1, 3, 5, 7, 9,... 2, 4, 6, 8, 10,... 1, 4, 7, 10, 13,... 2, 5, 8, 11, 14,... 3, 6, 9, 12, 15,...

Transcript

1 To Je rhma tou Dirichlet Dèspoina NÐka IoÔlioc 999 Majhmatikì Tm ma Panepist mio Kr thc

2 2

3 Prìlogoc Oi pr toi arijmoð, 2, 3, 5, 7,,..., eðnai ekeðnoi oi fusikoð arijmoð oi opoðoi èqoun akrib c dôo diairètec, ton eautì touc kai th mon da. Oi arijmoð autoð jewroôntai jemèlioi lðjoi thc Arijmhtik c kurðwc lìgw tou Jemeli douc Jewr matoc thc Arijmhtik c K je fusikìc arijmìc megalôteroc thc mon dac gr fetai me monadikì trìpo san ginìmeno pr twn arijm n. 'Ena apì ta kentrik zht mata thc JewrÐac Arijm n eðnai h melèth thc katanom c twn pr twn arijm n an mesa stouc fusikoôc arijmoôc. To pr to kai pasðgnwsto apotèlesma se autì to plaðsio eðnai tou EukleÐdh apì thn arqaiìthta Up rqoun peiroi pr toi arijmoð. H apìdeixh apoteleð upìdeigma suntomðac kai komyìthtac: èstw ìti to pl joc ìlwn twn pr twn arijm n eðnai peperasmèno. SqhmatÐzoume to ginìmenì touc kai prosjètoume mia mon da sto apotèlesma. O arijmìc pou prokôptei de diaireðtai apì kanènan apì touc pr touc arijmoôc pou ton sqhm tisan kai, epomènwc, den èqei apomeðnei lloc pr toc arijmìc gi na ton diaireð. Autì antif skei me to Jemeli dec Je rhma. To sônolo twn fusik n arijm n qwrðzetai se dôo arijmhtikèc proìdouc me b ma 2, thn arijmhtik prìodo twn peritt n arijm n, 3, 5, 7, 9,... kai thn arijmhtik prìodo twn rtiwn arijm n 2, 4, 6, 8, 0,.... EÐnai profanèc ìti, ektìc tou 2, kanènac lloc rtioc arijmìc den eðnai pr toc arijmìc: ìloi diairoôntai apì to 2. 'Ara oi peiroi pr toi arijmoð perièqontai, ektìc tou 2, ìloi sthn arijmhtik prìodo twn peritt n arijm n. Ta epiqeir mata duskoleôoun an qwrðsoume to sônolo twn fusik n arijm n stic treðc arijmhtikèc proìdouc me b ma 3, 4, 7, 0, 3,... 2, 5, 8,, 4,... 3, 6, 9, 2, 5,....

4 2 HtrÐth, ektìc tou 3, den perièqei kanènan pr to arijmì kai, epomènwc ìloi oi pr toi arijmoð moir zontai an mesa stic dôo llec. All den eðnai profanèc an kai oi dôo perièqoun peirouc pr touc arijmoôc, m pwc, k poia apì tic dôo perièqei peperasmèno pl joc apì autoôc. Gia stoiqei deic lôseic autoô, all kai llwn parìmoiwn problhm twn, o endiaferìmenoc anagn sthc mporeð na anatrèxei sto biblðo An Introduction to the Theory of Numbers twn Hardy kai Wright. To prìblhma eðnai, fusik, genikìtero. 'Estw b ma m kai oi, m sto pl joc, arijmhtikèc prìodoi stic opoðec qwrðzetai to sônolo twn fusik n arijm n a, a + m, a +2m, a +3m,... ìpou to a diatrèqei touc arijmoôc, 2,..., m. EÐnai profanèc ìti, an o a kai o m èqoun koinì diairèth megalôtero thc mon dac, tìte autìc o diairèthc diaireð k je ìro thc arijmhtik c proìdou kai, epomènwc, kanènac apì touc ìrouc thc, ektìc Ðswc tou Ðdiou tou a, den eðnai pr toc arijmìc. Opìte genniètai to er thma poièc apì tic upìloipec arijmhtikèc proìdouc, dhlad gia tic opoðec o antðstoiqoc a eðnai sqetik pr toc me ton m, perièqoun peirouc pr touc arijmoôc. H ergasða aut asqoleðtai me to di shmo Je rhma tou Dirichlet K je arijmhtik prìodoc a, a+m, a+2m,..., me a, m na eðnai sqetik pr toi, perièqei peirouc pr touc arijmoôc. To apotèlesma autì apodeðqjhke gia pr th for apì ton Dirichlet to 837 kai katìpin apodeðqjhke kai apì llouc majhmatikoôc me diaforetikoôc trìpouc apìdeixhc: Mertens 897), Selberg 949), Zassenhaus 949). Sthn ergasða aut ja parousi soume me k je leptomèreia thn apìdeixh tou Jewr matoc tou Dirichlet basismènoi ston trìpo parousðashc pou ektðjetai sto biblðo Introduction to Analytic Number Theory tou K. Chandrasekharan. Ja prèpei na tonisjeð ìti sth sugkekrimènh apìdeixh dðnetai idiaðterh èmfash stic mejìdouc thc migadik c an lushc. 'Oson afor stic apaitoômenec gn seic gia thn an gnwsh thc ergasðac mac autèc kalôptontai apì ta sun jh proptuqiak maj mata enìc majhmatikoô tm matoc. Opoiad pote apotelèsmata xefeôgoun apì ta kajierwmèna ektðjentai kai apodeiknôontai sthn Eisagwg thc ergasðac. Eidik tera, deqìmaste ta ex c.. Apì th JewrÐa Arijm n: Touc pr touc arijmoôc touc opoðouc, se aôxousa di taxh, sumbolðzoume p,p 2,p 3,.... To Jemeli dec Je rhma thc Arijmhtik c. Thn ènnoia tou mègistou koinoô diairèth gcda, b) dôo arijm n. Thn ènnoia twn isoôpoloðpwn mod m me to sqetikì sumbolismì a bmodm. Tic antðstoiqec kl seic isoôpoloðpwn [], [2],..., [m] kai p c tic prosjètoume kai tic pollaplasi zoume. Th sun rthsh tou Euler ϕm). To Je rhma twn Fermat - Euler. 2. Apì th JewrÐa Om dwn: Thn ènnoia thc t xhc miac om dac kai thn tautìthta a m = e gi ta stoiqeða a miac om dac, thn t xh thc m kai to monadiaðo stoiqeðo thc e. To ìti ìtan

5 3 a eðnai èna stoiqeðo mi c peperasmènhc om dac kai to b diatrèqei mða for ìla ta stoiqeða thc, tìte to ab diatrèqei mða for ìla ta stoiqeða thc. Thn ènnoia thc abelian c om dac kai to par deigma G m = {[a] : a m, gcda, m) = }. Thn ènnoia thc kuklik c om dac. To Jemeli dec Je rhma twn Peperasmènwn Abelian n Om dwn: k je peperasmènh abelian om da eðnai eujô ginìmeno kuklik n upoom dwn thc. 3. Apì ton Proqwrhmèno Apeirostikì Logismì: 'Ola ta basik gia akoloujðec, ìria kai sunèqeia sunart sewn, parag gish, olokl rwsh suneq n sunart sewn. Stoiqei deic idiìthtec arijmhtik n seir n kai eidik ìti h apìluth sôgklish sunep getai th sôgklish mi c seir c. Basikèc idiìthtec seir n sunart sewn: omoiìmorfh sôgklish kai h sqèsh thc me th sunèqeia, to krit rio tou Cauchy gia omoiìmorfh sôgklish kai to M test tou Weierstrass. Ta sômbola [x] kai {x} gia to akèraio mèroc kai to klasmatikì mèroc. Thn ènnoia tou genikeumènou oloklhr matoc + ft)dt kai to ìti h a apìluth sôgklis tou sunep getai th sôgklis tou. To Je rhma Mèshc Tim c gia sunart seic dôo metablht n. 4. Apì th Migadik An lush: Thn ènnoia thc m ost c rðzac thc mon dac. Thn ènnoia tou sumpagoôc uposunìlou tou migadikoô epipèdou C se sqèsh me anoiktèc kalôyeic kai me upo-akoloujðec. Thn ènnoia tou anoiktoô kai sunektikoô uposunìlou tou C kai eidik ìti: an Ω eðnai anoiktì kai sunektikì uposônolo tou C kai Ω = A B, ìpouta A, B eðnai anoikt kai xèna metaxô touc, tìte èna apì aut eðnai kenì. Thn ènnoia thc analutikìthtac kai tic exis seic Cauchy-Riemann. Thn anapar stash mi c analutik c sun rthshc san seir -Taylor sto megalôtero dðsko pou perièqetai sto sônolo analutikìtht c thc kai me dosmèno kèntro mèsa sto sônolo autì. Ta Jewr mata twn Cauchy, Morera kaj c kai touc TÔpouc tou Cauchy gia tic parag gouc mi c analutik c sun rthshc. Thn ènnoia tou pìlou, thc t xhc enìc pìlou kai tou residue mi c sun rthshc se ènan pìlo thc. Thn ènnoia thc merìmorfhc sun rthshc kai to ìti jroisma kai ginìmeno merìmorfwn sunart sewn eðnai merìmorfec sunart seic. Thn epitrop gia thn krðsh aut c thc diplwmatik c ergasðac apetèlesan oi M. Papadhmhtr khc, epiblèpwn kajhght c I. Antwni dhc E. Katsoprin khc touc opoðouc euqarist. Dèspoina NÐka IoÔlioc 999

6 4

7 Kef laio Eisagwg. AkoloujÐec kai Seirèc Analutik n Sunart sewn Je rhma. 'Estw Ω anoiktì uposônolo tou C kai akoloujða {f n } analutik n sunart sewn sto Ω. An f n f omoiìmorfa se k je sumpagèc uposônolo tou Ω, tìte hf eðnai analutik sto Ω kai f n k) f k) omoiìmorfa se k je sumpagèc uposônolo tou Ω gia opoiod pote k N. Apìdeixh: AfoÔ k je f n eðnai suneq c sto Ω, sunep getai ìti h f eðnai suneq c se k je sumpagèc uposônolo tou Ω. An p roume tuqìn s Ω kai δ>0 ste o kleistìc dðskoc s; δ) na perièqetai sto Ω, tìte h f eðnai suneq c sto dðsko kai, epomènwc, eðnai suneq c sto s. 'Ara h f eðnai suneq c se k je shmeðo tou Ω. JewroÔme tuqoôsa trigwnik kampôlh γ, thc opoðac h eikìna γ mazð me to eswterikì thc trðgwno perièqetai sto Ω. Tìte, apì to Je rhma tou Cauchy sunep getai ìti gia k je n isqôei f n s)ds =0. Lìgw omoiìmorfhc sôgklishc sto sumpagèc γ,èqoume f n s)ds fs)ds f n s) fs) ds γ γ γ γ max s γ f n s) fs) m kocγ) 0 ìtan n +. 'Ara γ fs)ds =0 5

8 6 KEF ALAIO. EISAGWG H kai, epeid h f eðnai suneq c sto Ω, sunep getai apì to Je rhma tou Morera ìti h f eðnai analutik sto Ω. Katìpin, jewroôme tuqìnta kleistì dðsko s 0 ; δ 0 ) uposônolo tou Ω. An sumbolðsoume Πs 0 ; δ 0 ) thn perifèreia tou s 0 ; δ 0 ), tìte apì ton tôpo tou Cauchy èqoume ìti gia k je n kai gia k je k isqôei f n k) k! f n ς) s) = dς 2πi Πs 0;δ 0) ς s) k+ kai f k) s) = k! fς) dς 2πi Πs 0;δ 0) ς s) k+ gia k je s ston anoiktì dðsko s 0 ; δ 0 ). PeriorÐzoume, t ra, to s ston anoiktì dðsko s 0 ; 2 δ 0) kai èqoume 'Ara f k) n s) f k) s) = k! 2πi k! 2π k! 2π = 2k k! δ k 0 Πs 0;δ 0) Πs 0;δ 0) max s s0; 2 δ0) f k) n s) f k) s) 2k k! δ k 0 f n ς) fς) ς s) k+ dς f n ς) fς) ς s k+ dς max ς Πs0;δ 0) f n ς) fς) 2 δ 0) k+ 2π 2 δ 0 max ς Πs0;δ 0) f n ς) fς). max ς Πs0;δ 0) f n ς) fς). Lìgw omoiìmorfhc sôgklishc thc {f n } sthn f sto sumpagèc Πs 0 ; δ 0 ), sunep getai ìti h {f n k) } sugklðnei sthn f k) omoiìmorfa sto s 0 ; 2 δ 0). 'Eqoume, loipìn, apodeðxei ìti gia k je shmeðo tou Ω up rqei dðskoc me kèntro to shmeðo autì ston opoðon h{f n k) } sugklðnei omoiìmorfa sthn f k). 'Estw, t ra, tuqìn sumpagèc K uposônolo tou Ω. Gia k je s K up rqei s; δ) ìpou h {f n k) } sugklðnei omoiìmorfa sthn f k). To K kalôptetai apì ta anoikt sônola s; δ) kaj c to s diatrèqei to K kai, epeid to K eðnai sumpagèc, up rqoun peperasmèna s,..., s m ste to K na perièqetai sthn ènwsh m j= s j; δ j ). Tìte max s K f n k) s) f k) s) max j m max s sj;δ j) f n k) s) f k) s), opìte h {f k) n } sugklðnei sthn f k) omoiìmorfa sto K.

9 .2. APEIRO-GIN OMENA MIGADIK WN ARIJM WN 7 Je rhma.2 'Estw anoiktì uposônolo Ω tou C kai {f n } akoloujða sunart sewn analutik n sto Ω. An h seir + f n sugklðnei sthn f omoiìmorfa se k je sumpagèc uposônolo tou Ω, tìte h f eðnai analutik sto Ω kai gia k je k N isqôei ìti h seir + f n k) sugklðnei sthn f k) omoiìmorfa se k je sumpagèc uposônolo tou Ω. Apìdeixh: Efarmìzoume to prohgoômeno Je rhma. sthn akoloujða sunart sewn {s n }, ìpou s n = f + + f n. K je s n eðnai analutik sto Ω kai h {s n } sugklðnei sthn f omoiìmorfa se k je sumpagèc uposônolo tou Ω. 'Ara h f eðnai analutik sto Ω kai h {s k) n } sugklðnei sthn f k) omoiìmorfa se k je sumpagèc uposônolo tou Ω. 'Omwc s k) n = f k) + + f n k) kai h apìdeixh eðnai pl rhc. H piì sunhjismènh genik perðptwsh efarmog c tou teleutaðou Jewr matoc.2 eðnai se sunduasmì me to polô gnwstì Je rhma.3 To M-test tou Weierstrass ) 'Estw akoloujða migadik n sunart sewn {f n } orismènwn seènasônolo A kai èstw akoloujða arijm n M n ste na isqôei f n a) M n gia k je n N kai gia k je a A. An h seir arijm n + M n sugklðnei, tìte h seir sunart sewn + f n sugklðnei se k poia sun rthsh omoiìmorfa sto A. H apìdeixh jewreðtai gnwst..2 Apeiro-ginìmena Migadik n Arijm n Onom zoume apeiro- Orismìc. 'Estw {z n } akoloujða migadik n arijm n. ginìmeno megenikììroz n tosômbolo + z n z z 2. Lème ìti to apeiro-ginìmeno sugklðnei sto migadikì arijmì P,an sumbaðnoun ta akìlouja. up rqei N ste z n 0gia k je n N kai 2. h akoloujða merik n ginomènwn z N,z N z N+,z N z N+ z N+2,... sugklðnei se k poio Q kai 3. to Q eðnai diaforetikì apì to 0 kai to kai 4. P = z z N Q. An k poio apì ta, 2, 3 den isqôei, tìte lème ìti to apeiro-ginìmeno apoklðnei. Lème ìti to apeiro-ginìmeno apoklðnei sto 0 sto, an isqôoun ta, 2, all to Q eðnai Ðso me 0 antðstoiqa.

10 8 KEF ALAIO. EISAGWG H Parat rhsh An to apeiro-ginìmeno eðte sugklðnei eðte apoklðnei sto 0 sto, tìte sunep getai ìti to polô peperasmèno pl joc apì ta z n mporeð na eðnai Ðsa me 0. EpÐshc, an to apeiro-ginìmeno sugklðnei se arijmì diaforetikì apì to 0, tìte sunep getai ìti kanèna apì ta z n den eðnai Ðso me to 0. Kai antistrìfwc, an to apeiro-ginìmeno sugklðnei kai kanèna apì ta z n den eðnai Ðso me to 0, tìte to apeiro-ginìmeno sugklðnei se arijmì diaforetikì apì to 0. Prìtash. An to + z n sugklðnei, tìte z n. Apìdeixh: QrhsimopoioÔme to sumbolismì tou orismoô kai jètoume Q N = z N,Q N+ = z N z N+,Q N+2 = z N z N+ z N+2,... Tìte Q n Q kai, epeid Q 0, sunep getai ìti z n = Q n Q n Q Q =. ApoteleÐ par dosh na qrhsimopoieðtai gia to n ostì ìro o sumbolismìc +a n,antðtou z n,opìtehprìtash. lèei ìti an to + + a n) sugklðnei, tìte a n 0. Je rhma.4 An h seir + a n sugklðnei apolôtwc, tìte to + + a n) sugklðnei. Apìdeixh: Epeid h seir + a n sugklðnei, sunep getai ìti a n 0 kai, epomènwc, to polô peperasmèno pl joc apì ta +a n mporeð na eðnai Ðsa me to 0. PerÐptwsh. + a n <. Tìte a n < gia k je n, opìte kanèna +a n den eðnai Ðso me 0. Jètoume P n =+a ) + a n ) kai ja deðxoume ìti h akoloujða {P n } sugklðnei se migadikì arijmì diaforetikì apì to 0. 'Eqoume ìti P n + a ) + a n ) exp a + + a n ) e. opìte EpÐshc, P n+ P n = P n + a n+ ) P n = P n a n+, P n+ P n e a n+.

11 .3. H ARQ H THS ANALUTIK HS SUN EQISHS 9 Epeid h + a n+ sugklðnei, sunep getai ìti h P + + P n+ P n ) sugklðnei. An P eðnai to jroisma thc seir c aut c, tìte P n = P +P 2 P )+ +P n P n ) P kai apomènei na deðxoume ìti P 0. 'Omwc P n a ) a n ) a + + a n ) a k = α>0. O α eðnai anex rthtoc tou n, opìte, an af soume to n na teðnei sto +, paðrnoume P α>0 kai, epomènwc, P 0. PerÐptwsh 2. + a n. Up rqei N ste a n <. n=n Tìte, apì thn pr th perðptwsh paðrnoume ìti to + n=n + a n) sugklðnei se migadikì arijmì diaforetikì apì to 0. 'Ara, b sei tou orismoô, to + + a n) sugklðnei. k=.3 H Arq thc Analutik c Sunèqishc Je rhma.5 'Estw f analutik se anoiktì kai sunektikì uposônolo Ω tou C. An up rqei akoloujða {s n } sto Ω kai s Ω me s n s kai s n s gia k je n, ste fs n )=0gia k je n, tìte hf tautðzetai me th mhdenik sun rthsh sto Ω. Me llh diatôpwsh: an oi rðzec thc f èqoun shmeðo suss reushc sto Ω, tìte h f mhdenðzetai pantoô sto Ω. Apìdeixh: A). 'Estw opoiod pote s Ω me thn idiìthta: up rqei {s n } sto Ω me s n s, s n s gia k je n kai fs n )=0gia k je n. PaÐrnoume δ>0 ste s; δ) Ω kai gr foume th seir -Taylor thc f sto dðsko autìn: fη) =a 0 + a η s)+a 2 η s) 2 +. 'Estw ìti den eðnai ìla ta a n Ðsa me to 0. Tìte up rqei k poio el qisto N ste a N 0. Dhlad h seir gðnetai fη) = a N η s) N + a N+ η s) N+ + = η s) N [a N + a N+ η s)+ ]=η s) N gη),

12 0 KEF ALAIO. EISAGWG H ìpou h sun rthsh g orðzetai sto dðsko s; δ) apì th dunamoseir gη) =a N + a N+ η s)+ kai, epomènwc, eðnai analutik ekeð. Epeid s n s, sunep getai ìti, gia meg la n, to s n perièqetai ston s; δ), opìte gia meg la n. 'Ara, afoô s n s, 0=fs n )=s n s) N gs n ) gs n )=0 gia meg la n, opìte, lìgw sunèqeiac thc g sto s 0 = lim gs n )=gs) =a N kai katal goume se topo. 'Ara ìla ta a n eðnai Ðsa me to 0, opìte h f mhdenðzetai pantoô ston s; δ). B). QwrÐzoume to Ω sto uposônolo A = {s Ω:h f mhdenðzetai se k poio dðsko me kèntro to s} kai sto sumplhrwmatikì B =Ω\ A. To A eðnai, profan c, anoiktì sônolo. 'Estw ìti to B den eðnai anoiktì. Tìte up rqei s B ste kanènac dðskoc me kèntro s na mhn perièqetai sto B. Epeid k poioc dðskoc me kèntro s perièqetai sto Ω, sunep getai ìti k je dðskoc me kèntro s tèmnei to A. 'Ara up rqei akoloujða Tìte {s n } sto A me s n s. s n s kai fs n )=0 gia k je n kai apì to mèroc A) sunep getai ìti s A. Autì eðnai topo, opìte to B eðnai anoiktì sônolo. Epeid to Ω eðnai sunektikì, k poio apì ta A, B eðnai kenì. Apì thn upìjesh tou jewr matoc kai apì to apotèlesma tou mèrouc A) sunep getai ìti to A den eðnai kenì, opìte Ω=A. 'Ara h f mhdenðzetai pantoô sto Ω. Je rhma.6 'Estw anoiktì kai sunektikì uposônolo Ω tou C kai f analutik sto Ω. An h f mhdenðzetai se k je shmeðo enìc eujôgrammou tm matoc enìc anoiktoô dðskou sto Ω, tìte h f mhdenðzetai pantoô sto Ω. Apìdeixh: 'Amesh sunèpeia tou Jewr matoc.5.

13 .4. ANALUTIK ES SUNART HSEIS POU OR IZONTAI AP O OLOKLHR WMATA.4 Analutikèc Sunart seic pou OrÐzontai apì Oloklhr mata Je rhma.7 'Estw Ω anoiktì uposônolo tou C kai f :[a, b] Ω C suneq c sto [a, b] Ω. Tìte orðzetai h sto Ω kai eðnai suneq c sto Ω. F s) = b a ft, s)dt Apìdeixh: To olokl rwma eðnai kal orismèno diìti h f,s) eðnai suneq c sto [a, b] gia k je s Ω. PaÐrnoume tuqìn s 0 Ω kai ja deðxoume ìti h F eðnai suneq c sto s 0. JewroÔme mèsa sto Ω ènan kleistì dðsko me kèntro to s 0. H f eðnai suneq c sto sumpagèc [a, b], opìte eðnai omoiìmorfa suneq c ekeð. 'Ara, gia k je ɛ>0 up rqei δ>0 ste ft,s ) ft,s ) <ɛ gia k je t,t [a, b] kai s,s me t t <δkai s s <δ. Tìte, gia k je s,s me s s <δ,èqoume F s ) F s ) = b a [ft, s ) ft, s )]dt b a ft, s ) ft, s ) dt b a)ɛ. 'Ara h F eðnai suneq c sto kai, epomènwc, eðnai suneq c sto s 0. Je rhma.8 'Estw Ω anoiktì uposônolo tou C kai f :[a, b] Ω C ste. h f eðnai suneq c sto [a, b] Ω, 2. gia k je t [a, b] h ft, ) eðnai analutik sto Ω kai 3. hmigadik ) par gwgoc f s eðnai suneq c sto [a, b] Ω. Tìte hf s) = b ft, s)dt eðnai analutik sto Ω kai a gia k je s Ω. F s) = b a f s t, s)dt ìti Apìdeixh: PaÐrnoume tuqìn s 0 Ω kai ja deðxoume ìti h F eðnai analutik sto s 0 kai F s 0 )= b a f s t, s 0 )dt.

14 2 KEF ALAIO. EISAGWG H JewroÔme th sun rthsh g :[a, b] Ω C pou orðzetai me ton tôpo { ft,s) ft,s0) gt, s) = s s 0, an s s 0 f s t, s 0 ), an s = s 0. An jewr soume shmeðo t,s ) me s s 0, tìte, epeid h f eðnai suneq c, èqoume ft, s) ft, s 0 ) lim gt, s) = lim = ft,s ) ft,s 0 ) = gt,s ). t t,s s t t,s s s s 0 s s 0 'Estw shmeðo t 0,s 0 ). Ja apodeðxoume ìti lim gt, s) =gt 0,s 0 ), t t 0,s s 0 all o prohgoômenoc upologismìc den isqôei, opìte k noume to ex c. Gr foume f = u + iv kai s = σ + iτ, opìte apì tic exis seic Cauchy-Riemann : Tìte f s t, ) =u σ t, )+iv σ t, ) =v τ t, ) iu τ t, ). gt, s) gt 0,s 0 ) gt, s) gt, s 0 ) + gt, s 0 ) gt 0,s 0 ) = ft, s) ft, s0 ) f s t, s 0 )s s 0 ) s s 0 = + f s t, s 0 ) f s t 0,s 0 ) [ut, s) ut, s 0 )] + i[vt, s) vt, s 0 )] s s 0 f s t, s 0 )[σ σ 0 )+iτ τ 0 )] + f s t, s 0 ) f s t 0,s 0 ). Apì to Je rhma Mèshc Tim c èqoume ìti gia k poia s,s 2 sto eujôgrammo tm ma [s 0,s] isqôei kai ut, s) ut, s 0 )=u σ t, s )σ σ 0 )+u τ t, s )τ τ 0 ) vt, s) vt, s 0 )=v σ t, s 2 )σ σ 0 )+v τ t, s 2 )τ τ 0 ). Apì tic treic teleutaðec sqèseic kaiapìticexis seic Cauchy-Riemann paðrnoume, telik, ìti gt, s) gt 0,s 0 ) [u σ t, s ) u σ t, s 0 ))σ σ 0 )+u τ t, s ) u τ t, s 0 ))τ τ 0 )] s s 0 +i[v σ t, s 2 ) v σ t, s 0 ))σ σ 0 )+v τ t, s 2 ) v τ t, s 0 ))τ τ 0 )] + f s t, s 0 ) f s t 0,s 0 ) u σ t, s ) u σ t, s 0 ) + u τ t, s ) u τ t, s 0 ) + v σ t, s 2 ) v σ t, s 0 ) + v τ t, s 2 ) v τ t, s 0 ) + f s t, s 0 ) f s t 0,s 0 ).

15 .5. ENALLAG H DIADOQIK WN AJRO ISEWN 3 H sunèqeia thc f s sunep getai th sunèqeia twn u σ,u τ,v σ,v τ kai, epeid s s 0 kai s 2 s 0 ìtan s s 0, paðrnoume ìti lim gt, s) =gt 0,s 0 ). t t 0,s s 0 ApodeÐxame, loipìn, ìti h g eðnai suneq c se k je shmeðo tou [a, b] Ω eðte to shmeðo èqei deôterh suntetagmènh diaforetik apì to s 0 eðte èqei deôterh suntetagmènh Ðsh me to s 0. 'Ara h g eðnai suneq c sto [a, b] Ω C, opìte apì to Je rhma.7 h b Gs) = gt, s)dt eðnai suneq c sto Ω. 'Ara F s) F s 0 ) lim = lim s s 0 s s 0 b s s0 a = Gs 0 ) = a gt, s)dt = b a lim Gs) s s0 gt, s 0 )dt =.5 Enallag Diadoqik n AjroÐsewn b a f s t, s 0 )dt. EÐnai gnwstì ìti, an oi seirèc + a n kai + b n sugklðnoun, tìte kai h seir + a n + b n ) sugklðnei kai a n + b n )= a n + Autì epekteðnetai me epagwg se peperasmèno pl joc seir n: an gia m =,..., M oi seirèc + a m,n sugklðnoun, tìte kai h seir + M m= a m,n) sugklðnei kai M M a m,n )= a m,n ). m= m= Je rhma.9 'Estw a m,n 0 gia k je m, n. Tìte Apìdeixh: 'Estw K = Ja apodeðxoume ìti m= m= a m,n )= m= a m,n ), L = K L. b n. a m,n ). m= a m,n ).

16 4 KEF ALAIO. EISAGWG H Autì eðnai profanèc an L =+, opìte upojètoume ìti L<+. Tìte, ìmwc, gia k je m isqôei ìti a m,n m= opìte gia k je m h seir + a m,n sugklðnei. 'Ara gia k je M èqoume ìti M m= a m,n )= M a m,n ) m= a m,n )=L<+, m= a m,n )=L. Epeid to M m= + a m,n) aux nei kaj c to M aux nei, sunep getai, af nontac to M na teðnei sto +, ìti K = Summetrik apodeiknôetai ìti m= L K, a m,n ) L. opìte Je rhma.0 'Estw K = L. Tìte m= a m,n )= m= a m,n )= m= m= a m,n ) < +. a m,n ). Apìdeixh: Kat' arq n parathroôme ìti h isìthtatwn seir n sthn upìjesh eðnai sunèpeia tou Jewr matoc.9. Gia k je a m,n orðzoume touc arijmoôc a + m,n =maxa m,n, 0) kai a m,n = mina m,n, 0) kai eðnai profanèc ìti gia k je m, n isqôei af' enìc af' etèrou a + m,n 0 kai a m,n 0, a + m,n a m,n = a m,n kai a + m,n + a m,n = a m,n.

17 .6. OT UPOS AJROISHS TOU ABEL 5 Epomènwc, sugkrðnontac me thn upìjesh kai qrhsimopoi ntac to Je rhma.9, paðrnoume ìti kai m= m= a + m,n )= + a m,n )= + Afair ntac kat mèlh brðskoume m= m= a + m,n ) < + a m,n ) < +. m= a m,n ) = = m= a + m,n) m= + a + m,n ) m= m= a m,n) a m,n ) = + m= a m,n )..6 O TÔpoc 'Ajroishc tou Abel Je rhma. 'Estw gnhsðwc aôxousa akoloujða pragmatik n arijm n {λ n }, h opoða apoklðnei sto + kai akoloujða migadik n {z n }. JewroÔme th sun rthsh A :[λ, + ) C me tôpo Ax) = z n. EpÐshc, èstw φ :[λ, + ) C. Tìte k n:λ n x k z n φλ n )=Aλ k )φλ k ) Aλ n )[φλ n+ ) φλ n )]. An, epiplèon, h φ èqei suneq par gwgo sto [λ, + ), tìte n:λ n x z n φλ n )=Ax)φx) An, epðshc, lim x + Ax)φx) =0,tìte x λ At)φ t)dt. + z n φλ n )= At)φ t)dt. λ Apìdeixh: UpologÐzoume Aλ n )= m:λ m λ n z m = z + + z n,

18 6 KEF ALAIO. EISAGWG H Aλ n )= z m = z + + z n. m:λ m λ n 'Ara z n = Aλ n ) Aλ n ) gia k je n 2 kai, fusik, z = Aλ ). Epomènwc k z n φλ n ) = Aλ )φλ )+ = = k [Aλ n ) Aλ n )]φλ n ) n=2 k Aλ n )φλ n ) k Aλ n )φλ n ) n=2 k k Aλ n )φλ n )+Aλ k )φλ k ) Aλ n )φλ n+ ) k = Aλ k )φλ k ) Aλ n )[φλ n+ ) φλ n )]. 'Estw k o megalôteroc akèraioc ste λ k x kai, epomènwc, Tìte n:λ n x z n φλ n ) = k z n φλ n ) λ k x<λ k+. k = Aλ k )φλ k ) Aλ n )[φλ n+ ) φλ n )] k = Ax)φλ k ) Aλ n ) λn+ λ n φ t)dt k = Ax)φx) Ax)[φx) φλ k )] x = Ax)φx) Aλ k ) φ t)dt λ k x = Ax)φx) = Ax)φx) λn+ λ n k λn+ At)φ t)dt λ k x λ At)φ t)dt. λ n k λn+ λ n Aλ n )φ t)dt Aλ n )φ t)dt At)φ t)dt

19 .6. OT UPOS AJROISHS TOU ABEL 7 'Ara, an Ax)φx) 0 kaj c x +, tìte + z n φλ n )= At)φ t)dt. λ

20 8 KEF ALAIO. EISAGWG H

21 Kef laio 2 H ζ-sun rthsh tou Riemann Je rhma 2. H seir + k= p k kai to peiro-ginìmeno + k= p k ) apoklðnoun sto +. Apìdeixh: ) GnwrÐzoume ìti, gia 0 u<, h gewmetrik seir + m=0 um sugklðnei. 'Ara, gia autèc tic timèc tou u sto u + u = m=0 u m +u + u u m. Jètoume u = p kai paðrnoume ) + p p + p p m. Epilègoume tuqìn x 2 kai èstw n o megalôteroc akèraioc ste p n x, opìte oi arijmoð 2, 3, 5,..., p n eðnai ìloi oi pr toi mèqri to x. 'Estw m opoiosd pote akèraioc me 2 m x. Apì thn prohgoômenh anisìthta brðskoume ) ) ) 2 3 p n ) 2 m ) 3 m + + p n p ) n p m n 9

22 20 KEF ALAIO 2. H ζ-sun ARTHSH TOU RIEMANN = 0 i,j,...,h m 2 i 3 j p h n Sto teleutaðo jroisma oi ekjètec i, j,..., h diatrèqoun o kajènac kai anex rthta o ènac apì ton llon tictimèc0,,..., m. Oi paronomastèc tou ajroðsmatoc eðnai arijmoð pou h anapar stas touc san ginìmena pr twn den perilamb nei kanènan pr to arijmì diaforetikì apì touc 2, 3,..., p n. Isqurizìmaste ìti an mesa s' autoôc touc paronomastèc up rqoun ìloi oi arijmoð, 2,..., [x]. Pr gmati, èstw y ènac apì toôc, 2,..., [x]. Tìte, h anapar stas tou den perièqei kanènan pr to diaforetikì apì touc 2, 3,..., p n,afoôk jepr toc sthn anapar stash tou y eðnai y x. 'Ara gia k poia i, j,..., h. Tìte, ìmwc, opìte kai, epomènwc, y =2 i 3 j p h n 2 i+j+ +h y x 2 m, i + j + + h m 0 i, j,..., h m. 'Ara to y eðnai ènac apì touc paronomastèc tou parap nw pollaploô ajroðsmatoc. 'Ara kai, epomènwc, 0 i,j,...,h m 2 i 3 j p h n logx +) [x] ) ) ) p n logx +). [x] An sumbolðsoume P n = tìte èqoume apodeðxei ìti ) ) ), 2 3 p n P n logx +), ìpou n eðnai o megalôteroc akèraioc ste p n x. H {P n } eðnai aôxousa akoloujða pragmatik n arijm n, opìte, an den apoklðnei sto +, tìte up rqei M ste P n M.

23 2 gia k je n. Autì sunep getai ìti logx +) M gia k je x 2, pr gma pou eðnai topo. 'Ara opìte P n +. 2) GnwrÐzoume ìti, gia 0 u<, isqôei to an ptugma-taylor log Jètoume u = p log u = u + u2 2 + u3 3 + u4 4 +, u u + u2 2 + u3 2 + u4 2 + = u + u 2 2 u). kai paðrnoume log ) p p + 2pp ). Prosjètontac gia p =2, 3,..., p n upologðzoume log P n p n 2 22 ) + 33 ) + + ) p n p n ) = p n p n ) p n p n 2. An sumbolðsoume tìte èqoume apodeðxei ìti S n = p n, S n log logx +) 2, ìpou n eðnai o megalôteroc akèraioc ste p n x. H {S n } eðnai aôxousa akoloujða pragmatik n kai, ìpwc prohgoumènwc gia thn {P n }, apodeiknôetai ìti S n +. Antijètwc, an σ>, h seir + + n σ p σ n sugklðnei, afoô h megalôterh seir sugklðnei. Se lðgo ja apodeðxoume ìti, an σ>, tìte kai to apeiro- ) sugklðnei. ginìmeno + k= p σ k

24 22 KEF ALAIO 2. H ζ-sun ARTHSH TOU RIEMANN Orismìc 2. 'Estw f : N C. H f onom zetai pollaplasiastik, an den eðnai h mhdenik sun rthsh kai isqôei fmn) =fm)fn) gia k je m, n N me gcdm, n) =. H f onom zetai pl rwc-pollaplasiastik an den eðnai h mhdenik sun rthsh kai isqôei h Ðdia isìthta gia k je m, n N. An h f eðnai pollaplasiastik kai f) = 0, tìte fn) =f)fn) =0gia k je n. Autì antif skei me ton orismì, opìte f) 0. T ra, f) = f)f) kai, epomènwc, f) =. Par deigma. H sun rthsh me tôpo fn) = n s, s C, eðnai pl rwcpollaplasiastik. Paradeigma 2. H sun rthsh ϕ thc JewrÐac Arijm n - ìpou ϕn) eðnai to pl joc twn fusik n oi opoðoi eðnai n kai sqetik pr toi me to n - eðnai pollaplasiastik all ìqi pl rwc-pollaplasiastik ) sun rthsh. Je rhma 2.2 'Estw pollaplasiastik f kai èstw ìtihseir + fn) sugklðnei apolôtwc. Tìte isqôei ìti fn) = + k= +fpk )+fp 2 k )+fp3 k )+ ). An, epiplèon, h f eðnai pl rwc-pollaplasiastik, tìte isqôei ìti fn) = + k= fpk ) ). Apìdeixh: 'Estw tuqìn N N. An k eðnai o megalôteroc akèraioc ste p k N kai m eðnai opoiosd pote akèraioc ste 2 m N, tìte k +fpl )+ + fp m l ) ) = l= = f2 i )f3 j ) fp h k ) 0 i,j,...,h m 0 i,j,...,h m f 2 i 3 j p h k). ParathroÔme ta ex c

25 23. An mesa sta ginìmena pou emfanðzontai sto teleutaðo jroisma perilamb nontai ìloi oi arijmoð, 2,..., N. Pr gmati, èstw y N. Tìte k je pr toc pou ufðstatai sthn anapar stash tou y san ginìmeno pr twn eðnai y N kai, epomènwc, eðnai ènac apì touc 'Ara o y gr fetai 2, 3,..., p k. y =2 i 3 j p h k gia k poia i, j,..., h 0. Tìte, ìmwc, 2 i+j+ +h y N 2 m. 'Ara kai, epomènwc, i + j + + h m 0 i, j,..., h m. 2. Diaforetikèc epilogèc twn i, j,..., h dðnoun diaforetikèc timèc sto ginìmeno 2 i 3 j p h k,lìgw thc monadikìthtac thc anapar stashc opoioud pote arijmoô san ginìmeno pr twn. Apì ta. kai 2. paðrnoume ìti ) k +fpl )+ + fp m l ) ) N fn) l= n=n+ fn). Epeid h + fn) sugklðnei apolôtwc, sunep getai ìti, gia k je p, hseir me ligìterouc ìrouc +fp)+fp 2 )+ sugklðnei. Epeid h ) isqôei gia k je m me 2 m N, af noume to m na teðnei sto + kai paðrnoume ) k +fpl )+fp 2 )+ ) N l fn) l= n=n+ fn), ìpou k eðnai o megalôteroc akèraioc ste p k N. T ra epilègoume N = p k kai af noume to k na teðnei sto +. Tìte N = p k + kai h dexi pleur thc ) teðnei sto 0. 'Ara + l= +fpl )+fp 2 l )+ ) = fn).

26 24 KEF ALAIO 2. H ζ-sun ARTHSH TOU RIEMANN An h f eðnai pl rwc-pollaplasiastik, tìte gia k je p +fp)+fp 2 )+fp 3 )+ =+fp)+fp) 2 + fp) 3 + kai, epeid h seir sugklðnei apolôtwc, sunep getai fp) < kai, epomènwc, +fp)+fp 2 )+fp 3 )+ = fp) ). 'Ara, an h f eðnai pl rwc-pollaplasiastik, tìte fn) = + k= fpk ) ). Sto ex c ja qrhsimopoioôme to sômbolo Π σ gia to dexiì hmiepðpedo An Rs >, tìte h seir Π σ = {s C : Rs >σ}. sugklðnei apolôtwc. 'Ara sugklðnei se migadikì arijmì kai autìc o arijmìc exart tai, profan c, apì to s. n s Orismìc 2.2 Hsun rthsh ζ :Π C me tôpo ζs) = onom zetai ζ sun rthsh tou Riemann. Prìtash 2. TÔpoc tou Euler ) IsqÔei ìti gia k je s Π. ζs) = n s, s Π, + k= ) p s k Apìdeixh: Efarmìzoume to Je rhma 2.2 me thn pl rwc-pollaplasiastik sun rthsh fn) =n s.

27 25 Prìtash 2.2 Hseir + n s de sugklðnei omoiìmorfa sto Π. Apìdeixh: 'Estw ìti h seir sugklðnei omoiìmorfa sto Π. Tìte gia ɛ =up rqei n 0, tètoio ste l n=k n s < gia k je k, l me n 0 k<lkai gia k je s Π. Stajeropoi ntac ta k, l kai paðrnontac ìrio ìtan s, brðskoume ìti l n=k n gia k je k, l n 0. PaÐrnontac, t ra, ìrio ìtan l +, brðskoume n=k n gia k je k n 0. Katal goume, ètsi, se topo, diìti h seir apoklðnei sto +. n=k n Prìtash 2.3 Hseir + n s sugklðnei omoiìmorfa sto Π σ0 gia k je σ 0 >. EpÐshc, h seir sugklðnei omoiìmorfa se k jesumpagèc uposônolo tou Π. Apìdeixh: Gia k je s = σ + iτ Π σ0 isqôei Epeid h seir + n σ 0 n s = n σ n σ0. sugklðnei, apì to M test tou Weierstrass Je rhma.3) sunep getai ìti h seir + n s sugklðnei omoiìmorfa sto Π σ0. 'Estw, t ra, opoiod pote sumpagèc K Π. Tìte up rqei σ 0 > ste K Π σ0. Apì to pr to mèroc sunep getai ìti h seir sugklðnei omoiìmorfa sto K.

28 26 KEF ALAIO 2. H ζ-sun ARTHSH TOU RIEMANN Prìtash 2.4 H ζ sun rthsh tou Riemann eðnai analutik sun rthsh sto Π. Apìdeixh: Gia k je n h sun rthsh n s eðnai analutik sun rthsh tou s sto C. Epeid h sôgklish thc seir c twn sunart sewn aut n eðnai, apì thn Prìtash 2.3, omoiìmorfh se k je sumpagèc uposônolo tou Π, sunep getai apì to Je rhma.2 ìti h sun rthsh pou orðzei h seir, dhlad h ζ sun rthsh tou Riemann,eÐnai analutik sun rthsh sto Π. opìte Efarmìzoume ton tôpo tou Abel Je rhma.) gia λ n = n, z n =,ϕx) = x s, Ax) = n: λ n x [x] z n = =[x]. H ϕ èqei suneq par gwgo sto [, + ) kai, an s = σ + iτ Π, tìte gia k je x [, + ) Ax)ϕx) = [x] x σ x + x σ 2 x σ 0 ìtan x +. 'Ara lim x + Ax)ϕx) =0kai epomènwc ϖ) ζs) =s UpenjumÐzoume to sumbolismì + {t} = t [t], t R, [t] t s+ dt, s Π. kai ìti, profan c gia k je t R. Prìtash 2.5 To olokl rwma 0 {t} < Gs) = + {t} dt ts+ sugklðnei gia k je s Π 0 kai orðzei sun rthsh analutik sto Π 0. Apìdeixh: 'Estw opoiod pote s = σ + iτ Π 0. Tìte σ>0, opìte + {t} + dt t s+ dt < +. tσ+

29 27 'Ara to olokl rwma pou orðzei thn Gs) sugklðnei gia k je s Π 0. OrÐzoume k+ {t} k+ t k F k s) = dt = k ts+ k t s+ dt kai apì to Je rhma.8 sunep getai ìti h F k eðnai analutik sto C. EÐnai fanerì ìti Gs) = k= F k s) opìte, sômfwna me to Je rhma.2, arkeð na apodeðxoume ìti h seir sugklðnei omoiìmorfa se k je sumpagèc uposônolo tou Π 0. 'Estw sumpagèc K Π 0. Tìte up rqei σ 0 > 0 ste K Π σ0. 'Ara, gia k je s = σ + iτ K isqôei ìti Fk s) k+ {t} k+ dt t s+ t σ0+ dt = σ 0 k ). σ0 k +) σ0 'Omwc k k= k σ 0 k ) = < + σ0 k +) σ0 σ 0 kai apì to M-test tou Weierstrass sunep getai ìti h + k= F k sugklðnei omoiìmorfa sto K. OrÐzoume, t ra, sun rthsh H :Π 0 \{} C me tôpo Hs) = s s sgs), s Π 0 \{}. Prìtash 2.6 Hsun rthsh H eðnai merìmorfh sto Π 0. 'Eqei èna monadikì aplì pìlo sto shmeðo s =me residue =sto shmeðo autì. Apìdeixh: To sumpèrasma eðnai meso apì thn Prìtash 2.5. O tôpoc ϖ) lèei ìti oi sunart seic ζ kai H tautðzontai sto Π. Pr gmati, an s Π,tìte + [t] ζs) = s dt = s ts+ + = s t s dt s + + t {t} t s+ {t} s dt = ts+ dt sgs) = Hs). s 'Ara h sun rthsh H mporeð na jewrhjeð epèktash thc ζ sto megalôtero pedðo orismoô Π 0.

30 28 KEF ALAIO 2. H ζ-sun ARTHSH TOU RIEMANN Gi' autìn tolìgo ja sumbolðzoume thn H me to sômbolo ζ kai ja jewroôme th ζ sun rthsh tou Riemann epektetamènh sto megalôtero pedðo orismoô Π 0. To epìmeno je rhma katagr fei ta mèqri t ra apotelèsmata. Prèpei na dojeð prosoq sta sônola sta opoða isqôei o k je tôpoc. Je rhma 2.3 H ζ sun rthsh tou Riemann eðnai merìmorfh sun rthsh tou s sto sônolo Π 0, èqei èna monadikì aplì pìlo sto shmeðo s =me residue = sto shmeðo autì kai de mhdenðzetai se kanèna shmeðotou Π. EpÐshc isqôoun oi tôpoi ) ζs) = s s s 2) ζs) =s 3) ζs) = 4) ζs) = + + k= + {t} t s+ dt, s Π 0 [t] t s+ dt, s Π n s, s Π p s k ), s Π. Apìdeixh: O tôpoc 3) den eðnai tðpote llo apì ton Orismì 2.2, o tôpoc 4) eðnai o tôpoc tou Euler Prìtash 2.) kai o tôpoc 2) eðnai o tôpoc ϖ). Tèloc, o tôpoc ) eðnai o orismìc thc sun rthshc H. To ìti h ζ eðnai merìmorfh sto Π 0 me monadikì aplì pìlo sto s = me residue =prokôptei apì thn Prìtash 2.6 kai ta sqìlia pou thn akoloôjhsan. To ìti ζs) 0gia k je s Π apodeiknôetai apì ton tôpo 4) kai apì thn Parat rhsh met apì ton Orismì., afoô kanènac ìroc p ) tou s k apeiro-ginìmenou den mhdenðzetai.

31 Kef laio 3 Seirèc-Dirichlet Orismìc 3. Oi seirèc-dirichlet eðnai seirèc thc morf c me migadikoôc suntelestèc a n. H seir pou orðzei th ζ sun rthsh tou Riemann eðnai par deigma seir c- Dirichlet me a n =gia k je n. Prìtash 3. 'Estw ìti h + a n n s sugklðnei gia k poio s = s 0. Tìte, gia k je θ me 0 <θ< π 2,hseir sugklðnei omoiìmorfa sthn kurt gwnða a n n s Γs 0 ; θ) ={s : args s 0 ) <θ}. Apìdeixh: 'Estw s = σ + iτ Γs 0 ; θ), s 0 = σ 0 + iτ 0 kai jètoume b n = a n n s0, opìte h + b n sugklðnei. EpÐshc jètoume r n = k=n b k, opìte r n 0 ìtan n +. 29

32 30 KEF ALAIO 3. SEIR ES-DIRICHLET 'Estw ɛ>0 kai jewroôme n 0 = n 0 ɛ), ste r n <ɛìtan n n 0. Tìte, an n 0 m<m : m n=m a n m = n s = = n=m m b n m = n s s0 r n n s s0 n=m n=m m + n=m+ r m m r m m + s s0 m + s s0 r n r n+ n s s0 r n n ) s s0 n=m+ r m m + r m + m σ σ0 m + σ σ0 r m + r m + + ɛ + ɛ + m n=m+ 2ɛ + ɛ cos θ = 2ɛ + ɛ cos θ 2ɛ + ɛ cos θ ɛ 2+ m n=m+ m n=m+ n=m+ r n n s s0 n ) s s0 ) r n s s 0 r n s s 0 n n n n dt t s s0+ dt t σ σ0+ r n s s 0 σ σ 0 n ) ) σ σ0 n σ σ0 n ) ) σ σ0 n σ σ0 m ) σ σ0 m σ σ0 m σ σ0 ). cos θ Apì to krit rio tou Cauchy sunep getai ìti h seir sugklðnei omoiìmorfa sto Γs 0 ; θ). Prìtash 3.2 An hseir + a n n s sugklðnei se k poio s 0 = σ 0 + iτ 0, tìte h seir sugklðnei se k je shmeðo tou anoiktoô hmiepipèdou Π σ0. EpÐshc, h seir sugklðnei omoiìmorfa se k jesumpagèc uposônolo tou Π σ0 kai, epomènwc, orðzei analutik sun rthsh sto hmiepðpedo autì. Apìdeixh: 'Estw sumpagèc K Π σ0 = {s : σ = Rs >Rs 0 = σ 0 }. EÐnai gewmetrik fanerì ìti up rqei θ me 0 <θ< π 2, ste K Γs 0 ; θ). Apì thn Prìtash 3. sunep getai ìti h seir sugklðnei omoiìmorfa sto Γs 0 ; θ), opìte sugklðnei omoiìmorfa kai sto K kai epomènwc h analutikìthta thc sun rthshc pou orðzetai apì th seir prokôptei apì to Je rhma.2. H kat shmeðo sôgklish eðnai, t ra, profan c.

33 3 Je rhma 3. Gia k je seir -Dirichlet up rqei α R {+, } ste. hseir sugklðnei se k je shmeðo tou hmiepipèdou Π α = {s : Rs >α} kai sugklðnei omoiìmorfa se k je sumpagèc uposônolo tou hmiepipèdou autoô. Epomènwc, orðzei analutik sun rthsh sto Π α. 2. h seir apoklðnei se k je shmeðo tou anoiktoô hmiepipèdou {s : Rs < α}. Gia thn eujeða {s : Rs = α} den up rqei genikì sumpèrasma. ShmeÐwsh: an α =+, tìte to èna hmiepðpedo eðnai kenì kai to llo eðnai to C.) Apìdeixh: An h seir sugklðnei se k je s C, tìte isqôei to. kai me tetrimmèno trìpo to 2. paðrnontac α =. Pr gmati, an K eðnai opoiod pote sumpagèc, tìte up rqei σ 0 ste to K na perièqetai sto anoiktì hmiepðpedo Π σ0. Apì thn Prìtash 3.2 sunep getai ìti h seir sugklðnei omoiìmorfa sto K, afoô sugklðnei sto shmeðo s = σ 0. An h seir apoklðnei se k je s C, tìte isqôoun ta. kai 2. me tetrimmèno trìpo, an jèsoume α =+. Mènei na exet soume thn perðptwsh pou h seir sugklðnei se k poio s = σ + iτ kai apoklðnei se k poio s 2 = σ 2 + iτ 2. Apì thn Prìtash 3.2 sunep getai ìti h seir sugklðnei se k je shmeðo tou Π σ kai omoiìmorfa se k je sumpagèc uposônolo tou Π σ ), opìte èqoume ìti σ 2 σ. DiakrÐnoume, t ra, tic peript seic. σ 2 = σ. An up rqe s me Rs <σ 2 sto opoðo hseir sugklðnei, apì thn Prìtash 3.2 ja eðqame ìti h seir sugklðnei kai sto s 2. Autì eðnai topo, opìte to je rhma isqôei an jèsoume α = σ 2 = σ. 2. σ 2 <σ. Tìte, ìpwc prohgoumènwc, sunep getai ìti h seir apoklðnei se k je shmeðo tou {s : Rs <σ 2 }. 'Estw T = {t : σ 2 t, h seir sugklðnei se k je s me Rs >t}. To T eðnai mh-kenì afoô σ T ), opìte jètoume α = infimumt ). Ja deðxoume ìti to je rhma isqôei me aut n thn epilog tou α. 'Estw sumpagèc K Π α. Tìte up rqoun t,t 2 me α t 2 <t, t 2 T kai K Π t.

34 32 KEF ALAIO 3. SEIR ES-DIRICHLET Apì ton orismì tou T sunep getai ìti h seir sugklðnei sto shmeðo t, opìte apì thn Prìtash 3.2 sunep getai ìti h seir sugklðnei omoiìmorfa sto K. EpÐshc, an h seir sugklðnei se k poio s me t = Rs < α, tìte, apì thn Prìtash 3.2 sunep getai ìti t T kai autì eðnai topo afoô α = infimumt ). Orismìc 3.2 To α, tou opoðouthn Ôparxh exasfalðzei to Je rhma 3.,onom zetai tetmhmènh sôgklishc thc seir c-dirichlet. To hmiepðpedo Π α = {s : Rs >α} onom zetai hmiepðpedo sôgklishc, en to hmiepðpedo {s : Rs < α} onom zetai hmiepðpedo apìklishc thc seir c-dirichlet. Gia dosmènh seir -Dirichlet me tetmhmènh sôgklishc α, h seir a n n s eðnai, epðshc, seir -Dirichlet. Epomènwc kai s' aut n antistoiqeð h tetmhmènh sôgklis c thc, α 0. Ja doôme, t ra, èna apotèlesma gia th sqetik jèsh twn α, α 0. 'Estw tuqìn σ>α 0. Tìte h seir + sugklðnei kai, epomènwc, h seir + a n n σ a n n s a n n σ sugklðnei. 'Ara σ α. Amèswc sunep getai ìti α 0 α. 'Estw, t ra, tuqìn σ>α. Tìte h seir + a n n σ a n n σ 0 ìtan n +. 'Ara up rqei M>0 ste a n n σ M gia k je n. Epomènwc, gia k je ɛ>0 èqoume opìte h seir + a n n σ++ɛ a n + n σ++ɛ M n +ɛ, sugklðnei. 'Ara σ ++ɛ α 0. sugklðnei, opìte

35 33 AfoÔ autì isqôei gia k je ɛ>0, sunep getai ìti σ α 0. SumperaÐnoume, loipìn, ìti α α 0. 'Ara apodeðqjhke h Prìtash 3.3 α α 0 α +gia k je seir -Dirichlet. Orismìc 3.3 To α 0 onom zetai tetmhmènh apìluthc sôgklishc thc seir c- Dirichlet. To hmiepðpedo Π α0 = {s : Rs > α 0 } onom zetai hmiepðpedo apìluthc sôgklishc kai to hmiepðpedo {s : Rs <α 0 } onom zetai hmiepðpedo apìluthc apìklishc thc seir c-dirichlet. Hkatakìrufh z nh {s : α<rs <α 0 },andeneðnai ken, onom zetai z nh upì-sunj khn sôgklishc kai eðnai pl touc metaxô 0 kai. Par deigma : Gia th seir h opoða orðzei th ζ sun rthsh tou Riemann h tetmhmènh sôgklishc kai h tetmhmènh apìluthc sôgklishc, profan c, tautðzontai kai α = α 0 =, afoô h seir sugklðnei an s R kai s> kai apoklðnei an s R kai s. Par deigma 2: Gia th seir n s ) n h tetmhmènh apìluthc sôgklishc eðnai α 0 =,afoô + ) n n = + s n. s 'Omwc h tetmhmènh sôgklishc eðnai α =0. Pr gmati, gia k je σ>0h + ) n n sugklðnei, diìti eðnai seir me ìrouc σ pou fjðnoun proc to 0 kai me enallassìmena prìshma. Epiplèon, an σ =0, h seir gðnetai + )n kai apoklðnei. Orismìc 3.4 'Estw seir -Dirichlet + a n n s me tetmhmènh sôgklishc α R. Apì to Je rhma 3. sunep getai ìtihseir orðzei analutik sun rthsh fs) = sto hmiepðpedo Π α. 'Estw s 0 me Rs 0 = α. Tìte to s 0 onom zetai kanonikì shmeðo thcf, an up rqei δ>0 ste hf na epekteðnetai san analutik sun rthsh sto n s a n n s Π α s 0 ; δ), ìpou s 0 ; δ) eðnai o anoiktìc dðskoc me kèntro s 0 kai aktðna δ.

36 34 KEF ALAIO 3. SEIR ES-DIRICHLET Je rhma 3.2 Landau ) An a n 0 gia k je n kai α R eðnai h tetmhmènh sôgklishc thc seir c + a n n s, tìte to α den eðnai kanonikì shmeðo thc sun rthshc f pou orðzetai apì th seir sto Π α. Apìdeixh: 'Estw ìti to α eðnai kanonikì shmeðo thc f. Tìte up rqei δ > 0 ste h f na epekteðnetai san analutik sun rthsh sto Ω=Π α α; δ). JewroÔme pragmatikì arijmì megalôtero tou α. Gia par deigma ton α +. T ra, h f anaptôssetai san seir -Taylor sto megalôtero dðsko me kèntro α + o opoðoc perièqetai sto Ω. 'Omwc, eðnai gewmetrik profanèc ìti o dðskoc autìc perièqei k poion pragmatikì arijmì σ<α. Epomènwc, gia autì to σ h seir -Taylor dðnei fσ) = m=0 f m) α +) σ α ) m. m! Epeid to α + perièqetai sto hmiepðpedo sôgklishc thc seir c-dirichlet, sunep getai apì to Je rhma.2 ìti gia k je m f m) α +)= Apì tic sqèseic autèc paðrnoume fσ) = = = m=0 + m=0 + m=0 σ α ) m m! log n) m a n n α+. + σ α ) m m! log n) m a ) n n α+ log n) m a ) n n α+ [α + σ)logn] m a ) n m! n α+ parathr ntac ìti ìloi oi ìroi tou teleutaðou ajroðsmatoc eðnai mh-arnhtikoð kai efarmìzontac to Je rhma.) = = = = a n + n α+ m=0 a n eα+ σ)logn nα+ a n nα+ σ nα+ a n n σ. [α + σ)logn] m ) m!

37 35 SumperaÐnoume ìti to + a n n isoôtai me to migadikì arijmì fσ). Autì σ eðnai topo, diìti o arijmìc σ perièqetai sto hmiepðpedo apìklishc thc seir c- Dirichlet. Je rhma 3.3 Ginìmeno seir n-dirichlet. ) 'Estw a + n n s, b n n s duo seirèc-dirichlet oi opoðec sugklðnoun apolôtwc sto Ðdio shmeðo s 0. An orðsoume c n = a k b k gia k je n N, tìte h seir -Dirichlet k,k : kk =n sugklðnei apolôtwc sto s 0 kai c n n s Apìdeixh: I) Gia N N èqoume N a n n s0 a n b n n s0 n s0 N = + b n n = s0 k,k N c n n s0. a k b k kk ) s0. OmadopoioÔme, t ra, touc ìrouc tou teleutaðou ajroðsmatoc an loga me tic timèc tou n = kk, parathr ntac ìti. An n N, tìte sto jroisma autì emfanðzontai ìla ta zeug ria k, k me kk = n kai, epomènwc, an prosjèsoume touc antðstoiqouc ìrouc ja p roume to cn n s An N + n N 2 to ginìmeno kk de mporeð na uperbeð to N 2 ), tìte to kk = n sunep getai ìti èna toul qiston apì ta k, k eðnai [ N +]. 'Ara N a n n s0 N b n n N c n a kb k s0 n s0 k,k : k,k N kai k [ N+] k [ kk ) s0 N+] a kb k kk ) s0 k,k : k N,[ N+] k N

38 36 KEF ALAIO 3. SEIR ES-DIRICHLET = + k,k : k <[ N+],[ N+] k N N a k k s0 k= N + k= + k=[ N+] a k k s0 k=[ N+] N k =[ N+] a k k s0 + k =[ N+] b k k s0 [ N+] k = b k k s0 b k k s0 a k + b k. k s0 k s0 k = a kb k kk ) s0 Af nontac to N na teðnei sto + paðrnoume ìti h teleutaða par stash teðnei sto 0,opìte a n b n n s0 n = + c n s0 n. s0 II) An s 0 = σ 0 + iτ 0,tìte h apìluth sôgklish twn seir n isodunameð me th sôgklish twn OrÐzoume kai èqoume amèswc ìti d n = a n n, + σ0 k,k : kk =n c n d n n s0 n σ0 b n n σ0. a k b k gia k je n. 'Ara, an efarmìsoume to mèroc I), paðrnoume kai epomènwc h + c n + d n n s0 n σ0 c n n s 0 = + a n n σ0 sugklðnei apolôtwc. b n < + σ0 n

39 Kef laio 4 Qarakt rec Orismìc 4. 'Estw G peperasmènh abelian om da kai èstw χ : G C me tic idiìthtec. χab) = χa) χb) gia k je a, b G 2. χ den eðnai h mhdenik sun rthsh. Tìte h χ onom zetai qarakt rac thc G. Prìtash 4. 'Estw qarakt rac χ thc peperasmènhc abelian c om dac G.. An e eðnai to monadiaðo stoiqeðo thc G, tìte χe) =. 2. Gia k je a G isqôei χa) χa )= χa) gia k je a G. 4. An m eðnaiht xhthcg, tìte gia k je a G to χa) eðnai m-ost rðza thc mon dac. 5. χa) =gia k je a G. Apìdeixh. 'Estw a 0 G ste χa 0 ) 0. Tìte χa 0 )= χa 0 e)= χa 0 ) χe). 'Ara χe) =. 2. An gia k poio a G eðnai χa) =0,tìte = χe) = χaa )= χa) χa )=0. 'Atopo. 3. Apì ton prohgoômeno upologismì èqoume amèswc ìti = χa) χa ) kai, epomènwc, gia k je a G. χa )= χa) 37

40 38 KEF ALAIO 4. QARAKT HRES 4. Epeid a m = e gia k je a G, sunep getai χa) m = χa m )= χe) =. 5. Apì thn prohgoômenh sqèsh paðrnoume χa) m =,opìte χa) =. 'Ara k je qarakt rac eðnai sun rthsh χ : G T, ìpou T = {z C : z =}. To sônolo T eðnai om da me thn pr xh tou pollaplasiasmoô migadik n arijm n kai parathroôme ìti o orismìc tou qarakt ra isodunameð me to ìti h χ eðnai omomorfismìc om dwn. Orismìc 4.2 Me χ sumbolðzoume to qarakt ra me tôpo χ a) =, a G. Ton χ onom zoume kôrio qarakt ra. Me Ĝ sumbolðzoume to sônolo twn qarakt rwn thc G. Tèloc, sto Ĝ orðzoume pr xh wc ex c: gia k je χ, ψ Ĝ o χ ψ eðnai o qarakt rac me tôpo χ ψ)a) = χa) ψa), a G. To ìti o χ ψ eðnai stoiqeðo thcĝ, dhlad qarakt rac thc G, apodeiknôetai eôkola: χ ψ)ab) = χab) ψab) = χa) χb) ψa) ψb) = χ ψ)a) χ ψ)b) gia k je a, b G kai, epiplèon, χ ψ)e) = χe) ψe) =, opìte h χ ψ den eðnai h mhdenik sun rthsh. ParathroÔme ìti gia k je χ, ψ, ω Ĝ:. χ ψ)a) = χa) ψa) = ψa) χa) = ψ χ)a) gia k je a G kai, epomènwc, χ ψ = ψ χ. 2. [ χ ψ) ω]a) = χ ψ)a) ωa) = χa) ψa) ωa) = χa) ψ ω)a) =[ χ ψ ω)]a) gia k je a G, opìte χ ψ) ω = χ ψ ω).

41 39 3. χ χ)a) = χ a) χa) = χa) gia k je a G, opìte χ χ = χ. 4. An orðsoume χ : G C me tôpo χ a) = χa)), a G, tìte χ χ)a) = χ a) χa) == χ a) gia k je a G kai, epomènwc, χ χ = χ. EpÐshc, h sun rthsh χ eðnai stoiqeðo thcĝ. Pr gmati, χ ab) = χab)) = χa) χb)) = χa)) χb)) = χ a) χ b) gia k je a, b G kai χ e) = χe)) = =. Je rhma 4. Gia k je peperasmènh abelian om da G to sônolo Ĝ twn qarakt rwn thc G apoteleð peperasmènh abelian om da kai h t xh thc eðnai Ðsh me thn t xh thc G. Apìdeixh Mìlic prohgoumènwc eðdame ìti h Ĝ apoteleð abelian om da me thn pr xh pou orðsjhke. To monadiaðo stoiqeðo thc eðnai o kôrioc qarakt rac χ kai to antðstrofo stoiqeðo tou χ eðnai to χ, ìpwc autì orðsjhke sto 4. pio p nw. 'Eqoume, epomènwc, na apodeðxoume ìti h Ĝ èqei thn Ðdia t xh me thn G. PerÐptwsh. 'Estw ìti h G eðnai kuklik t xhc m me genn tora a. Dhlad G = {a, a 2,..., a m,a m = e}. K je χ Ĝ kajorðzetai apì thn tim χa), afoô apì th sqèsh χak )= χa)) k sunep getai ìti k je tim tou χ prokôptei monos manta apì thn tim χa). JewroÔme to sunolo twn m-ost n riz n thc mon dac, ìpou Λ m = {ω, ω 2,..., ω m,ω m =} ω =cos 2π m + i sin 2π m. H Prìtash 4.4) mac epitrèpei na sqhmatðsoume thn apeikìnish Ĝ Λ m

42 40 KEF ALAIO 4. QARAKT HRES me tôpo χ χa) kai ja èqoume oloklhr sei thn apìdeixh, an apodeðxoume ìti aut h apeikìnish eðnai amfimonos manth. 'Estw χ, ψ Ĝ ste χa) = ψa). Tìte χa k )= χa)) k = ψa)) k = ψa k ) gia k je k kai, epomènwc, 'Ara h apeikìnish eðnai èna-proc-èna. 'Estw η Λ m. OrÐzoume χ : G T me tôpo χ = ψ. χa k )=η k gia k je k =, 2,..., m. Tìte χa) =η, opìte mènei na apodeðxoume ìti h χ eðnai qarakt rac. Kat' arq n h χ, profan c, den eðnai h mhdenik sun rthsh. Katìpin, an a k,a l me k, l m eðnai opoiad pote stoiqeða thc G, diakrðnoume tic peript seic. an k + l m, tìte χa k a l )= χa k+l )=η k+l = η k η l = χa k ) χa l ), 2. an m + k + l 2m, tìte a k+l = a k+l m kai k + l m m, opìte χa k a l )= χa k+l )= χa k+l m )=η k+l m = η k+l = η k η l = χa k ) χa l ). PerÐptwsh 2. H genik perðptwsh me m na eðnai h t xh thc G. Apì to Jemeli dec Je rhma twn Peperasmènwn Abelian n Om dwn sunep - getai ìti h G gr fetai G = G G 2... G n, ìpou k je G j eðnai kuklik upoom da thc G t xhc m j. Dhlad, k je a G gr fetai me monadikì trìpo san ginìmeno a = a a 2 a n me a G,a 2 G 2,..., a n G n. Epomènwc, m = m m 2 m n. OrÐzoume thn apeikìnish Ĝ Ĝ Ĝ2 Ĝn

43 4 me ton tôpo χ χ ), χ 2),..., χ n) ), ìpou χ j) eðnai o periorismìc tou χ sthn upoom da G j. 'Estw χ, ψ Ĝ ste χj) = ψ j) gia k je j =, 2,..., n. Tìte paðrnoume opoiod pote a G kai to gr foume me monadik kajorismèna a j G j. Tìte a = a a n χa) = χa ) χa n ) = χ ) a ) χ n) a n ) = ψ ) a ) ψ n) a n ) = ψa ) ψa n ) = ψa). 'Ara χ = ψ kai apodeðqjhke ìti h apeikìnish eðnai -. 'Estw ìti dðnontai ta χ ) Ĝ,..., χ n) Ĝn. Ja broôme χ Ĝ tou opoðou o periorismìc se k je G j na eðnai o χ j) kai ja èqoume apodeðxei ìti h apeikìnish eðnai epð. Gia tuqìn a G gr foume a = a a n me monadik kajorismèna a j G j kai orðzoume opìte An a = a a n,b= b b n,tìte χa) = χ ) a ) χ n) a n ). ab =a b ) a n b n ), χab) = χ ) a b ) χ n) a n b n )= χ ) a ) χ n) a n ) χ ) b ) χ n) b n )= χa) χb). EpÐshc e = e e kai, epomènwc, 'Ara o χ eðnai qarakt rac thc G. Gia k je a j G j, profan c, isqôei χe) = χ ) e) χ n) e) =. a j = e ea j e e, opìte χa j )= χ ) e) χ j) a j ) χ n) e) = χ j) a j ).

44 42 KEF ALAIO 4. QARAKT HRES 'Ara o periorismìc tou χ sthn G j tautðzetai me ton χ j). 'Eqoume, loipìn, apodeðxei ìti h parap nw apeikìnish eðnai amfimonos manth. Ĝ Ĝ Ĝ2 Ĝn Apì thn pr th perðptwsh gnwrðzoume ìti k je Ĝj èqei t xh m j, opìte to kartesianì ginìmenì touc èqei plhj rijmo m m n = m. 'Ara kai h Ĝ èqei ton Ðdio plhj rijmo m. Prìtash 4.2 An G eðnai peperasmènh abelian om da kaia G, a e, tìte up rqei qarakt rac χ thc G ste χa). Apìdeixh: Ja qrhsimopoi soume to sumbolismì thc apìdeixhc tou Jewr matoc 4.. 'Estw ìti h G eðnai kuklik G = {a, a 2,..., a m,a m = e}. An ω =cos 2π m + i sin 2π m, tìte o qarakt rac χ pou orðzetai mèsw tou tôpou χa k )=ω k èqei, profan c, thn idiìthta naapeikonðzei k je stoiqeðo a k thc G pou den eðnai to e dhlad k m) searijmì. Sth genik perðptwsh G = G G 2 G n, an p roume a = a a n e, tìte èna toul qiston a j eðnai e. 'Eqontac telei sei me thn eidik perðptwsh, gnwrðzoume ìti, gia to Ðdio j, up rqei χ j) qarakt rac thc G j ste χ j) a j ). JewroÔme, t ra, to monadikì qarakt ra χ thc G, tou opoðou o periorismìc sthn G j tautðzetai me ton χ j) kai o periorismìc tou se k je llh G i eðnai o kôrioc qarakt rac thc G i. H kataskeu tou χ perigr fetai sthn apìdeixh thc perðptwshc 2. tou Jewr matoc 4..

45 43 Tìte, fusik, χa) = χ ) a ) χ j) a j ) χ n) a n )= χ j) a j ) kai h prìtash apodeðqjhke. 'Estw peperasmènh abelian om da G t xhc m. Ja upologðsoume gia k je qarakt ra χ thc G to jroisma S = a G χa). StajeropoioÔme b G. Tìte ) χb)s = χb) χa) = χba) = χc) =S a G a G c G kai diakrðnoume tic dôo peript seic:. An χ χ,tìte up rqei b G ste χb). Qrhsimopoi ntac autì to b sthn ) paðrnoume S =0. 2. An χ = χ,tìte S = a G χ a) = a G =m. EpÐshc, ja upologðsoume gia k je a G to jroisma T = χ Ĝ χa). StajeropoioÔme ψ Ĝ. Tìte ) ψa)t = ψa) χa) = χ Ĝ ψ χ)a) = ω Ĝ ωa) =T χ Ĝ kai diakrðnoume tic peript seic. An a e, tìte apì thn Prìtash 4.2 èqoume ìti up rqei ψ ste ψa). Qrhsimopoi ntac autì to ψ sthn ) paðrnoume T =0. 2. An a = e, tìte T = χ Ĝ χe) = χ Ĝ =m apì to Je rhma 4..

46 44 KEF ALAIO 4. QARAKT HRES 'Eqoume, loipìn, apodeðxei to Je rhma 4.2 'Estw G peperasmènh abelian om da t xhc m. Tìte gia k je qarakt ra χ thc G isqôei { 0, an χ χ χa) = m, an χ = χ. a G EpÐshc, gia k je a G isqôei ìti χa) = χ Ĝ { 0, an a e m, an a = e. QARAKTHRES mod m 'Estw m jetikìc akèraioc. JewroÔme to sônolo G m twn kl sewn upoloðpwn mod m oi opoðec eðnai pr tec proc to m. G m = {[a] : a m, gcda, m) =}. EÐnai gnwstì ìti h G m apoteleð abelian om da me pr xh tonpollaplasiasmì [a][b] =[ab]. H t xh thc G m eðnai Ðsh me ϕm). Par deigma: 'Estw m =9, opìte ϕ9) = 6 kai G 9 = {[], [2], [4], [5], [7], [8]}. EÐnai eôkolo na elègxoume ìti h G 9 eðnai kuklik me genn tora to [2]. 'Ara oi timèc opoioud pote qarakt ra χ Ĝ9 kajorðzontai apì thn tim χ[2]). To sônolo twn èktwn riz n thc mon dac eðnai to {ω, ω 2,ω 3,ω 4,ω 5,ω 6 =}, ìpou ω =cos 2π 6 + i sin 2π 6 kai gnwrðzoume apì thn Prìtash 4.4) ìti oi timèc pou mporeð na p rei to χ[2]) eðnai k poia apì autèc tic èktec rðzec thc mon dac. Sthn perðptwsh thc apìdeixhc tou Jewr matoc 4. eðdame p c kataskeu zetai qarakt rac thc G 9 tou opoðou h tim sto [2] eðnai opoiad pote proepilegmènh èkth rðza thc mon dac. MporoÔme, epomènwc, na broôme ìlouc touc qarakt rec thc G 9 twn opoðwn to pl joc eðnai 6) kai na fti xoume ton pðnaka [] [2] [4] = [2] 2 [5] = [2] 5 [7] = [2] 4 [8] = [2] 3 χ χ 2 ω ω 2 ω 5 ω 4 ω 3 χ 3 ω 2 ω 4 ω 0 = ω 4 ω 8 = ω 2 ω 6 = χ 4 ω 3 ω 6 = ω 5 = ω 3 ω 2 = ω 9 = ω 3 χ 5 ω 4 ω 8 = ω 2 ω 20 = ω 2 ω 6 = ω 4 ω 2 = χ 6 ω 5 ω 0 = ω 4 ω 25 = ω ω 20 = ω 2 ω 5 = ω 3

47 45 Orismìc 4.3 An χ : G m T eðnai qarakt rac thc G m,tìte orðzoume mia nèa sun rthsh χ : Z T {0} me ton tôpo { χ[a]), an gcda, m) =opìte [a] Gm ) χa) = 0, an gcda, m) >. H sun rthsh aut onom zetai qarakt rac mod m. Oorismìc eðnai kalìc, diìti, an èna stoiqeðo miac kl shc mod m eðnai sqetik pr to meto m, tìte k je llo stoiqeðo thc eðnai epðshc sqetik pr to meto m. Kai, an èna stoiqeðo miac kl shc mod m den eðnai sqetik pr to meto m, tìte kai k je llo stoiqeðo thc den eðnai sqetik pr to meto m. H sun rthsh χ dðnei thn Ðdia tim se ìlouc touc akèraiouc pou brðskontai sthn Ðdia kl sh mod m. H tim aut eðnai akrib c h tim pou dðnei h sun rthsh χ sthn koin kl sh pou perièqei autoôc touc akèraiouc, an autoð eðnai sqetik pr toi me to m. An autoð den eðnai sqetik pr toi me to m, tìte h tim pou paðrnoun apì thn χ eðnai 0. Merikèc idiìthtec tou χ eðnai. χ) = χ[]) =. 2. 'Estw a bmodm. An gcda, m) >, tìte gcdb, m) >, opìte χa) = χb) = 0. gcda, m) =,tìte gcdb, m) =kai χa) = χ[a]) = χ[b]) = χb). Dhlad, se k je perðptwsh En, an χa) =χb). 3. χa)χb) =χab) gia k je a, b Z. Aut h idiìthta apodeiknôetai wc ex c. An èna toul qiston apì ta a, b den eðnai sqetik pr to me to m, tìte oôte kai to ab eðnai, opìte kai oi duo pleurèc thc isìthtac eðnai 0. 'Estw, loipìn, ìti gcda, m) =gcdb, m) =. Tìte gcdab, m) =kai χa)χb) = χ[a]) χ[b]) = χ[a][b]) = χ[ab]) = χab). 'Ara kai s' aut n thn perðptwsh apodeðqjhke h isìthta. Hapeikìnish χ χ eðnai èna-proc-èna. Pr gmati, èstw χ, ψ qarakt rec thc G m me χ = ψ. Dhlad χa) =ψa)