EXAMENSARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET
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(3) Rings of arithmetic functions with regular convolutions. Elin Gawell. Examensarbete i matematik 20 po¨ang Handledare: Jan Snellman 2005.
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(47) ïLðSíóïî ÷'öúø3ï ì÷!è ÷'ð`î ïìõï ï í ï êxé)õ¬ölè_ö ÷*í ü ï ë ní ï?î'õk÷Cø'ì÷'ñ í ïWõlð ïLìè%ïWö õlñ?õ[íRë ïLì ë ví ï ÿ ìï M "ON ê`ìëCî'é ü%ílè?ë ní ï ÿ ìèLí ÿ në ïLíóìJïYê`í ìëC÷ î'í=é í ü%í ï_õ¬ð ïLìè%ïcö õlñ?õ[í õ¬èJê ïLìYî3ï ÿ ðõ[í[õ[ë'ð|õ¬èë ñ_ë ìqê õ[ü
(48) íqë\í ïcõlð ÿ ðSõlíóï σi : lim Γ[Vi ] → Γ[Vi ] ←−. (f0 , f1 , f2 , . . . ) ∈ lim Γ[Vi ] ←−. σi. fi ∈ Γ[Vi ]. σi = σji ◦σj. αi : Γ[W ] → Γ[Vi ] P P∞ f = n∈Vi f (n)en n=0 f (n)en. σji αi = σji ◦ αj. i ≤ j. α : Γ[W ] → lim Γ[Vi ] ←− fW ∈ Γ[W ] f = (f1 , f2 , f3 , . . . ) ∈ lim Γ[Vi ] α ←− 0 0 α f ∈ lim Γ[Vi ] f = (f1 , f2 , f3 , . . . ) ←− fi ∈ V i σji (fj ) = fi i < j f ∈ lim Γ[Vi ] ←− fW ∈ Γ[W ] fW ∈ Γ[W ] W w ∈ W ∪Vj = W V 1 ⊂ V2 ⊂ . . . w ∈ V j j j fW (w) = fj (w) k >j TS fk (w) = fj (w) σkj (fk ) = fj fW >S αi (fW ) = fi α . VU p ∈P B 1 , B2 ∈ πp W Bi0 = pj j ∈ Bi 3 α. ! >#. "!. X&. Y+. !. #. Γ[Bi0 ] ∼ = C[[x]] O #. 9#. B1 6= B2 !. !. 9#. Bi0. Γ[Bi0 ] ∼ =. 6. Γ[B20 ] 6 Γ[B10 ∪ B20 ] ∼ = Γ[B10 ] × / C $ C R R → C ← S3 !. !. Γ[{pi |i ≥ 0}] ∼ (× Γ[Bj0 ]) 6 = lim ←− C 9# ! n3 [. !. /. !. C[x] # x`. . Bi0. ×C(Bj ∈πp ) Γ[Bj0 ] 3. \. `. & !. S6 . 3. 3. R ×C S 3. n. !. . !. !. 9#. !. !. !. /. /. Z.
(49) ìëCë P[$'oM1 *$'N Y F )Y1 C !&1¹S ! ,+_] +,-%1QS ! H¥$'$'Q 1T$'Y [ P .01$' CY9v !$ PQLr5] F%1T1,+& w"%$'R&)1TT ! !Y,+&R1J Z,+& (*w%".0$'$' -4v1]S7V '"%nZ.=$a,+1T %\9\7S 'S1 !,-1 ô ,+1= - 7V#,+&6K& ,+ FQs#LK& K. *#. Bi ∈ π p a. 3. pj ∈ Bi0. Bi = {0, a, 2a, . . . } j = ka. k ∈ N Γ[Bi0 ] Bi. FGb&$'P[$' H F. Bi = {0, a, 2a, . . . , ra} k∈N pj = (pa )k {epj |j ∈ Bi }. ϕ : Γ[Bi0 ] → C[[x]]. ,-1?,-1c v$'Y$'Y$'] ,-11,+&"% !&4 ,+&6K& ,+a,+_] +,-%1cS ! ,+ó&'c%,-1?".o,+(* + JFÔ4 )L,+K&&"%% 4fH ! !&&,-4t1w4 ,+1&6,+&K"%& a,+;Pk\$'a$'& ! +&)99 ,+& /S !_
(50) 8 !]L1cr5,-1$ 1a ,-1 HQ,+ ä 1 F 1"8S ! H,qFú F Pk$'2 !&)9 ?Lr5,-11 cP[ &",-$'& 1"8S ! 1"C 'Y1 1CHx
(51) !<*%1 1 F0[F %" ,-,+1O(*H` HI !& !4;&48n&Pk '"%"?b,+ S,- !1Z !&/,-1$'Y !$' -1$w] "%,-$'1&3;T !F,+&51 $a,+,+&6&/K& ,+,-1 : $O.e - 7xFQUK: & $O,+. 4 .=L,+K& n -2YPk$' +&3 -$C1O.=F«,+&
(52) & $'Y$'Y$'S] ' 1«,- 01 7S '1,-1 j. epj 7→ x a. e pj ∗ e. pm. = epj+m. Bi. j = ka. j, l ∈ Bi Bi j+m j m a ϕ(epj+m ) = x = x a x a = ϕ(epj )ϕ(epm ) 0 0 k ∈ N j ∈ Bi xk e pj. j. Γ[Bi0 ]. ϕ(epj ) = x a = xk ϕ 0 ∼ Γ[Bi ] = C[[x]] Bi0. Γ[Bi0 ]. `. . . . , ep(`−1)a }. ϕ : Γ[Bi0 ] →. {e1 , epa , ep2a ,. C[x] x`. q$' P 4 ¹$=
(53) !<*H5 7 .0 + )4 LK&%4fH H5,qFú F 5H «F G & 11 Y ),+&"% ,-1=.o + «4 LK&%4W !&4v '1b !7V$C(*n.oZ*=S ! F M 7`N^s#LK& j. j, m ∈ Bi. ϕ. j +m ∈ / Bi j. epj 7→ x a. j + m ≥ `a ϕ(epj )ϕ(epm ) = 0. m. ϕ(epj )ϕ(epm ) = x a x a = x. j+m a. `a+n. H 1$?,+& F H. e pj ∗ e pm = 0 j+m = `a+n n. n. = x a = x` x a = 0 · x a = 0 ϕ C[x] Γ[Bi0 ] ∼ = x`. g1 : Γ[B10 ] → C. #s LK& !&S,pP ! -$'*$'1OF ,pP F qP & !&4 !&4 F 31 !&4 {e !Zn.o"C !&; + «&4 $OL. K&46%T4v %. 1 [%Z"P[,-$'$' +& -$O1C.=F ,+& w"%$'_ T !,+(*?46, ! T ! f 7→ f (1). g2 g1 (epi ∗ epj ) = g1 (epi+j ) = epi+j (1) = 0 = i + j ∈ B1 , i + j 6= 0 g1 (epi ∗ epj ) = g1 (0) = 0 epi (1)epj (1) i+j ∈ / B1 i+j = 0 i = j = 0 g1 (ep0 ∗ ep0 ) = g1 g2 g1 (e1 ∗ e1 ) = g1 (e1 ) = e1 (1) = 1 = 1 · 1 = g1 (e1 )g1 (e1 ) R. Γ[B10 ∪ B20 ]. OOOq2 OO'. q1 oo. wooo. Γ[B10 ] P. PPP P g1 PPP( ]. C. nn nnn v nn g2 n. Γ[B20 ].
(54) !&4 !n4 LK&%4\,+&vZ&S ! T ! I.0 C9. q1. ^R. q2. X. q1 (. f (n)en ) =. n∈B10 ∪B20. X. f (n)en. n∈B10. {eZS %(*R c$'Y$'Y$'] ,-1. p1 : Γ[B10 ] ×C Γ[B20 ] → Γ[B10 ]. e{ Z,+ !& -146$c"%S% C4\(*$' cY$'$'YY$'$'Y] $',-1] ,-1 ,+&; 4 2 LKPk$'& +% -4W$C.= !,+&S& !a -$'46*, $' ! T1J ! $ F ,-1= ,+(*%1 (f1 , f2 ) 7→ f1 p2. R. ψ. p1. Γ[B10 ] ×C Γ[B20 ] O. NNN NNNp2 NNN ψ NN' 0 0 / Γ[B 0 ] Γ[B1 ∪ B2 ] q2 2 p. p1 pppp. Γ[B10 ] o. ψ. ppp wppp. ,-1J4 LK&%4;,+&;2P[$' + -$O.=,+&
(55) . %9 q1. _R. ψ(. X. f (n)en ) = (. X. f (n)en ,. X. f (n)en ).. -, 1J !&,+&'4 %",+(*HSR !&14;,+&"% &"%,+J '"<61b1 ] ,+_]x$'] + ,-%12,+& S ! '"Fo<6«1 $
(56) 1 1] ]x$O. $' S,+&v ! 7V$',- 1 ,-1 F 1 & [,-~3%&" +,+9n(* 4 H*/T ! !<*_W,+& %4%8739Z]x$ 1T1,+( 7 !,+ + +,+%,-%1I1C,+H¹T !<*%FX1¹)$'&,+&"%J(*H 9 H¥T !<* $' H v F ó P U H H p , P U H ,pP &IH«1,+&"%Y.=,+Y + 46, ! LT !Pk$'J"%7V$' _.oR + pjó%4 1 LK&%4fF &"%J !,-1&4|L$;r5,-1 -1 !&4\,-1= & ,-~) H &"% ,-1=1 [%",+(* F n∈B10 ∪B20. n∈B10. ψ(f ) = (0, 0). ψ B10. B20. (f1 , f2 ) ∈. Γ[B10 ]×C Γ[B20 ]. n∈B20. f B10 ∪ B20. ψ. Γ[B10 ∪B20 ] w ∈ B10 ∪B20 B20 \ {1}. f∈ B10 ∪B20 w ∈ B10 \ {1} w ∈ f (w) = f1 (w) w ∈ B20 f (w) = f2 (w) f1 (1) = f2 (1) GS f. w =1 w ∈ B10 w =1 f (1) = f1 (1) = f2 (1). ψ. ∴ Γ[B10 ∪ B20 ] ∼ = Γ[B10 ] ×C Γ[B20 ]. M "ON^s#LK&Z#Pk$' + -$C.=,+&
(57) $'Y$'Y$'] ,-1Y1
(58) !]1 $. σji : Γ[B10 ] ×C · · · ×C Γ[Bj0 ] → Γ[B10 ] ×C · · · ×C Γ[Bi0 ],. σji. σj. (f1 , f2 , . . . , fi , . . . , fj ).
(59) !]1. (f1 , f2 , . . . , fi ). F. σj : lim(×C Γ[Bj0 ]) → Γ[B10 ] ×C · · · ×C Γ[Bj0 ] ←− (f1 , (fk )2k=1 , (fk )3k=1 , . . . ). $. (fk )jk=1. F. αj : Γ[{pi |i ≥ 0}] → Γ[B10 ] ×C · · · ×C Γ[Bj0 ]. B. i≤j.
(60) -, 1J4 LK&%4\,+&;Z1 !YZ.0 C9\ '1 +, & M 7`NLF ,-1b" -C ! +9\ c1 [%",-$'&IF %1Z,+&46"% ?$'Y$'Y$'] ,-1. αj. ψ. αj. α : Γ[{pi |i ≥ 0}] → lim(×C Γ[Bj0 ]) ←−. $'1T !73Y(5T !,-Z$'<*. %1Z1; %9; !S&ù ' !1= ,+ -&\Y,-1#&)Z,+&' ] %"$5$!,+PU(* F?$
(61) S !%$',-1Z¯61F° ¯6[F $ %",+(*_"C !&/7xc1$O.=&e,+F&/óä 1 f 0 ∈ Γ[{pi |i ≥ 0}] α. α. α. ¢£¥´ X´SO§¬²O§l´`÷ ð`© î ®fç+¼Iç ïí . ∴ Γ[{pi |i ≥ 0}] ∼ (× Γ[Bj0 ]) = lim ←− C. ïð. p, q ∈ P, p 6= q 3 1 ∈ B10 ⊂ {pi |i ≥ 0} 6 1 ∈ B20 ⊂ "! 0 0 i j i 0 j B1 B2 = {p q |p ∈ B1 , q ∈ B20 } 3. ÷'è ÷'öúø3ï ì÷!è õ ï /. $. ïí. U. 2U. {q i |i ≥ 0}. C. (α1 (f 0 ), α2 (f 0 ), . . . ). 6. 3. 3. b C Γ[B20 ] Γ[B10 B20 ] ∼ = Γ[B10 ]⊗. Γ[B10 B20 ]. è%ë'ö ïèRí ï ë ö¬ö+ë õlð)øYé5ðõ ïLìè÷'ö ê`ìë ö+ïñ . ! >#. '&. /. . 6. Γ[B 0 ]. 1C t CC ϕ σ1 ttt CC 1 t CC tt C! ztt ψ 0 0 Γ[B1 B2 ] _ _ _ _ _ _ _ _=/ D dJJ { JJ {{ JJ { { {ϕ σ2 JJJ {{ 2. Γ[B10 ]. ï÷üë'ìñ ï ê`ö-ïíóï ÷'ö ø)ï ì÷ ÷'ð`î ÷'ìï ë'é5ð`î3ïî ÷'ö ø)ï ì÷ ë'ñYë'ñYë'ìó÷'ê ìï õ¬èñ?ë é)èRð`õlî)ðSïíqîë L ÷'öúø3ï ë ì÷í ë'ñYë'ñYë'÷'ìóð`ê î õ¬èñ?è ÷ ìï ÿ ðõ[íóï í ïvüë ñbêxö+ïíóï8íóïð èë'ì?êxìëCî é6ü%í_ü÷ ð ï ìïóê`ö+÷*üTïî ýaë ìî'õlð`÷'ìLý
(62) íóïLðèë'ì0ê`ìëCî'é ü%íJë ïLì. ìëCë s#LK& /. & !. C. $. !. /. O #. !. /. K. $. /. !. ϕi : Γ[Bi0 ] →/ D, i = 1, 2 C $ 0 C D σi : Γ[Bi ] → Γ[B10 B20 ], i = 1, 2 6 /. *#. !. Γ[B10 ]. 3. Γ[B20 ]. . 6. !. /. /. !. C3. 3. σi : Γ[Bi0 ] → Γ[B10 B20 ],. X. cj ej 7→. X. cj ej ,. i = 1, 2. $' 1% +1T9tY 8 !"%_$'_$'73] (5 -,-$' 1 +9/jd !7x +$'* 7 &T4 6%F4 d&jd !$' +*4 7 YT ^$/$'(*Y$',pP[Y9A$'] ^ ,- 1Y& ,+1C(*F 1T ! b] $']x,-1nd9A$'73Pk(5$',pj H) - HI.=4 ,-&"$'A2,-1Z7V $' &jq4 7 %,+ +4|,+&73C9 !
(63) !] FYóP $'&Y"C !&] $C(*YS ! ,-1Z,+&6j 46"%%10 # & ,-~)=7x$' &4 %4 jd ! +*7 T Z$'Y$'Y$'] ,-1 1"T
(64) S !X=P[$' + -$O.=,+& . Γ[B10 B20 ] Φ ϕ1 (f1 )ϕ2 (f2 ). j∈Bi0. Γ[B10 B20 ]. C. C. C. C. j∈Bi0. Φ : Γ[B10 ] × Γ[B20 ] → D, (f1 , f2 ) 7→ |ϕ1 ||ϕ2 | ψ. @Cy.
(65) 46"%$', !_ RT ! }T !"%$',+(*_ RF %1CH &? ,-1 .=,+ + ! -1$
(66) !<*Q 46, ! T !Ô,+&Ro] $']V$ 1,+,-$'& ψ. ∗. Γ[B10 ] × Γ[B20 ]. LLL LLΦ LLL LLL &. ψ. v. v. / Γ[B 0 B 0 ] 1 2 v v. ,-1? 4 7 ,+ +& !,+&$'RC%"% !1
(67) $'
(68) &3 !(*&$' 4A + + ,+7x] ,-$'$' +,-&I "C&H !4 !,-%&$'4f&I4/FH ,q Fú &Ft"%w_7 ,+#,+\ +,+,-&1"%C$' ' !&31(*1,+d$'$59 +" , Pk !$',-$' +,+ -(*&I$C_.JF !1?ó&cP[4eR$'46 ,-11Y,+7V7 Y^ Pq '(*,+"(*c ,pF_KSS% !4S !S !,-1_,- 1 7V:2$'$C .n&H4 -%4\ ,-1=(*,pKS%4;739; !7V&;#C '1n9^1"C !J -"$! PX ! ! + ¹,-$'&I -F Y&31 $'&;#Pk$' D. {v. ∗. ∗. ∗. ∗. W ⊂ Γ[B10 B20 ]. N M X X. cij epi qj ∈ Γ[B10 B20 ]. .= FX &;.0 !ZaSK %(*& n,+ F¿#ãIP[$' + -$O.=,+& w46, ! T ! i=0 j=0. Γ[Bi0 ]f in ⊂ Γ[Bi0 ]. M, N. Γ[Bi0 ]. Γ[B10 ]f in × Γ[B20 ]f in. 7x\\1TY$!P K& ,+;1 Y1_,+&. ∗. OOO OOOΦ OOO OOO O'. ψ. . . . /W . &A1O,+Fn{e,-1#"S -C % !(*nSS ! ! ,-14 _,+&%4A739/_( ! +%1$!P $!P= ä !&4v1,+&"%#n46, ! T !òS '1b$c7xZ"%$'_R T !,+(*.0ZS %(*S ! D. ψ(f ), f ∈ W. e pi q j. ψ. Φ(epi , eqj ) = ϕ1 (epi )ϕ2 (eqj ). =.z0 0,-"T1W,+&1T"% C 9wS !bnLr5,-1.0b ?"C !&\& ,-~3 !] ] +9_] 1$'"]x;$ 1S,+ !,-b$'&^ DZ,-1JPl46$', ! T"S ! !] "%$'Z_@ RF+@ F Z%,+1 & °¯ ψ(epi qj ) = ϕ1 (epi )ϕ2 (eqj ). `\. W = Γ[B10 B20 ]. A B. ψ. ψ. W _. 2&"% S. . Γ[B10 B20 ]. Γ[B10 B20 ]. " -C !qbP S ! !&4. ψ. /D. 1$' +(*%1=Z & ,+(*1 ! I] $'7 - !&4\&"% /D. b C Γ[B20 ] Γ[B10 B20 ] ∼ = Γ[B10 ]⊗. !#K& ,+ 1JF1<5,+]^Z '1J1]\,+&;#] $5$!PX !&4\,+=,-1. Γ[B10 ] Γ[B20 ] 0 Γ[B1 B20 ] ∼ = Γ[B10 ] ⊗C Γ[B20 ]. @ @.
(69) ¢£¥´ X´SO§¬²O§l´`©®Vç ÃVç ïí . HU. Γi = Γ[{pji |j ≥ 0}] 3. "!. ïLð. b ...⊗ b C Γn Γ ⊗ Γ∼ = lim ←− 1 C n→∞. 1TìYëCë ,+&46ãI" ,-$'&t$'&A] $']V$ 1,+,-$'&¯6F ûv.oFYw *?&S73 !98 ] ,+& &$O. .o"C !&W !] ] +9^%$'¯6F°¯c.= ,-"v ,+(*%1JZ%1 +CF K. *#. 3. Γ1...k = Γ[{pj11 , . . . pjkk }]. Γ1...k. @C¯. F\z0& 4 !. Bi0 = {pji |j ≥ 0} ∼ bC . . . ⊗ b C Γk = Γ1 ⊗.
(70) Ç ÈÊÉtËÍÌRÎWÏ b. a. Õ Ô ×Ó Ò × cN. d. cN. fe. hg. "%Ft$'&) T !,+&& 1 G2@ F11s# YLKW&;S 8 ! ]S ! , !,+(* b%$'1\4 !\"%$'$'&)& (*$' + 73,-$'9 &IH# !&4ùS !, , b 1 ? 5 $ C " ! + + w 9 K & + , n x ] $ 1 C H T 1 c $ o . n C " ! v & 4 L K & + , Z 1 l õ ` ð ü k õ 3 î L ï ` ð T ü c ï ÷ ö 3 ø ï ì ÷ ]V] $'$5,+&346d.=H ".=,- 1\ ,-"T 'w4 46"%,+$'&,-$'1&t,-1 !1&$!4APf ! + +jq,+] ( ! +,- +"C !%4?,-$'Pl&¿ &$!"Pb,-1T$'"C& !1o !$'&Y1CH" ! -$ &14%4c.=,+,+&3( !
(71) -1X"%,+$'& &)(*$' +H .=,-,+$'& M E6F+@ON d=,-1= ,+w(*ì&;ïî'é üTnïî|$'õ¬]xð`$'ü% -õ[$'î3 ï9^ð`üT$!ïYPU]V÷ ö $'ø3,+ï &3dì.=÷ ,-1"%$'&)(**$!P &"% F "%$'&1,-11#$!Poc1 7S ! +*7 T $! !Po&4 Pl &",-$'& !1n.="% $',-&"1|,-4 T !<*%c4v%~)_ ,+1T(' ! !Y -?&)J( !,pP +_$'&/%~) ,+F( ! -&3Z,+&3( ! -1CH¹.= õ¬èZõ è%ë'ñYë'ìóê õküíqë ½ ¹¤)´S£!¤)¨ ¶Iç[¡Sç èR÷wíqëê6ë öpëø õ[ü÷ ö ÷'ö ø)ï ì÷. ìëCë b%4 ,pP &8%b4 %4 ,-1 n1%46"T\"%%S4W ! ,+&",-4 &"%? ! +*7 T w$!P $C(* F= n
(72) !] ] ,+& A. ≤=≤A. W = (W, ≤A ) I(W ). m ≤A n. W. ji. W ⊂ N+ m ∈ A(n). C. /. W. f ∗ g([a, b]) = /. X. f ([a, c])g([c, b]). a≤c≤b. Red(W ). W. [a, b]. [c, d]. @? K. b/a = d/c. HF. *#. C. /. $. 6. 3. (W ) = {f ∈ I(W )|f ([x1 , y1 ]) = f ([x2 , y2 ]). ϕ:. (W ) (W ) → Γ[W ]. W. f 7→. ,-1 ?7 , %",-$'&IFX &IHS,pP !&4;,pP. !. Red(W ). %4. ∞ X. y1 y2 = } x1 x2 C. f ([1, n])en. n=1. f, g ∈ (W ) X X n (f · g)([1, n]) = f ([1, d])g([d, n]) = f ([1, d])g([1, ]) d d∈A(n). ∞ X. n=1. d∈A(n). ∞ ∞ X X cn en an en ∗ bn en =. @CE. n=1. n=1. Γ[W ] 3.
(73) & cn en =. X. ad ed ∗ b nd e nd =. X. ad b nd en. .="%$' ,-"T%\1]x,+_$'&] 4 +1Q,-%1 $n +,+] +,-"C !,-$'&^,+& F ,-1=F1d$OP[.J$' +1 -$O.JS1Q ! S ! 0 +b,+] %4 +,-"C !,-$',-&;1,+,-&\1$'bY%$'4 ] ,-" $ F åS '¤31?æ©U§¬a²O§l ´`&©4 ¶Iç +®V95ç ,+& / !1T&4 ! $' !]x&4T !,-$'^&10$'$'4 &
(74) ]Vc$ ,-11c4 1XLK4 &LK%4t&% '1 1 '1Pk$' + -$C.J,1CF ,+ !&S4 '1 ,+& '1bH c$' &4 +95,+&
(75) !1&4CF ,+,& |,+F !&4 ,+& H $' !&4 ,+& H $' F Pkå $'b¤3æ !&3©U9 §¬²O§l´`©¶fç-¶fç H5GÔ,+ & 4 +9),+& ^$'1 $!FPo _]S !, ! + +98$'4 %481Tb,-1# Wü ÷'õ¬ð|,pP å -¤3Yæ©U&)§¬²O§l´`©Ô¶fç+¼IçP[$'G b !o + ïîCø)ï\ë H w1ü "v÷'õ¬ðSè
(76) !,- 1R v1$!P[PJ$'b"S ! ! + ,+&1 F .=,+ !& cn =. Γ[W ]. P. ⊕. u, v ∈ P P ⊕Q. u≤v P ∪Q. d∈An. d∈An. an b nd. (W ). Γ[W ]. (W ). + P ∪Q P u, v ∈ Q. • u, v ∈ P. u≤v. P. • u, v ∈ Q. u≤v. Q. d∈An. u≤v u≤v. u ≤ v. Q. ji. • u ∈ P, v ∈ Q u, v ∈ B. k&. w ∈ Ci. u≤v 9#. !. !. B v≤u. i∈I. P +Q. ji. {Ci | i ∈ I} u ∈ Ci. w≤u. ô ,+ E6F+@ Xur !_] -%1 $!P c"TS !,+&8 !&4v ?.0%46*$!PU"TS !,+&1 {eZ ,+(* c"%$' &3]S !2$ $']V$ 1,+,-$'&W¯6F°E ¢J÷'£¥ìL´ í[Xõk÷'´SölOöþýa§¬²Oë'§l´`ìî)©ÔïLì ¶Iç-Ãfç ÷'ð`ïî í ÷'ð`÷!èîW÷ ö-ïíë ï ïí î3ïð`ë ö+íóï%ïRí í ïð`÷ í[é5ì÷ öIðé5ñ Lï ìTè í õ[ï í %ë'í ölöpïë õ¬ì2õ¬ð`ð)ø÷ í[é5ë'ì÷ ö+î öfë ìî3ïì ÷'ðxî í ï2õlð`î'é üTïîYèé ê6ë!èCïíoë'ð Lï ð õ õ¬èZõ¬ð ÿ ðõ[íóï ÷'ð`î õ ÷!è ïLö+ïñYïðSílè M *N @û XR. ml. . VU. K. &. !. ! >#. Bi0. !. X&. Bi0 ∼ =N. !. Y+. #. Bi0. p ∈P B10 ∪ B20 6 [n] 6. nR. !. /. B 1 , B2 ∈ πp W U N/ 3. ∼ = [`]. #. Bi0. !. `. Bi0 = !. . pj j ∈ B/ i. {1, 2, . . . , n} 3 3. "!. 3.
(77) M 7`N. O #. K. *#. . B10 ∪ B20 ∼ = [1] ⊕ (B10 \ {1}) + (B20 \ {1}). <. !. 9#. B10. pi i ≥ 0 V&. !. !. 6. *o. B20. "!. 3. !. õë ï í ï ÷'ðxî ÷'èTè%ï?î'ðxõk÷Cï ø'í?ì÷'íqñ ëëï÷*ü ë í ïLì õ èí ë ïLðíq÷'õkõ¬î3ðVïLïðSî í[õ Lý#ý!õlê`ð)ö+ø÷*üí õlð)ïWøaí í Qëtï?î'ïõkö+÷CïLø'ñ
(78) ìïL÷'ð`ñ?ílèè üë'ìLìïLèqê6ë'ðxî'õ¬ð5ø\íqë ¬ õ a è ÷ o ï C î 3 ø W ï ë ^ ü ' ÷ l õ ð è ' ë x ð ï ' ë \ ì ï * ÷ ü + ö C ë ü % ' ë l õ x ð ï î íqëø3ïí ïLì õ[í í ï?ïö+ïLñ
(79) ïLð`í ÷!èní ïñ?õlðõlñcé)ñ ïü ÷'Lõlìð ïwõ í è
(80) ï÷ ìíbïRñYõ¬èë!÷wèLí2üë'ë ðxñ?ï
(81) ñYõ¬ë'ð ðÿ ðõ[ë íóé)ïYð`ü î^÷ ë nõ¬ðtí õlðïnö-ïLü ð)ø*í ÷ èRð`îWë nõ Yí í ï?ïë ìí ïwïõ¬ìRè
(82) ü ÷'ðA÷'õlõlð ð èÿ ðSõlíóï H Fs#LK& M *N^ã¹ 7V#,+&6K& ,+ F0),+&"% 3 3 9#. M "ON M 4SN. ìëCë. í ïLð. B10 6= B20. 8. A&. !. B10 ∪ B20 !. !. /. /. #. 6. !. !. !. (&. 3. 9#. !. !. !. 1. 2#. 6. !. 9#. Bi0. Z. 3. - .. !. /. /. !. !. 6. #. !. 9#. Bi ∈ π p 6. P. !. !. !. !. 3. Bi ∈ πp Bi0 = {1, pa , p2a , . . . }. ϕ : Bi0 → N. ,-1J,-1J$'7)(),-$'1 +9^ c]V$ 1=,-1$'Y$'] ,-1;F ãI 7x2K& ,+ Fo: $O.. pra 7→ r + 1. Bi0. Bi0 = {1, pa , . . . , p(`−1)a }. FXs#LK&. ϕ : Bi0 → [`]. Gb -1T$Y !&v$'73(5,-$'1J]V$ 1=,-1$'Y$'] ,-1;F ,-1nR]V$ 1#.=,+| &4 +95,+& v1 M `7 N !&4 , /,+ H$' !&4 H !&4 F s#LK& pra 7→ r + 1. [1] ⊕ (B10 \ {1} + B20 \ {1}) ji u≤v p. B10 ∪ B20. u=1. p. p. u, v ∈ B10 u, v ∈ B20. u≤v u≤v. ψ : B10 ∪ B20 → [1] ⊕ [(B10 \ {1}) + (B20 \ {1})]. YóR$',-1R] $' 7),-1() ,-$'] 1R%P[1T$'(*%1=^n !7V$'$C4 (*wCF S2 !R&"%Zw,+ ,+ä&31b(* !&;1a]V$ $!1PJ= ,-1,-1R$'Y7 ,$'%] " ,+,-(*1a;F $'Y$!j M O" N ô $' + -$C.J1bP[$' %$'¯?,+& +@ óF B1 ∪ B2 3 pra 7→ pra ∈ [1] ⊕ [(B10 \ {1}) + (B20 \ {1})] >S. . @. q. H.
(83) M 4SN^G 1T1 Y#S !Jn46, ! T !òS '1Jó.o$
(84) ,+&6K& ,+"TS !,+&1CH !&4 F & B10. B20. B1 = {0, a, 2a, . . . }. 0z b1,+&"% !&4 H .onS %(* _"%$'&)T '46,-",-$'&IF G 1T1 YaS !?F \46&W, ! T !? "%$'_S 'Y1_$'$'&/&^7x$',+&6 K&& 48,+$!\PX"T -S& ! ,+&I/HQ1T$! CPQ9 $'FeãIZ "S !,+&1 S %(*$¿7V 7V% !"C& !4\1|,pP
(85) &v ,+="T.oS$' ! ,+&I -4;H ,+&)H#1S% '"1; -& F & ¢÷ £¥´ ïcXí ´SO溺²Oè%§l÷'´`ñ
(86) © ïR¶Iö+ïç-µfð)ø ç í 8í÷'ö¬ïLö2ðeü í ÷'õlïRð ö-èvïLð)õlø ð í ÷ ÷ ÷!ïcèè%íqï/ë î'õkï ÷Cø'ì÷'ñØë ìRë cí ê6ïcë!èCî'ïõkíl÷Cè ø'ì÷'ñ õ¬ölö ï üTë'ñ ê6ë!è%ïî\ë ë'ðö ýwë'ðxïõlð ÿ ðõ[íóïnü ÷'õlð -$' & & ì +ëC 9aë .o%{e46H *v%&S1 ! ! +Y.=C,+ ' +v9469t"TS<5 !\&,+&$C 1b.ò& $!,+TP ! -;9¿& L "%r6$',- 1&3(*$'& +"% ;F ,-$'$!&IPF / "%$'&)(*$'LPk + $'W,-$'& .=,-1Y,+] $5 !$! +PZ "%"S$'& !,+1&,-4 1a$!1 P 1T$cb"CZ '1%J1.=J,+$!P
(87) $'& Z +9_] $'$)&$!bP,+$'&6& K +& 9^,+4 bC !"S -1J !,+.=&w,+,-^1XK& =,+.0n"T + pSjq<) !&,+&$O1C.=F &\s ,+,-"T -o"%$'&)(*$' + ,-$'& ã¹ ! + V"S !,+&1 7Vb$!PI -& F &w2"S !,+&1J !2"%$'&1"%4\ '1XPk$' + -$C.J1 B2 = {0, b, 2b, . . . }. ba ∈ B1 ab ∈ B2. ba = ab. {0, a, 2a, . . . } a {0, b, 2b, . . . , ab} !. . !. 9#. K. *#. <. !. 6. B1 =. a+1. Bi =. B1. rO #. . !. Bi0. B10. !. !. !s!. !. 9#. /. . 26 3. {pi |i ≥ 0} / k&. !. 3. 3. 2. >2. r>2. {0, 1, 2, . . . , r − 1} {0, r, 2r, . . . , (r − 1)r} {0, r + 1, 2(r + 1), . . . , (r − 1)(r + 1)} ... {0, s, 2s, . . . , rs} .... =.G211 Y ,-1=,-1c2(*K&I1bH&3 & 7Vb&$'=HX,+&81$W !&39^] ^L()r5,-,-$'11?1b""TSS ! !,+,+&I&F1c.=,+ !&4 '1 K1 !&4¿&$'&6jq>C,+
(88) $/ -C ' -1
(89) Y"%$'&3_CFY$'& !&4 +,+] -v.= !,+ +
(90) J7x(* & !&4F zo&"% ww.o1,+^&"%"C\ !&e7V$'Pq ' "$'c !$'& 4 ! ? % 1 2 1 S ! & ? C ' 1 # % " ' $ _ Y ' $ | & + + , ] c = . + , + x 7 ? + , 8 & x 7 ' $ "TS(* !&v,+& -1Z& !& 4;;,-1=&"%RZ & ,+T$' !& +9\9\"C.o '%146 *F ?$!PX"S !,+&12.=c ! + «"S !,+&1# !?$!PR1T !Y. : O $ . ' 1 1 Y , = 1 5 $ 4 f 4 F 0 ' $ & 1 , 4 ! & 4 F. Z L 5 r , 1 = 1 T " S ! + , & 1 .=$!P ,+ !&4 !&4 ,-1 '1JK1b&F$')&6,+jq&>C"%2$^7V$' - Y&3CF2 !& 4 -C ' !1 "%$'_Y$'&/ b + -C,+] '1 - ".=%$',+_8Y !$' + I&A"S ! ,+& +12,+] $! -wPU.=,+Z + X1T !7VY_,+&e$)4 7V4^$' -& " S ! ,+&1c !&F 4|&"%YwLr5,-1Z&$;46, ! T ! s. r. r > 3 r r+2. r. r(r+2) 2. 2. r+2 2. =. r 2. +1. r−1 ut. r 6= 3 r+1 r+3 r+1 r + 3 (r+1)(r+3) 2. r+1. r+1 2. >3. @CD. r+2 r 2. r+3 r+3 2. ≤ r−1.
(91) {&. n"TS !,+&1J.=,+ + I7V. XR. r=3. {0, 1, 2} {0, 3, 6} {0, 4, 8} {0, 5, 10} {0, 9, 18} .... %.="dY$',+&) +,- «1w,+"&3 -"C -C !? ! +
(92) 9w$Y&ST$' ! !J<*YR7V#%,+n&W1W& !L"&)r6S9wb !,+] &)& 1w(5.=,-7V$',+ + b#1b&&"TS$'(*b !,+,+w&I&|HS,+&3 !1,+&3&9\"%1 ] %"C(5H=,-$'$' 1 1"T,-,p4 PS !,+&IHxHo17VF %%9 "C !HS1v,pP#& .0 0. s. 2s 6= 2t. 2s. s 6= t. ô$c,+ ]V $C.0?E6F°1 ¯ $!PXU c$ 11,+& 12 -$!ZP] s2,+,+Y,+ "T -CHS & ,+T !9v !&4v&S !9;"%$'&3(*$' + ,-$'&IHx%1,-"%4 ¢õlð £¥ÿ ´ ðSXõlíó´Sï Oö-§¬ïL²Oð)§l´`ø í©Ô¶Iõ ç xï ç í ölï öoZî'õkõl÷Cìø'õküì÷'ñ?ö-ïíXè üë ï ð *üïóë'êxö é5í í[%õkë'ë'ð ì
(93) í ñ?ï^é)ë'èLðVíXï üTë'ðSõ[íqí ÷'õlðWé)ü èLí#÷ ë'õ¬ðxð èZï^ë ü 2÷'ö+ïLõlð5ðø íë ë'ì. FnìëCã¹ë ZG21?1 "TYS !#,+&eS"% !$'J&3T !n,+& 46,+& , ! T !S C(*4 c$)% -1b& & $'b"%$'F &3T !,+,-&/1 K !& &3,+9\ "HIS1 !,+,+&&"%12c$!$'P -& .=,- 1c.0$' S"TS C !(*,+n&1b$!nPs2 -,+& ,-"T -2$'"C '1RFX ! &4\&;Y$'.0Z%$C.=(*,+ + IS C(*ZHS1n,+&""%SZ !,+&1 46, ! T !4 $5%1& ä JS %(* nRGl. . v wF. !. 2 K. -. 3. 3. ! yx. 6. !. *o. f#. !. X.. . 6. 7&. !. P. *#. !. !. 9#. !. 33. *#. 2. 3. 3. 1. 2. 3. r r r≥4. {0, 1, . . . , r − 1} {0, r, . . . , sr} {0, r + 1, . . . , t(r + 1)} {0, r + 2, . . . , u(r + 2)} {0, r + 3, . . . , v(r + 3)} .... .=S,+ !oó.0$"S !,+&10S C(*bFb1$'z0Y Jb -Y&8&),+JoPk,+$'&a + -"%$C$'.J_1ZY '1 $'&I,+&WH3.= ,-"Tw] ,-$)10$! ZPQ"%$!$'PX&3] T$' ']V46$ ,-"1,+,-$',-$'&I&|F E6F°D s, t, u, v, · · · ≥ 3. @. z\.
(94) ¢£¥´ X´SO§¬²O§l´`©¶fç ïfð ç ïí . . qi. 10 2U. i≥0. "!. 3. U. p, q ∈ P 6 p 6= q 3. ïí. 1 ∈ B1 ⊂. . pi i ≥ 0. 6. 1 ∈ B2 ⊂. Lï ìï?í ï î3ïLð`ë íóïLèí ï ÷'ìLíóïLèõk÷'ðYêxìëCî é6ü%í=ë ní ï ê ë!è%ïílè '÷ ð`î. !&ì4ëCë
(95) $'FR 4 R_,+1T& !Y_$'4 ,-1Z4 ,-L1#K&1%%4A4/ ',+1& 1S ! qH«,qFú HVF ,qFú F , !&4 Fs#LK& & !. K. !. *#. B1 B2 ∼ = B1 × B 2 ,. ! |{. ×. 3. qb ≥ qs a p ≥ pr qb ≥ qs. 9#. !. B1. B2 3. ,. ji. pa q b ≥ p r q s pa ≥ p r ji (pa , q b ) ≥ (pr , q s ). B1 B2. B1 × B 2. ϕ : B 1 B2 → B 1 × B 2. -,4 )1
(96) ,+L&K"%"%&$' _n]x$'Z&,+&3&)(*ó.=,-11nv$ H +,+] ,-1J +,-"C R !,-$'$'Y&IFù$'Y $' ] ,-1_ ,-
(97) 1 !<*1%,+1 &"% 8R 7 +, ,+%] " +,-,-"C$' !&IF¿,-$'&v{e,+\& "C !& ! -1$ pr q s 7→ (pr , q s ). ϕ(pa ) = (pa , 1) ϕ. B 1 × B2. ϕ. ϕ. #,+&)(*1Z,-1=.0 + pjó4 LK&%4W ! '&"%4"%$'46,+& n$Z=4 LK& ,+,+,-_$'&w] $! +,-PV%1b$'4 S !o !7V$C(* F 2&"% ,+ ä ,-1 1 ? !&?]x$ $'14 b,-1] $'Y%1$'] (5 ,+& ,-1#;7 F , %",-$'&c.=$ 10,+&)(*10,-1U$'4 ] %1T(),+& n !&4?&"% ϕ−1 ((pr , q s )) = pr q s. pr q s ϕ. ≤. pn q m. =. (pr , q s ) ≤ (pn , q m ). ϕ((pn , q m )). ϕ((pr , q s )) = fS. ∴ B 1 B2 ∼ = B1 × B 2. õ èwõ è%ë'ñYë'ìqê õ[ü;íqë/í ï^õlð ÿ ðõ[íóïaî'õlìïüCí ¢ £¥´ X´SO§¬²O§l´`©}¶Iç fç ïnê6ë'è%ïí êxìëCî é6ü%íXë bí ïXê6ë!èCïílè ïLð õ¬èJõ è%ë'ñYë'ìóê õkü íqë ÿ í ï ^ ð í 2 ï ¬ õ ` ð ó í L ï ì ' ÷ ö õ J è ¬ õ è ' ë Y ñ ' ë ó ì ê k õ 2 ü q í _ ë b ÷ ` ê ì C ë ' î é % ü Q í ë ð l õ ó í ï þ ö R ý _ ñ ÷ ð ý ÿ ðõlíóïRü ÷ õ¬ð è. ìëCë ãI S C(*n#] ,+YnPk '"$',+>O !,-$'& s#LK& . 14 } "!. *#. O #. K. n∈ *#. N+ 6 !. !. !. !. (No+ , ≤A ) ( pji j ≥ 0 , ≤A ) 3 n. . [1, n]. "!. !. !. (N+ , ≤A ). !. 9#. S3. 3. n = pa11 pa22 · · · pann. n ∈ N+. 3. !. ,-1 !
(98) ] !1 ] ] ,+& $R^] 2%(*1%"(*$'%=12.=,+R $'4 $'&^Pk$' + -$C2.J,1 Pl^$'] '"% %1J% !$'&4 '$ @ &;F+@û !F + f$',-12" -1CCF ! +S9v !
(99) 7 , %",-$'&IFX Z,+&)(*1n$!P &vLr5,-11J !&4 ϕ : (N+ , ≤A ) → ({pj1 |j ≥ 0} × {pj2 |j ≥ 0} × . . . , ≤A ). ϕ. ~R. pai i. n. 0. ϕ. ϕ. ,+_] +,-%1. ai ≥ b i. P[$'b ! + . F &"%. (pa11 , pa22 , . . . ) ≥ (pb11 , pb22 , . . . ) i∈N. S. ϕ−1 ((pa11 , pa22 , . . . )) = pa11 pa22 · · · ≥ pb11 pb22 · · · = ϕ−1 ((pb11 , pb22 , . . . )). 2&"% ,-1a |]V$ 1Y,-1$'Y$'] ,-1 !&4 S. ϕ 0} × . . . , ≤A ). @. j j (N+ , ≤A ) ∼ = ({p1 |j ≥ 0} × {p2 |j ≥. ].
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(103)
(104) K]V& $ ,+1nb !,-1&$'4^Y !$' + ] ,-1 '"%"%$'46,+& ^ !$
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(106) Lr5,-1CH31,+&"% õ¬è õ b÷'ölö ÷'ìïXê`ìõlñcõlí[õ ïê`ìLõ¬ñ
(107) ïê6ë oïLìè ½ ¹ë í¤)´Sï£!ì ¤)¨õ¬è%ï ¶Iç[¡5Âfç. ìëCë ãI 7x#] ,+_,+,+(* Fo & 1≤n∈N pi pi. pk ≤ n plii. li ∈ N. n. ϕ : ([1, n], ≤A ) → ({pj1 |0 ≤ j ≤ l1 } × · · · × {pjk |0 ≤ j ≤ lk }, ≤A ) {pji |0 ≤ j ≤ li }. P. n∈W. @? . &. !. K. 1 ∈ W ⊂ N+ en ∈ Γ[W ] ζ(n) ≡ 1 µ = ζ −1. *#. µ(pa11 · · · par r ). 3. #. # (−1)r. Γ[W ]. 5A /. n∈W. . pai i. ζ =. #. ζ(1) 6= 0 '&. 6. pa. 3. µ ∗ ζ(pa ) =. X. µ(d)ζ(. .= ,-"Tv,+_] +,-%1=S !. d∈A(pa ). µ ∗ ζ(p2a ) =. pa )= d. p , P & ' $ 6 & q j ] + , _ + , + , * ( F !&4 ! Z] ,+_,+,+(* FQ &. µ(d) = µ(1) + µ(pa11 ) + µ(pa22 ) + µ(pa11 pa22 ) =. 2& !"% #] ,+_,+,+(* FoãI 2&ãI"% . H. d∈A(pa ). µ(pb ) = 0 pb pa11 pa22. d∈A(n). pai i. µ(d) = µ(1) + µ(pa ) = 1 + µ(pa ) = 0. ,pP ,-1=] ,+_,+,+(* FQ &IH,pP. d∈A(p2a ). n = pa11 pa22 X. S. X. µ(pa ) = −1 pa pa ∈ A(p2a ) X µ(d) = µ(1) + µ(pa ) + µ(p2a ) = 1 − 1 + µ(p2a ) = 0. d Pk$'ãI + -$C .J1J7)9w,+&46".=,-$'&WZS !7V $'. S. . FQG211 Y.=ZS ! ! + !Z] ,+_,+,+(* FX & ,pPX ! + . = 1 − 1 − 1 + µ(pa11 pa22 ) = 0. a. r−1 ) = (−1)r−1 µ(pa11 pa22 ) = 1 = (−1)2 µ(pa11 · · · pr−1 ai a a n = p1 · · · p r r pi i r−1 X X r (−1)i + µ(pa11 · · · par r ) = 0 µ(d) = i. .= -, ,p1=P& !$' + &6jq] ,+_ !,+#,+] (*c,+_ !&,+4 ,+(* F ,-1=] ,+_,+,+(* FX &. d∈A(n). i=0. µ(pa1i · · · par r ) = (−1)r pai i 2b a 2b n = p 1 p2 p1 pa2 , pb1 X b a 2b a µ(d) = µ(1) + µ(pa2 ) + µ(pb1 ) + µ(p2b 1 ) + µ(p1 p2 ) + µ(p1 p2 ) =. d∈A(n). .=S !,- "T;,+_] +,-%1=S ! ,pPI$'&=$!PfF ó&461",-1,-$'&&v$'&6$O(*jq] =,+_,+#,+(*&3 FXRh87x$'b%$!$CP«(*Pk 'o",+&$'461 " ,+(*,-$'%& 1 ,p$OP(*1$' YZR$!&3P R 7xZ1J$!PX !&Z$'&6&jq$'] &6jq,+] _,+,+_,+(*,+?,+(*Pq ' "F $'1b&/1$O.J1 S ! @CB a = 1 − 1 − 1 + 0 + 1 + µ(p2b 1 p2 ) = 0. µ(pa1i. a µ(p2b 1 p2 ) = 0 · · · par r ) = 0. pai i. pai i. µ(pa1i · · · par r ) = 0.
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