
ramanujan kaa yog

sankhya siddhant mein, ramanujan kaa yog, jise aamtaur parr "cq" ("n") kahaa jaataa haye, sootra dwara paribhaashit dou sakaratmak purnank charr "q" aur "n" kaa ack falan haye.
jahan ("a", "q") = 1 kaa arth haye kii "a" kewal "q" key liye saha-moolya letaa haye.
srinivas ramanujan ney 1918 key ack lekh mein raashi kaa ullekh kiya thaa. iss lekh mein charcha kiye gaye vistaaron key alawa, ramanujan key yog kaa upayog vinogradov key pramey key pramaan mein kiya gayaa haye kii pratyek paryaapt roop seey badee visham sankhya teen abhaajya kaa yog haye.
sanketan.
purnank "a" aur "b" key liye, sootra _ 2 koo "a" vibhajit kartaa haye "b" padhaa jaataa haye aur iska matlab haye kii ack purnank "c" haye jaise kii sootra _ 3 isi tarah, sootra _ 4 koo padhaa jaataa haye "a" vibhajit naheen kartaa haye "b". yog prateek
iska matlab haye kii "d" "m", e key sabhi sakaratmak bhajakon seey hocker gujarta haye.jee.
sootra _ 7 sabse badaa samanya bhaajak haye,
sootra _ 8 euler kaa totient function haye,
sootra _ 9 mobius falan haye, aur
sootra _ 10 reeman jeeta falan haye.
c" "q" ("n") key liye sootra.
trikonamiti.
yeye sootra paribhasha, euler key sootra sootra _ 11 aur praathamik trikonamitiya pehchaan seey aatey hain.
aur isi tarah (,,,..,...). "cq" ("n") hamesha ack purnank hotaa haye.
cleuver.
maan leejiye sootra 13 taba sameekaran kaa mool haye. apani pratyek shakti,
yeh bhee ack jarh haye. isaliye, choonki unamen seey "q" hain, way sabhi jadein hain. sankhya sootra _ 15 jahan 1 ≤ "n" ≤ "q" koo "q"-ekataa kii jadein kahaa jaataa haye. ise ack adim "q" kahaa jaataa haye-ekataa kaa chautha mool kyonki sootra _ 16 banaane vaalaa "n" kaa sabse chhota maan "q" haye. anya adim "q"-ekataa kii jadein sankhya sootra _ 17 hain jahan ("a", "q") = 1. isaliye, φ ("q") adim "q"-ekataa kii jadein hain.
iss prakaar, ramanujan yog "cq" ("n"), adim "q"-ekataa kii jadon kii "n"-th shaktiyon kaa yog haye.
yeh ack tathya haye kii "q" key sabhi bhajakon key liye shaktiyaan sateek roop seey adim mool hain.
udaaharan key liye. maan leejiye "q" = 12. tou
isaliye aggar
sabhi jadon kii "aine"-veen shaktiyon kaa yog haye, joe adim aur dosharahit haye,
aur mobius vyutkram dwara,
yeh pehchaan "x" q "−1 = (" x "−1) (" x "q" −1 + "x" q "−2 + seey anusarega... + "x" + 1) kii
aur yeh sootra kii or leey jaataa haye
1906 mein cleuver dwara prakaashit.
yeh darshaata haye kii "c" "q" ("n") hamesha ack purnank hotaa haye. isaki tulanaa sootra seey karein
wonn sternack.
yeh aasaani seey paribhasha seey dikhaaya jaataa haye kii "c" "q" ("n") gunan haye jaba "n": i key nishchit moolya key liye "q" key falan key roop mein maanaa jaataa haye.ee.
paribhasha (yaa cleuver kaa sootra) seey yeh saabit karna seedha haye kii, yadi "pee" ack abhaajya sankhya haye,
aur yadi "pee" "key" ack pramukh shakti haye jahan "key"> 1,
iss parinam aur gunak guna kaa upayog saabit karane key liye kiya jaa sakata haye
ise wonn sternak kaa ankganitiya falan kahaa jaataa haye. isaki samaantaa aur ramanujan kaa yog holder key kaaran haye.
c" "q" ("n") key anya guna. 
sabhi sakaratmak purnankon key liye "q",
q" key ack nishchit moolya key liye anukram sootra kaa nirpeksh moolya _ 39 φ ("q") seey ghiraa huaa haye, aur "n" key ack nishchit moolya key liye anukram sootra kaa nirpeksh moolya _ 40 "n" seey ghiraa huaa haye.
yadi "q"> 1
maan leejiye "m" 1 ", m" 2> 0 ", m" = lcm ("m" 1 ", m" 2). taba ramanujan key yog ack orthogonality guna koo santusht karate hainh
maan leejiye "n", "k"> 0. phir
jise bahadur-redmaker pehchaan key roop mein jaanaa jaataa haye.
yadi "n"> 0 aur "a" koee purnank haye, tou hamaare paas bhee haye
kohain key kaaran.
ramanujan vistaar.
yadi "f" ("n") ack ankganitiya falan haye (i.ee. purnank yaa praakritik sankhyaon kaa ack jatil-moolya vaalaa kaary), phir roop kii ack abhisari anant shrinkhalaah
yaa roop seh
jahan, "f" ("n") kaa ramanujan vistaar kahaa jaataa haye.
ramanujan ney sankhya siddhant key kuchh prasiddh karyon key vistaar paae. yeye sabhi parinam "praathamik" tareeke seey saabit hotey hain (i.ee. kewal shrinkhala key aupchaarik herfer aur abhisaran key baare mein sabse saral parinamon kaa upayog karna).
shoonya falan kaa vistaar abhaajya sankhyaon key vishleshanatmak siddhant key parinam parr nirbhar kartaa haye, arthaat kii shrinkhala
0 mein parivartit hotaa haye, aur "r" ("n") aur "r" ("n") key parinam pahale key pepper mein pramey parr nirbhar karate hain.
iss khand key sabhi sootra ramanujan key 1918 key lekh seey hain. 
kaary utpann karna.
ramanujan yog key utpann karane vaale kaary dirichlet shrinkhala hainh
cq" (1), "cq" (2), anukram key liye ack utpann karane vaalaa falan haye... jahan "q" sthir rakhaa jaataa haye, aur
c" 1 ("n"), "c" 2 ("n"), anukram key liye ack utpann karane vaalaa falan haye... jahan "n" sthir rakhaa jaataa haye.
double dirichlet shrinkhala bhee haye.
σ "k" ("n").
σ "k" ("n") bhaajak falan (i) haye.ee. n" key bhajakon kii "k"-th shaktiyon kaa yog, jisamen 1 aur "n" shamil hain. σ0 ("n"), "n" key bhajakon kii sankhya, aamtaur parr "d" ("n") aur σ1 ("n") likhi jaati haye, "n" key bhajakon kaa yog, aamtaur parr σ ("n") likhaa jaataa haye.
yadi "s"> 0,
setting "s" = 1 detaa haye
yadi reeman parikalpana sahee haye, aur sootra _ 53
"dee" ("aine").
"d" ("n") = σ0 ("n") "n" key bhajakon kii sankhya haye, jisamen 1 aur "n" shamil hain.
jahan γ = 0.5772.. euler-masteuroni sthiraank haye.
"φ" ("n").
euler kaa totient falan φ ("n") "n" seey cumm dhanaatmak purnankon kii sankhya haye aur "n" key liye saha-samaank haye. ramanujan isake saamaanyikaran koo paribhaashit karate hain, yadi
n" kaa abhaajya gunankhandan haye, aur "s" ack jatil sankhya haye, maan leejiye
taaki "φ" 1 ("n") = "φ" ("n") euler kaa falan hoe.
vaha saabit kartaa haye kii
aur iska upayog yeh dikhaane key liye kartaa haye kii
s" = 1,
dhyaan dein kii sthiraank σ ("n") key sootra mein ack kaa vyutkram haye.
λ ("n").
wonn mangolt kaa falan jaba taka kii "n" = "pk" ack abhaajya sankhya kii shakti naheen haye, iss mamle mein yeh praakritik laghuganak logg "p" haye.
shoonya.
sabhi "aine"> 0 key liye,
yeh abhaajya sankhya pramey key barabar haye.
"r" 2 "" s "(" "n" ") (vargon kaa yog).
"r" "2" "s" "(" "n" ") 2" "s" "vargon key yog key roop mein" "n" "kaa pratinidhitva karane key tareeke kii sankhya haye, vibhinn aadeshon aur sanketon koo alag-alag (e".jee., r" 2 (13) = 8,13 = (±2) 2 + (±3) 2 = (±3) 2 + (±2) 2.)
ramanujan ack falan δ2 "s" ("n") koo paribhaashit karate hain aur ack pepper key sandarbh mein bataate hain jisamen unhone saabit kiya kii "s" = 1,2,3 aur 4 key liye "r" 2 "s" ("n") = δ2 "s" ("n"). "s"> 4 key liye vaha darshaata haye kii δ2 "s" ("n") "r" 2 "s" ("n") kaa ack achchha anuman haye.
"s" = 1 kaa ack vishesh sootra haih
nimnalikhit sootron mein sanket 4 kii avadhi key saath dohrate hain.
aur isaliye,
sootra _ 65 (trikonon kaa yog).
sootra _ 65 una tarikon kii sankhya haye jinhen "n" koo 2 "s" trikoniya sankhyaon (i.) key yog key roop mein darshaaya jaa sakata haye.ee. sankhyaaen 1,3 = 1 + 2,6 = 1 + 2 + 3,10 = 1 + 2 + 3 + 4,15,...n"-th trikoniya sankhya sootra "n" ("n" + 1)/2 dwara dee gaee haye.)
yahaan vishleshan vargon key liye samaan haye. ramanujan usee kagaj koo sandarbhit karate hain jaisa unhone vargon key liye kiya thaa, jahan unhone dikhaaya kii ack falan sootra _ 67 haye jaise kii sootra _ 68 "s" = 1,2,3 aur 4 key liye, aur "s"> 4 key liye, sootra _ 67 sootra _ 70 kaa ack achchha anuman haye.
phir seey, "s" = 1 key liye ack vishesh sootra kii aavashyakta hoti haih
yadi "s" 4 kaa gunak haye,
isaliye,
raashi.
jaane dou.
taba key liye,

