
purnataya gunak falan

sankhya siddhant mein, sakaratmak purnank key kaary joe utpaadon kaa sammaan karate hain, mahatvapurna hain aur jinhen poori tarah seey gunak kaary yaa poori tarah seey gunak kaary kahaa jaataa haye. kewal saha-maulik sankhya key utpaadon kaa sammaan karate hue ack kamjor sthiti bhee mahatvapurna haye, aur aise karyon koo gunak kaary kahaa jaataa haye. sankhya siddhant key baahar, "gunak falan" shabd koo aksar "poori tarah seey gunak falan" kaa paryaay maanaa jaataa haye jaisa kii iss lekh mein paribhaashit kiya gayaa haye.
paribhasha.
ack purnataya gunak falan (yaa purnataya gunak falan) ack ankganitiya falan haye (arthaat, ack falan jiska kshetra praakritik sankhya haye), jaise kii "f" (1) = 1 aur "f" ("a") = "f" ("b") sabhi "dhanaatmak purnank" a "aur" b "key liye" dhaaran kartaa haye.
f" (1) = 1 kii aavashyakta key binaa, kissy key paas abhee bhee "f" (1) = 0 hoe sakata haye, lekin phir sabhi sakaratmak purnank "a" key liye "f" ("a") = 0 hoe sakata haye, isaliye yeh bahut majboot pratibandh naheen haye.
uparokt paribhasha koo beejganit kii bhasha kaa upayog karke phir seey prastut kiya jaa sakata haih ack poori tarah seey gunak falan ack samroopata haye joe ack sootra _ 1 (yaani, gunan key tahat sakaratmak purnank) seey kissy anya ekaatmak mein haye.
udaaharan.
purnataya gunak falan kaa sabse aasaan udaaharan agrani gunank 1 key saath ack ekmusht haih kissy vishesh sakaratmak purnank "n" key liye, "f" ("a") = "a" n "koo paribhaashit karein. taba "f" ("bc") = ("bc") "n" = "b" "n" "c" "n" = "f" ("b") "f" ("c"), aur "f" (1) = 1 "n" = 1.
leuville falan poori tarah seey gunak falan kaa ack gair-tuchchh udaaharan haye kyonki dirichlet varna, jacoby prateek aur kinvadanti prateek hain.
guna.
ack poori tarah seey gunak falan poori tarah seey abhaajya sankhyaon parr isake moolyon dwara nirdhaarit kiya jaataa haye, joe ankganit kii maulik pramey kaa parinam haye. iss prakaar, yadi "n" vishisht abhaajya rashiyon kii shaktiyon kaa utpaad haye, tou kahein kii "n" = "p" a "" q "b"..., phir "f" ("n") = "f" ("p") "a" "f" ("q") "b"...
jabaki dou gunak karyon kaa dirichlet convolution gunak haye, dou poori tarah seey gunak karyon kaa dirichlet convolution poori tarah seey gunak honey kii aavashyakta naheen haye.
ack falan key baare mein vibhinn prakaar key kathan hain joe isake poori tarah seey gunan key barabar hain. udaaharan key liye, yadi koee falan "f" gunak haye tou yeh poori tarah seey gunak haye yadi aur kewal tabhi jaba iska dirichlet vyutkram sootra _ 2 haye jahan sootra _ 3 mobius falan haye.
poori tarah seey gunak kaary bhee ack vitaran kaanoon koo santusht karate hain. yadi "f" poori tarah seey gunansheel haye tou
sootra _ 4
jahan "*" dirichlet utpaad kaa pratinidhitva kartaa haye aur sootra _ 5 bindu kii disha mein gunan kaa pratinidhitva kartaa haye. iska ack parinam yeh haye kii kissy bhee poorna roop seey gunak falan "f" key liye ack haye
sootra _ 6
jise donon sootra _ 7 lagaakar uparokt seey nikala jaa sakata haye, jahan sootra _ 8 sthir falan haye.
yahaan sootra _ 9 vibhaajak falan haye.
diriclet shrinkhala.
purnataya (yaa purnataya) gunak dirichlet shrinkhala sootra kaa ell-falan _ 11 santusht kartaa haye
jiska arth haye kii praakritik sankhyaon parr yog sabhi abhaajya sankhyaon parr utpaad key barabar haye.