
1 + 2 + 3 + 4 + <unk>

anant malika jyachya sandnya naisargik sankhya aahet tee eka bhinn malika aahe. malikechi 'aine' vee aanshik berries hii trikoni sankhya aahe.
jeey 'aine' anantteparyant jaat asatana bandhanashivay vadhate. aanshik rakmecha kram maryadit maryadet ekroop honyat ayashasvi jhalyamule, malikechi berries nasate.
jaree pahilya najaret yaa malikeche konatehi arthpurn moolya nasale taree ganitiyadrushtya manoranjak parinam midvinyasathi tyat ferfar keela jaau shakato. udaharanarth, ganitat bhinn malikela dekhil sankhyatmak moolye niyukt karnyasathi anek berries paddhati vaparlya jaatat. visheshtah, zeeta karya niyamit karnyachya paddhati aanee ramanujan yog yaa malikeche moolya tharavatat, jeey eka prasiddha sutradvare vyakt kele jaate.
jethe davya bajucha arth, vareel yog paddhatinpaiki eka vaparoon midavalele moolya mhanoon lavala pahije, aanee nehamichya arthatil anant malikechi berries mhanoon nahee. yaa paddhatinche guntaguntiche vishleshan, quantum kshetra siddhant aanee string siddhant yasarkhya itar kshetranmadhye upayog aahet.
alberta vidyapithachya ganitshastradnya terry gannon yanni chandrachya prakash siddhantavaril eka monographmadhye yaa samikaranala "vidnyanatil sarvat ullekhaneey sutranpaiki eka" asey mhatale aahe.
aanshik rakkam.
malikechi aanshik berries ityadi aahet. aine" chee aanshik berries eka sopya sutradvare dilee jaateh
heey samikaran pythagoriansana ii. sha. puu. sahavya shatkachya suruvatilach maheet hote. yaa swarupachya sankhyanna trikoni sankhya mhanatat, kaaran tyanna sambahu trikon mhanoon mandata yete.
trikoni sankhyancha anant kram + φ madhye vibhagala jaato, mhanoon vyakhyenusar, anant malika dekhil + φ madhye vibhagali jaate. vichalan haah malikechya swarupacha eka sadha parinam aaheh sandnya shunyaparyant pohochat naheet, mhanoon chachani yaa sandnyedvare malika vegali hote.
saransh.
shastriy bhinn malikepaiki, maryadit moolyat hatalane tulanene katheen aahe. bhinn malikanna sankhyatmak moolye niyukt karnyasathi anek berries paddhati vaparlya jaatat, kahee itaranpeksha adhik shaktishali asatat. udaharanarth, cesaro yog hii eka suprasiddha paddhat aahe jee grandychi malika, saumyapane bhinn malika, hii ektrit karate. abel yog hii eka adhik shaktishali paddhat aahe jee keval grandychya malikechi berries karat nahee tarr klisht malikechi berries dekhil karate.
vareel malikechya ulat, cesaro saransh kinwa aabel saransh nahee. tyaa paddhati dolayman vichalit malikevar kaam karatat, parantu teey + φ paryant vichalit honanya malikesathi maryadit uttar deu shakat naheet. bhinn malikechya bereejchya bahutek adhik prathamik vyakhya sthir aanee resheey asatat aanee sthir aanee resheey asha konatyahi paddhatichi berries maryadit moolyashi houu shakat nahee; khalee pahaa. zett function niyamit karane kinwa ramanujan yog yasarkhya adhik pragat paddhatinchi aavashyakta aahe. yaa paddhatinshi sambandhit kahee sthool shodhshastracha vapar karnyachya mulyabaddal vaad ghalane dekhil shakya aahe.
shodhshastra.
srinivas ramanujan yanni tyanchya pahilya pustakachya 8 vya adhyayat 'chey done vyutpann sadar kele. khalilpramane, sopee, qamee kathor vyutpatti done payyanmadhye pudhe jaate.
pahili mahattvachi antardrushti ashee aahe qee sakaratmak sankhyanchi malika bariksarik malikeshi jawalun midtejulate. nantarchi malika dekhil bhinn aahe, parantu tyasah kaam karane khoop sope aahe; anek shastriy paddhati aahet jyaa tyala moolya detat, jyaa 18 vya shatkapasoon shodhalya gelya aahet.
malikeche rupantar karnyasathi, ekhadi vyakti dusya termmadhun 4, chauthya termmadhun 8, sahavya termmadhun 12 aanee asech itar vaja karoo shakate. vaja karanyachi ekoon rakkam mhanje mool malikechya 4 putt. heey sambandh bijganitacha vapar karoon vyakt kele jaau shakatat. malikechi "berries" kahihi asso, tyala asey mhana aanee nantar yaa samikaranala 4 neey gunakar qaraa aanee pahilya samikaranatun dusare samikaran vaja karah
dusari mahattvachi antardrushti ashee aahe qee paryayi malika hii karyacha aupcharik shakti malika vistar aahe parantu 'x' chee vyakhya 1 ashee kelee aahe. (heey 'x' chya nakaratmak nasalelya shaktinchya paryayi berijashi samatulya karoon aanee nantar samikaranachya donhi bajunna vegale karoon aanee nakarun pahile jaau shakate.) tyanusar ramanujan lihitat
donhi bajunna −3 neey bhagalyas, ekhadyala "c" = milate.
sarvasadharanpane sangayache tarr, anant malika maryadit berries asalyapramane hatalane chukiche aahe. udaharanarth, jarr bhinn malikechya aniyantrit sthananmadhye shoonye ghatali gelee tarr itar paddhatinshi susangat rahoo dya, tarr swa-susangat nasalelya parinamanvar pohochane shakya aahe. visheshtah, heey paul keval yogatmak olakh kayadyadvare nyayya nahee. eka tokachya udaharanasathi, malikechya samore ekach shoonya jodalyane vegala parinam houu shakato.
yaa paristhitivar upay karnyacha aanee shoonye samavisht karata yeteel asha thikananna maryadit karnyacha eka marg mhanje kahee karyanvar avalambitva jodoon maliketil pratyek sandnyecha magova ghene. yaa maliket, pratyek sandnya "aine" hii fakt eka sankhya aahe. jarr 'aine' yaa sandnyela 'aine' − ess 'yaa karyat padonnati dilee gelee, jethe' ess 'heey eka jatil chull aahe, tarr ekhadi vyakti heey sunishchit karoo shakate qee keval samaan sandnya jodalya gelya aahet. parinami malika adhik kathor paddhatine hatalali jaau shakate aanee chull "s" nantar −1 varr seet kele jaau shakate. yaa dhoranachya amalbajavanila zeeta function regularization mhanatat.
zeeta karya niyamit karane.
zeeta function niyamitkaranamadhye, malika sootra _ 6 chee jaaga malika sootra _ 7 neey ghetali aahe. nantarchi malika hii diriclate malikeche udaharan aahe. jevha 's' chaa vastavik bhag 1 peksha motha asato, tevha diriclate malika ekatra yete aanee tyachi berries reeman zeta function 'ζ' ('s') asate. dusarikade, jevha 's' chaa vastavik bhag 1 peksha qamee kinwa samaan asato tevha diriclate malika vegali hote, mhanoon visheshtah, 's' =-1 seet kelyane udbhavnari malika ektrit hott nahee. reeman zeeta karya sadar karnyacha fayda assa aahe qee teey vishleshanatmak satatyadvare "s" chya itar mulyansathi paribhashit kele jaau shakate. tyanantar ekhadi vyakti 'ζ' (−1) chee zeeta-niyamit berries paribhashit karoo shakate.
yaa tappyapasoon, heey siddha karnyache kahee marg aahet qee eka paddhat, yularchya tarkshastrachya dhartivar, reeman zeta karya aanee dirichlet etaa karya "η" ("s") yanchyatil sambandh vaparte. etaa karya paryayi diriclate malikedware paribhashit kele jaate, mhanoon hii paddhat purvichya huristicsla samantar aahe. jethe donhi diriclate malika ekatra yetat, tithe aekhadyachi olakh asateh
jevha donhi karye vishleshanatmak satatyadvare vadhavali jaatat aanee jyasathi vareel malika bhinn hotat asha "s" chya mulyancha samavesh keela jaato tevha olakh sootra _ 9 tikoon rahate. aata, badalalyas, "η" (−1) chee ganana karane sope kaam aahe, kaaran etaa karya tyachya paribhashit malikechya ebel bereejchya barobariche aahe, jee ektarfi maryada aaheh
donhi bajunna −3 neey bhagalyas, eka milate 
niyamitata bund qaraa.
cutoff functioncha vapar karoon niyamit karanyachi paddhat malika 'suralit' karoo shakate. chikat karane haah jatil vishleshanavar avalambun asalelya zeeta karya niyamitkaran aanee euler-mclaurin sutrasathichya shortcuts ramanujan yogamadhil eka sankalpanatmak pool aahe. tyaeivaji, hii paddhat pratyaksh vishleshanatun paddhati vaparoon, malikechya puranmatvadi parivartananvar thet karya karate.
ayogya vaganuk asalelya _ 11 yaa swatantra malikechya sutrala gulgulit aavruttine badalanyachi kalpana aahe.
jethe "f" heey yogya gunadharmansah cutoff karya aahe. cutoff karya samanya karane aavashyak aahe; heey vibhedak samikarananmadhye vaparlya jaananyapeksha vegale samanyikaran aahe. cutoff functionmadhye maliketil surkutya gulgulit karnyasathi purese bounded derivatives asavet aanee malika vadhanyapeksha teey 0 paryant vegane kshay jhaale pahije. soyisathi, ekhadyala 'eff' gulgulit, baddh aanee sankshiptapane samarthit asane aavashyak asoo shakate. tyanantar ekhadi vyakti heey siddha karoo shakate qee hii gulgulit berries asymptotic aahe, jethe "see" haah sthirank aahe joou "eff" varr avalambun aahe. asymptotic vistarachi sthir sandnya "f" varr avalambun nasateh vishleshanatmak satatyadvare dilele samaan moolya asane aavashyak aahe.
ramanujan saransh.
ramanujanchi berries dekhil aahe. ramanujan yanni tyanchya dusya patrat jee. etch. hardy, dinank 27 februwari 1913:
ramanujan yog hii euler-mclaurin sutratil sthir sandnya malikechya aanshik berijsathi vegale karanyachi eka paddhat aahe. f" yaa karyasathi, malika sootra _ 13 chee shastriy ramanujan berries ashee paribhashit kelee jaateh
jethe "f" (2 "k" −1) heey "f" chey (2 "k" −1)-th vyutpann aahe aanee "b" 2 "k" hii 2 "k"-th bernauli sankhya aaheh,, aanee asech. setting, "f" chey pahile vyutpann 1 aahe, aanee pratyek itar sandnya nahishi hote, mhanoon
visangati talhanyasathi, ramanujan yogachya aadhunik siddhantanusar, 'eff' heey 'niyamit' asane aavashyak aahe, yaa arthane qee 'eff' chey uchch-kramache vyutpann yular-mclaurin sutratil urvarit sandnyansathi 0 paryant jaanyaitke lavkar kshay hotat. ramanujanane maunane haah gunadharm gruheet dharala. niyamitatechi aavashyakta antaral-baherchya malikevar ramanujan yogacha vapar karnyas pratibandhit karate, kaaran konatehi niyamit karya tee moolye ghet nahee. tyaeivaji, asha malikecha arth zita karya niyamit karoon lavala jaane aavashyak aahe. yaa karanastav, sambandhit malikechya berries shodhanyasathi dnyat malikechya ramanujan berries lagoo kartana hardy "khoop savadhgiri" balganyachi shifaras karato.
sthir resheey berries paddhatinche apayash.
resheey aanee sthir asaleli berries paddhat malikechi berries konatyahi maryadit moolyashi karoo shakat nahee. (sthir mhanje malikechya suruvatila sandnya jodalyane jodalya gelelya sandnyechya mulyanusar berries vadhate.) heey khalilpramane pahile jaau shakate. jarr
sootra _ 16
mug donhi bajunna 0 jodalyane milate
sootra _ 17
sthiratene. reshatmakatedvare, dusare samikaran pahilya pasoon vaja kele jaau shakate (dusya olichya pratyek ghatkala stambhatil pahilya olitoon vaja karane)
sootra _ 18
donhi bajunna punha 0 jodalyane milate
sootra _ 19
aanee shevatchya done malikanna vaja kelyane milate
sootra _ 20
sthiratechya viruddha.
mhanoon, bereejala maryadit moolya denari pratyek paddhat sthir kinwa resheey nasate.
bhautikshastra.
bosonic string siddhantat, stringchya sambhavya urja patalichi ganana karnyacha prayatna aahe, visheshtah, sarvat qamee urja patali. anaupacharikpane sangayache tarr, stringchya pratyek harmonikkade "d"-2 swatantra quantum harmonic ausilatorcha sangrah mhanoon pahile jaau shakate, pratyek aadvya dishene eka, jethe "d" heey spacetimeche pariman aahe. jarr moolbhut dolan varanvarta "ω" asel, tarr dolankartyatil urja "n"-th harmonikmadhye yogdan denari "n ω"/2 aahe. mhanoon bhinn malika vaparoon, serve harmonikvaril berries aahe. shevati heey tathya aahe, goddard-konta prameyasah ektrit kele, jyamule bosonic string siddhant 26 vyatirikt itar parimananmadhye susangat rahnyas ayashasvi hoto.
eka parimanatil scalar kshetrasathi casimir shaktichi ganana karnyat dekhil niyameet karane samavisht aahe. malika suralit karnyasathi ghatankiy cutoff karya purese asate, jeey heey darshavate qee vahak platesdware aniyantritpane uchch-urja mode avarodhit kele jaat naheet. samasyechi avkashiy samamiti hii vistarachi dvighat sandnya radd karnyasathi jababdar aahe. fakt sthir sandnya-1/12 shillak aahe aanee yaa nikalache nakaratmak chinh casimir shakti aakarshak aahe heey darshavate.
riman zeta karyachya jaagi epstein zeta-karya vaparoon, ashich ganana teen parimananmadhye samavisht aahe.
itihas.
leonhard yularne yaa malikecha saransh deela qee nahee heey spasht nahee. mauris clinechya matey, yoolarche bhinn malikanvaril suruvatiche kaam karya vistaravar avalambun hote, jyatoon tyane nishkarsh kadhala. raymond ayoobchya matey, bhinn zeeta malika ebel-saranshyogya nahee yaa vastusthitimule ulerla zeeta karyacha iita karyapramane muktapane vapar karnyapasoon rokhale gele aanee too yaa malikeshi "arth jodu shakala nasata". itar lekhakanni ulerla yaa bereejche shrey dile aahe, asey suchavat qee ulerne zita aanee iita functionsmadhil sambandh nakaratmak purnankanparyant vadhavala asata. prathamik sahityat, yoolarchya 1760 chya prakashanat bhinn bhaumitik malikesah yaa malikecha ullekh aahe. euler asey soochit karato qee yaa prakarchya malikanmadhye maryadit, nakaratmak berries asatat aanee bhaumitik malikesathi yacha arth kaay aahe heey too spasht karato, parantu too charcha karnyasathi parat yet nahee. tyach prakashanat, euler lihito qee berries anant aahe.
lokpriy madhyamanmadhye.
david levit yanchya 2007 salchya 'daad indian clerk' yaa kadambarimadhye eka drushya samavisht aahe jyat hardy aanee littlewood yaa malikechya arthachi charcha karatat. ramanujanane 'ζ' (−1) punha shodhun kadhala aahe aanee ramanujan tyanchyabarobar khelat aahe yache lakshan mhanoon teey tyachya dusya patratil 'vede aashray' ola ghetat assa tyancha nishkarsh aahe.
sayman mcburneyche 2007 madheel natak 'aaa discipearing number' heey suruvatichya drushyatil malikevar kendrit aahe. mukhya patra, rooth, eka vyakhyangruhat jaate aanee "mea tumhala kharokharach kahitari romanchak dakhavnar aahe", asey ghoshit karnyapurvi eka bhinn malikechya kalpanechi olakh karoon dete. jevha rooth zett functionchya karyatmak samikaranachya vyutpattimadhye prarambh karato, tevha dusra abhineta prekshakanna sambodhit karato, teey abhinete asalyache kabool karatoh "parantu ganeet vastavik aahe. teey bhayanak aahe, pann teey kharey aahe."
janewari 2014 madhye, numberfilene yaa malikevar eka youtube video tayar keela, jyala tyachya pahilya mahinyat 15 lakhanhun adhik vela pahile gele. 8 minitancha haah video nottingham vidyapithatil bhautikshastradnya toney padila yanni kathan keela aahe. padilachi suruvat ramanujanchya yuktivadapramanech padd-darr-mudatichya vajashi hote aanee nantarchyashi sambandhit aahe. numberfilene nottingham bhautikshastradnya edd copeland yancha samavesh asalelya vhidiochi 21 minitanchi aavrutti dekhil jaaree kelee, joou ebel berries mhanoon aanee "ζ" (−1) mhanoon adhik tapshilvar varnan karato. pahilya vhidiomadhye kathoratechya kamataratebaddal takrari midalyanantar, padilane tyachya vebprushthavar sambandhit dirichlet malikechya vishleshanatmak satatyanmadhil olakhinshi sambandhit videomadhil hatalanishi sambandhit spashtikaran dekhil lihile.
daad newyork times' madhye, numberfile vhidiochya kavharejmadhye, ganitshastradnya edward frankel yanni tippani keleeh "hii ganana ganitatil sarvat changlya gupt goshtinpaiki eka aahe. baherchya konalach yabaddal mahitee nahee."
smithsonian" masikat yaa vishayachya kavharejmadhye numberfile vhidioche varnan dishabhul karnare mhanoon kele aahe aanee asey namood kele aahe qee bereejcha arth vishleshanatmak satatya tantratoon, "samaan" chinhachya vishesh arthavar avalambun asato, jyamadhye "samaan" mhanje "sambandhit aahe".