
ramanujan tow function

ramanujan tow function, padicha, inipparayunna aidantitti nirvachicha function formula _ 1 aannu
ivide, formula _ 3 ennathu euler functionaanu, dedeckind ettaa functionaanu, koodaathe function 12, level 1 ennivayude bharam ulla oru holomorphic cusp roopamaanu, ithu discriminant modular phoom ennariyappedunnu (chila rachayithakkal, prathyekichu apostol, formula _ 5 nuu pakaram formula _ 4 ezhuthunnu). oru poornnasankhyaye 24 chathurangalude koottamaayi prakadippikkunnathinula vazhikalude ennam kanakkaakkunnathil ulppedunna oru "error tame" maayi ithu prathyakshappedunnu. iyaan jii kaaranam oru soothravaakyam. mcdonald akathekku kayatti.
moolyangal.
tow functionte aadya chila moolyangal inipparayunna pattikayil nalkiyirikkunnu
ramanujante oohangal.
 thaazhepparayunna moonnu gunangal nireekshichenkilum theliyikkanaayilla
aadyathe rund gunangal ramanujan anumaanamenn vilikkappedunna moonnaamathethu 1974 ll dilligne theliyichu, ithu addehathinte veyil anumaanangalude thelivukalude falamaanu (prathyekichu, avaye oru kuga-saatto inathil prayogichukondu addeham athu anumaanichu).
tow functionulla kongroons.
for and, ennathine, athinte vibhaajakangalude th shakthikalude aakethukayaayi nirvachikkuka. tow function niravadhi kongruvans bandhangale thripthippeduthunnu; avayil palathum ithinte adisthaanathil prakadippikkaan kazhiyum. avayil chilath ithaa
priminu, njangalkk undu
vyakthamaaya soothravaakyam.
1975ll dougless neiber ramanujan tow functionte vyakthamaaya soothravaakyam theliyichu
τ" ("n") ennathinekkurichulla anumanangal.
athu oru bhaara-poornnasankhya puthiya roopamaanennum forrier gunakangal poornnasamkhyakalaannennum karuthuka. prashnam pariganikkuka 
theerchayaayum, mikka primukalkkum eee property undaayirikkanam, athinaal avaye "oordinari" ennu vilikkunnu. galois praathinidyangalil dealine, sere enniva valia munnettangal nadathiyittum, avaa coprime nirnnayikkunnu, engane kanakkukoottanamenn vyakthamalla. ithumaayi bandhapetta eaka sidhaantham modular deerghavrithaakrithiyilulla vakrangalkkulla elkkeesukalude prashasthamaaya falamaanu, ithu ananthamaaya niravadhi primukal undennu urappunalkunnu, athinaal avaa 0 moduloik yojichathaan. 2-ll kooduthal bharamullah sii. emm. allaathavaykku ariyappedunna udaaharanangalonnumilla, athinu ananthamaaya niravadhi primukal (mikkavarum ellaavarkkum ithu shariyaanenkilum). ananthamaaya niravadhi perumaayi ariyappedunna udaaharanangalonnumilla. chila gaveshakar ananthamaaya palarkkum undo ennu samshayikkaan thudangi. thelivaayi palarum ramanujante (12 bharamullah cases) rekhakal nalki. 1010 vareyulla samavakyathinulla eaka pariharam 2,3,5,7,2411, ennivayaanu.
 ellaavarkkum, chilappol lehmarinte anumaanam ennariyappedunna oru avakaashavaadam ennu addeham anumaanichu. lehmer eee anumaanam ithuvare parishodhichu (aposthalan 1997, peju. 22). eee vyavastha ellaavarkkum baadhakamaaya valia moolyangal kandethunnathil undaaya purogathi thaazhepparayunna pattika samgrahikkunnu.
ramanujante "ell"-function.
ramanujante "ell"-function nirvachichirikkunnathu
soothravaakyam _ 18 aanenkil, vishakalanaparamaaya thudarchayiloode. ithu pravartthana samavaakyathe thripthippeduthunnu
koodaathe euler uthppannavum undu
21 enna soothravaakyathile ellah nissara poojyangalkkum 22 enna soothravaakyathinu thulyamaaya yadhaartha bhaagamundennu ramanujan anumaanichu.

