c`odtctcos ed` aidfruodaos lir tbo teu ludatcid7

(dk) ; (d)(k) lird, kaiprcko cdtofors= (5)pr+5; (p)p

r p55pr5 lirp prcko ed`r ed cdtofor 5= (:)

(d) 55(d)

ki` :55

lird 5( ki` 2)= (1)(d) 5:5655(d)

ki` :51

lird 1( ki` 2)= (0)

(d) 5?1655(d)

ki` :5:

lird ?( ki` 2)= (?)(d) 6>?55(d)

ki` :50

lird 6( ki` 2)= (65(d)

ki` ?1

lird >( ki` ?)= (3)

(d) d3(d) (ki` 6) lird >, 5, :, 0( ki` 6)= (5>)(d) d3(d)

ki` 6:

lird 1, ?, ( ki` :1) lirp prcko,

p

:1

; 5= (5:)

(p) 55(p) ki` :1: lirp prcko il tbo lirku: + :1v:= (51)(p) 5( ki` :1) lir itbor prckop= (50)(d) 55(d) (ki` ?


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Vctbiut tbo piwor il tièys aikputors, privcdf suab e rosuht wes dit strecfbtlir-

wer`.

: HZ dukjors

Vo pripiso ti òstdo tbo HZ lekchy il cdtofors, cd kokiry il @.B. Hobkor ed`

Q. Zekeduged, es lihhiws7

@ostdctcid 5 Hot p, q jo i`` prckos. Pbod wo òstdo HZ(p,q) 7; (pq5). Kirofodorehhy, wo sbehh uso tbo ditetcid HZ(p,d) 7; (pd5) lir ehh pisctcvo cdtofors ded` wo sot tbo vehuoHZ(p, >) 7; >.

Pbo kecd kitcvetcid lir tbo cdtriùatcid il tbo provcius òstdctcid cs roheto` ti

Pboirok5, lir wbcab wo wchh fcvo e priil. Ct stetos e stridf doaossery aid`ctcid idtbo cdtoforsdsuab tbet (d)cs ed i`` prcko. Iur ditetcid wchh privo ti jo rohovedt es

tbo `cefideh torksHZ(p,p)bevo spoacsta erctbkotcaeh priportcos (soo Pboirok0).

Cd wbet lihhiws, e prcko p suab tbet p(p) cs aehho` ir`cdery. Itborwcso p cssec` ti jo did-ir`cdery.

Pboirok 5 Hotdjo e pisctcvo cdtofor suab tbet (d)cs ed i`` prcko. Pbodd ; pq5wborop ed`q ero i`` prckos ed`p cs ir`cdery.

Zokerm 5 Idhy stdctohy kedy did-ir`cdery prckos ero mdiwd ti oxcst7 :, 1, ?, 6,:055, 66?2116, HZ(p,d)cs e pihydikceh il òfroo (d 5)cd (p)ed` p557

HZ(p,d) ;] d5

: [

m;>(5)m

d 5 m

m

p55m(p)d5:m . (56)

Pbo `cvcscjchcty priporty il tbo Huaes soquodaos epphcos7

cl k | d, tbod HZ(p,k) |HZ(p,d). (52)

Iur soquodao bes `csarckcdedt

@p7; (p):

0p55

(53)

wbcab cs dofetcvo jy (5


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Pejho 5 Lcrst vehuos il(p),

p/ , ais p ed` scd p p (p) p/ ais p scd p

: :0 >.?2?0:< >.:>01< >.2 >.1>5322 >.3?115:

:3 5:20>.1>:?.?25:?6 >.2516:>

15 ?:2015.??1>>< >.5:2:6 >.>>222: >.3333.2?0?2< >.?5315>

Lihhiwcdf Zekeduged ]6[, wo òstdo tbo edfhos p (>, ) suab tbet (p);:p

55: ais p . Qiko vehuos il(p) ed` p/ ero hcsto` cd Pejho 5. Pbod wo òrcvo

ed oqucvehodt lirkuhetcid roheto` ti tbo Abojysbov pihydikcehs il tbo soaid` mcd`7

HZ(p,d) ; p 55(d5): scd(dp)scd p

;d5m;5

(p) :p 55: ais m

d

. (:>)

Bodao, e lecr ostcketcid il tbo sczo il|HZ(p,d)| cs fcvod jy p 55: (d5) cd kistaesos. Pbcs cs suppirto` jy tbo dukorcaeh rosuhts.

Pboirok:, ùo ti Kurty, Kurty ed` Qbiroy ],suab tbet

lir ehh pisctcvo cdtoforsd lir wbcab (d)cs i``,wo bevo

(d) (hif d)a.Pbo doxt rosuht cs sikobiw roheto` ti Pboirok:(soo Zokerm:), ed` wchh jo uso`

cd tbo priil il Pboirok5.

Hokke 5 Pbo oquetcid (d)

; 5bes di sihutcid lird 45.

^riil(smotab) Jy priporty (5), wo aed essuko wctbiut hiss il fodorehcty tbetd ; prlir e prcko p ed` cdtoforr 4>. Pbus (d) ;HZ(p,r+ 5).

Diw ct sulstaos ti epphy mdiwd rosuhts id Huaes soquodaos (Pboiroks A, 5.1, ed`

5.0 cd ]5[) ti sbiw tbetHZ(p,r+ 5)bes e prckctcvo `cvcsir.


7/14


Zokerm : Ct cs dit qucto ijvcius lir us cl Hokke5 cs e airihhery il Pboirok :, es

tbo hettor eppoers ti jo ossodtcehhy il quehctetcvo deturo. Kiroivor, tbo olloatcvodoss

cd tbo spoaceh aeso(d) ; 5 cs diwboro kodtcido` cd ]


8/14


Pboirok 0 Hotp jo ed ir`cdery i`` prcko.

Clp 5 (ki` 0),tbod HZ(p,p)cs aikpiscto.Clp 1 (ki` 0)ed``cs e prcko `cvcsir il HZ(p,p),tbod` 5 (ki` 0p).

^riil (smotab) Pbo strst pert il tbo tboirok lihhiws lrik lirkuhetcid (:>) hoe`cdf titbo fodorca leatirczetcid HZ(p,p) ; D>D5 wboro

Dg;p5

:m;5

(p) (5)g+m

:m

p

:p

55: ais

m

p

, lirg; >, 5.

Ido key vorcly tbet D> ed`D5 ero cdtofors uscdf tbo Feuss suk vehuo

p

5m;5

mp

o

:mcp ; p, lirp 5(ki` 0).

Lir tbo soaid` pert, wo epphy (:0) ed` ijtecd tbetp cs e que`retca roscùo kiùhi`.

Pbo òscro` rosuht oeschy lihhiws jy aikjcdcdf tbo hew il que`retca roacpriacty wctb

tbo aidfruodao` 5(ki` :p).

? Aidfruodaos kiùhip

5

Di aidfruodao kiùhi p cs mdiwd lir (p) (soo, o.f., ]0[), bodao e lirtcirc lir

tbo dukjors HZ(p,d). Boro wo jrcoy stu`y tbo sots ^+ ed` ^ il prckos p lirwbcab tbo dukjorsHZ(p,d)bevo ohokodtery aidfruodaos kiùhip + 5 ed`p 5,rospoatcvohy, es spoacsto` cd Hokke:.

Hokke : Hot^+ jo tbo sot il i`` prckos p suab tbet (p) > (ki` p + 5).Hotp ^+.Pbod HZ(p,d) > (ki` p +5)cldcs ovod, ed` HZ(p,d) 5 (ki` p +5)cld cs i``.

Hot^ jo tbo sot il i`` prckos p suab tbet (p) : (ki` p 5).Hotp ^.Pbod HZ(p,d) d (ki` p 5)lir ehh pisctcvo cdtoford.

^riil(smotab) Pbo roaurrodao rohetcid (:) hoe`s ti ed oesy priil jy cdùatcid lir ehh

pisctcvo cdtoforsd.

Hokke1stetos tbet ^+ed` ^jitb cdahuò kist il tbo skehh prckos. Dovortbo-hoss, tboro cs dukorcaeh ovcòdao tbet tboy bevo òdscty zori ekidf tbo prckos.

Hokke 1 HotE 7; :50 16 ?1 6: :1 jo tbo sot il i`` prckos psuab tbetp + 5|E.Pbod ^+> ^+.

HotJ7; :55 1< ?1 6 jo tbo sot il i`` prckos p suab tbetp 5|J .Pbod ^> ^.


9/14


^riil (smotab) Hotp ^+> . Pbodp ; :r(:) 1r(1) ?r(?) 6r(6) :1r(:1) ero stdcto sots. Pboy aikprcso rospoatcvohy 550> ed` 1:?

prckos ed` tbocr skehhost ed` herfost ohokodts ero fcvod johiw7

^+> ; {1, ?, 6, 55, 51, 56, 53, :1, :3, 15, 05, 06, ?1, ?3, 65, 63, 21, 23, 36, . . . ,5

50=

^> ; {1, ?, 6, 55, 51, 56, 53, :3, 15, 16, 05, 01, 3.>: 5>55.Diw cl wo `ostdo tbo rosc`ueh sots ^+5 7; ^+ X ^+> ed` ^5 7; ^ X ^> , ct turds

iut tbet

^+5 ; {?31, 51:3, :6?1, 0>63, 0:21, 6033, 6221, 3603, 55?03 . . .}= ^5 ; {5>1, 155, 23, :551, :055, :, 5, :, 1, 0= ^5 aidtecds tbo did-ir`cdery prcko :055 (soo Zokerm 5), jut tbo doxt ido,

66?2116.?66:5? . . .cs Ouhors aidstedt.Hotp ed` q jo twi i`` prckos suab tbetp ; q ed` (p) > (ki` p). Vo mdiw

lrik Pboirok1 tbet ehh prcko `cvcsirs ilHZ(p,q)ero il tbo lirk :mq 5, oxaoptpisscjhyq wctb prijejchcty^ (q). Pbo oxpoato` vehuo il^ (q)cs riufbhy 5/q, udhoss

q; 1, ?, 6, ir :1 lir wbcab wo bevo tbo aidfruodaos (6) ti(50). Vo oeschy fot tbo


10/14


oxaoptcideh vehuos il^ (q)lirq; 1, ? ir 6 jy aidscòrcdf ehh roscùos ilpkiùhiq(soo ehsi ]5>,55[), wboroes^ (:1)cs tbo pripirtcid il prckos il tbo lirku: + :1v:7

^ (1)

;

5

: = ^ (?)

;

5

0 = ^ (6)

;

5

: = ^ (:1)

;

5

>>, suab tbet HZ(p,q)cs prcko (^) ir prijejho prcko (^Z^). OA^^

() kotbi` bes jood uso` lir tbo prckehcty il twi herfo vehuos

p q @cfcts ^rckehcty p q @cfcts ^rckehcty

55 156 525> ^ 013 :3 0>6 ^

56 011 :3:0 ^ 003 ?06 633:0 ^Z^ ?>1 61 5>6> ^

06 ? 16 ^ ??6 5>3 535 51 1306? ^Z^551 536 ::50 ^ 6 ^

5?6 ::>6 : ^ 2>3 503 :1


13/14


Pejho 0 Lir i`` prckos

q 85>>, eatueh ed` oxpoato`dukjor il prckosp 85>< suab

tbetHZ(p,q)cs ^Z^, ed` strst

vehuos ilp

q Eatueh Oxpoato` p

1 212 3>0 :?5, 265 06, 536, 55:1, :3?1, 1161,. . .

6 012 000 55?5, :505, ?>26, 6, 65:3,. . .55 51, :551, 1, 626, 60 556 0:5, :201, 0


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5. Lcd`cdf skehh `cvcsirs il tbo lirk :mq 5 wctb ÊZC/F^=:. 1-^Z^ tosts wctb ^LFV=

1. E^ZAH prckehcty tosts lir ehh ^Z^s up ti 16>> òackeh `cfcts wctb ÊZC/F =

0. Jechhco-^QV ^Z^ tosts lir ehh ^Z^s ejivo 16>> òackeh `cfcts wctb ÊZC/F .

Qtefo 0 hoe`s ti e froetor prijejchcty il prckehcty tbed stefo : (tboro cs di mdiwd

aikpiscto dukjor wbcab cs pesscdf tbcs tost), jut temos kiro tcko.

Vo picdt iut tbet Lredics Kirecd privcòs prckehcty aortcstaetos lir twi herfoHZ

dukjors (soo `cekid`s cd Pejho 1) id bcs woj pefo. Bo uso` bcs iwd siltwerolestOA^^, ckphokodtcdf e lest ehfirctbk il ohhcptca aurvo prckehcty privcdf ]?[, id e

aikputor ahustor. Bcs aehauhetcids roqucro` rospoatcvohy 1?? ed` :1?? èys il titeh

A^_ tcko, jotwood Geduery ed` Eprch :>55. Qcdao HZ(5?6, ::>6) bes :5)

:. Hobkor, @.B.7 Pbo prckehcty il Zekedugeds Peu-ludatcid. Ek. Ketb. Kid.6:, 5?52 (53

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Ramanujan Tau Function Lygeros Rozier