ON THE DISTRIBUTION OF k-th POWER FREE INTEGERS, II

Transcription

1 Duy, T.K. a Taaobu, S. Osaa J. Math. 5 (3), O THE DISTRIBUTIO OF -TH POWER FREE ITEGERS, II TRIH KHAH DUY a SATOSHI TAKAOBU (Receive July 5,, revise December, ) Abstract The iicator fuctio of the set of -th ower free itegers is aturally etee to a raom variable X () ( ) o ( Ç, ), where Ç is the rig of fiite itegral aeles a is the Haar robability measure. I the revious aer, the first author ote the strog law of large umbers for {X () ( )}, a showe the asymtotics: E [(Y () ) ] as, where Y () () Ï È X () ( ) (). I the reset aer, we rove the covergece of E [(Y () ) ]. For this, we reset a geeral roositio of aalytic umber theory, a give a roof to this.. Itrouctio Let Ç be the rig of fiite itegral aeles; B the Borel -fiel of Ç ; the Haar robability measure o (Ç,B). I [4, ], the trilet ( Ç,B, ) is itrouce i the followig way: For a rime umber, the -aic metric o is efie by (, y) Ï if{ l Á l ( y)},, y ¾. The comletio of by is eote by. By eteig the algebraic oeratios a i cotiuously to those i, the comact metric sace (, ) becomes a rig. I articular, (, ) is a comact abelia grou with resect to. Thus, there is a uique Haar robability measure with resect to o (,B( )), where B( ) is the Borel -fiel of. Puttig i i-th rime umber (i,, ), we set Ç Ï Ï i For ( i ), y (y i ) ¾ Ç, we efie i, (, y) Ï i i i i ( i, y i ), i. y Ï ( i y i ), y Ï ( i y i ). Mathematics Subject Classificatio. Primary 6F5; Secoary 6B, 6B5, 37, K4.

2 688 T.K. DUY AD S. TAKAOBU By these efiitios, Ç becomes a rig, which is just the rig of fiite itegral aeles state above. (Ç, ) is agai a comact metric sace, a both a are cotiuous. I articular, this is a comact abelia grou with resect to, a its Haar robability measure is othig but. By ietifyig with the iagoal set {(,, ) ¾ Á ¾ } Ç, it is see that is a ese subrig of Ç. Thus Ç is a comactificatio of. Let be a iteger,. Let B () be the set of all elemets i Ç havig o -th ower factors, i.e., B () Ï { ¾ Ç Á (: rime)}, where ¾ Ç (, so ¾ Ç ÒÇ ), a X () Ï B () fuctio of B () ). The followig are results of Duy []: ( the iicator È Fact (Strog law of large umbers). lim () X () () (), -a.e.. Here ( ) is the Riema zeta fuctio. () For each ¾ Æ, we set Y () () Ï () X () ( ) (). Fact. E [(Y () ) ] as. Fact 3. A sequece {Y () } (Ç i L,B, ) has o limit oit. amely, for ay subsequece { i } i (), {Y i } i is ot coverget i L as i. Fact follows at oce from the ergoicity of the shift È a E [X () ] (). From this fact, we have the followig questio: Whe (X () ( ) ()) is ormalize aroriately, is its istributio wealy coverget as? ¾ Fact tells us that a ormalizig costat must be (), a that a sequece {(Y () of istributios o Ê is tight. Fact 3 is a fuctioal aalytical result a brigs )} o ews for the behavior of Y () as. But, for this, we eect to have a limit theorem i robability theory. (Ufortuately, we still have o iformatio o this limit theorem.) I this aer, we mae some remar about Fact a Fact 3. Cf. Æ i the roof of Claim.

3 Theorem. lim E [(Y () ) ] -TH POWER FREE ITEGERS II 689 ( ) () () si(()). Theorem. (i) lim lim M E [(Y () M Y () ) ] É ( )( ) ( )(() ()si()). Fact 3 above is a cosequece of this. (ii) But, a whole sequece {Y () } (Ç i L,B,) is wealy coverget to as. Throughout È É this aer, the letter eotes a rime umber, a the symbols a are a rouct a a summatio etee over all rime umbers, resectively. Theorems above will be rove i Sectio 4. I Sectio, a aother comutatio of E [Y () M Y () ], which is ifferet from oe i Duy [], is give. A, i Sectio 3, to rove Theorem, we reare Proositio. This is a geeral roositio of aalytic umber theory, a will be rove i Sectio 5. The authors woul lie to tha the referee for goo avice which eable us to mae roofs clear a cosierably short.. Comutatio of E [Y () M Y() ] By a ifferet aroach from Duy [], we comute E [Y () M Y () ]. Claim. For M, E [Y () M Y () ] M () () c (c) c M c c M c c. Here ( ) is the Möbius fuctio a {a} is the fractioal art of the real umber a. Proof. Æ First E [Y () M Y () ] E M () () Fi M. We ivie the roof ito three stes: M m X () ( m) () X () ( ) () Duy s metho is origially ue to [4]. The same i of comutatio i the roof of Claim aears i early stuy of [4]. So, a hrase ifferet aroach may be too much to say.

4 69 T.K. DUY AD S. TAKAOBU M () () otig that mm, E [X () ( m)x () ( )] () (E [X () ( m)] E [X () ( )]) (). () X () (y) ( (y)), where, for ¾ Æ, (y) Ï, y ( y ¾ Ç ),, otherwise, (3) (4) we have { } is ieeet, ( ), ( ), E [X () ( m)] E [X () ()] (by the shift ivariace of ) () (by Euler s rouct of ( )), a thus Sice, by () E [Y () M Y () ] M () () mm, E [X () ( m)x () ( )] X () ( m)x () ( ) ( ( m)) ( ( )) (). ( ( ) ( m) ( m) ( )) ( ( ) ( m) (m ) ( )) (by a ietity: ( m) ( ) (m ) ( )),

5 we see from (3) a (4) that -TH POWER FREE ITEGERS II 69 E [Y () M Y () ] M () () mm, (m ) (). Æ By Euler s rouct of ( ) () () () c ( ) (c) c (where () Ï #{Á } the umber of ifferet rime factors of ) () c c, (c ) ( ) (c) c (c ) ( ) ( ) c (there eists a oe-to-oe corresoece betwee the set {(c, )Á is square free a c } a the set {(c, )Á c is square free}; a corresoece from the former to the latter is (c, ) (c, c) a oe from the latter to the former is (c, ) (c, c ). Here (c, ) a (c, ) eote a air of c a, a that of c a, resectively) (c ) ( ) ( ) c (c, ) c, (where (c, ) is the greatest commo ivisor of c a. ote that (c ) if (c, ) ) (c )( ) ( ) ( ) c (c, ) c,

6 69 T.K. DUY AD S. TAKAOBU (by the multilicativity of ) (c ) c c (c ) c c ( ) (c, )( ) ( ) (c, )( ) () () (by the multilicativity of ( ) (c, )( ) ( ) ) c (c ) c. c Similarly, sice (m ) (c ) c c c (m ), c we have c (m ) (c) c c () c (m ) c. Thus, by Æ a 3 Æ below E [Y () M Y () ] M () () mm, (c) M () () c c (c) M () () c c This is the assertio of the claim. c (c) c c (m ) c c M c m Mc c c (m ) c M c c.

7 -TH POWER FREE ITEGERS II Æ Fi u ¾ Æ. Let Q a s be a quotiet a a remaier of ivie by u, resectively. Thus Qu s, where Q u 3, s { u}u ¾ {,,, u }. The Q u (m ) q j Q q j Q u u (m ((q )u j)) s u u q j Q u j Q s j u (m (Qu j)) u (m j (q )u) s u (m j Qu) j u (m j) s u (m j) j u (m j) s u (m j) j u (m j) j (by a ietity: u (y u) u (y)) È (first u j u(m j) È ju u(m j) È ju u(m mo u j), where m mo u Ï the remaier of m ivie by u. Secoly, otig that for j u, u (m mo u j) j m mo u (mo u) j È u m (mo u), we see j u(m j)) u Therefore u M m u u s M m j u (m j). u (m ) u M m u u (m ) M u s j u (m j) M 3 For a ¾ Ê, a Ï ma{ ¾ Á a} a a Ï mi{ ¾ Á a }. We call Ï Ê a Ï Ê the floor fuctio a the ceilig fuctio, resectively. ote that {a} a a ¾ [, ). u

8 694 T.K. DUY AD S. TAKAOBU u u u u M u s M j m M u s M u r j i M u M s u r s i j M u M s u r s u ( j m) M u u ( j i) M u u (i j) M M u u (where r {Mu}u) (for i, j u, u i j u. Also u (i j) i j (mo u). Thus u (i j) i j) r s M u u u r s r s u u u u M u Claim. u u M u u s u M u (because {Mu} r u, { u} su) M u u For each ¾ Æ, lim M E [Y () M Y () ]. Proof. Let M. Sice {Mc }, { c }, M c M c c c c c. (by a ietity: ab (a b)(a b)). Multilyig both sies by (M () )( () )(c) É c ( ), a the aig them over c ¾ Æ yiel that E [Y () M Y () ] M () () M c (c) () E [(Y () ) ]. c From this, the assertio of the claim follows. c c

9 3. Presetatio of Proositio -TH POWER FREE ITEGERS II 695 (5) By Claim E [(Y () ) ] c c É (c) c ( ) f (c) c c c c, where (6) f () Ï () É ( ), ¾ Æ. To show the covergece of E [(Y () ) ] as a to fi the value of this limit, we reset a geeral roositio: Proositio. coitio (7) or (8): (7) (8) Let a arithmetic fuctio f, i.e., fï Æ satisfy the followig () f, su ¾Æ f (), È f has the mea-value M( f ), i.e., lim () f () is coverget to a fiite limit M( f ). The, it hols that for ¾ (, ) 4 a h ¾ C [, ] with h() (9) lim f ()h M( f ) h({}). Before rovig this roositio, we give some commets o the coitios (7) a (8): Claim 3. If f Ï Æ satisfies the coitio (7), the f has the mea-value M( f ) () f. 4 Here may be a real umber,, though was a iteger, at the begiig of this aer.

10 696 T.K. DUY AD S. TAKAOBU Proof. For simlicity, we efie f Ï Æ by () f () () f, ¾ Æ. Sice, by the Möbius iversio formula () f () f (), we have for, y ¾ [, ) f () Á f () f () f () y f () { } f () f () { } f (). y { } f () Trasosig the first term of the last right-ha sie, a the taig the absolute value, we see that () f () f () y f () y f () f () y y y { f () y } f () f (). y { } f () f () y By lettig a y, the assertio of the claim follows. REMARK. Schwarz Siler [3] calls Claim 3 Witer s theorem.

11 -TH POWER FREE ITEGERS II 697 We give a eamle of f satisfyig the coitio (7): Let fï Æ be multilicative, i.e., f a f (m) f (m) f () EXAMPLE. rovie that (m, ). If, i aitio, (3) f () the f satisfies the coitio (7)., l f ( l ) l, Proof. Multilicativity of a f is iherite to f, a so f. I geeral, multilicativity of a arithmetic fuctio imlies a rouct reresetatio of Dirichlet series associate with the fuctio. Thus Sice, by () (4) (5) f () e f () f ( ) f () µ f ( ) l l f () () f ()() f () f f () (ote that f () ), l (by a iequality: e ( ¾ Ê)). f ( l ) () f ( l )() f ( l ) (ote that ( j ) ( j )) f ( l ) f ( l ) (l ), we have f () f () f () l f ( l ) l l f () l f ( l ) f ( l ) l f ( l ) l l l f ( ) l l f () f ( l ) l

12 698 T.K. DUY AD S. TAKAOBU 3 f () (by (3)). Therefore f satisfies the coitio (7). l f ( l ) l The coitio (7) oes ot always imly the coitio (8). EXAMPLE. Let f Ï Æ be multilicative, a satisfy f () «, f (l ) (l ) for each rime, where «¾ (, ). Sice f (), «f ( ) satisfies the coitio (7) from Eamle. Also, sice f ( m ) f ( ) f ( m ) we see that This imlies that m i «i m m i i e «i e È m i «i, e («i )(«i ) (by a iequality: log( ) () ( )) È m e i («i )(«i È ) e («( «m )) i «i, lim f ( m if «, ) m if «. su f () if «, if «.

13 4. Proof of two theorems Proof of Theorem. satisfies Also, ote that -TH POWER FREE ITEGERS II 699 f Ï Æ, efie by (6), is clearly multilicative, a f () l f ( ) l l. f () e È 4 ( ¾ Æ), (because, by ( )4, É ( ) É (4 ) É e4 e È 4 e È 4 ). Hece, this f ( ) satisfies both the coitio (7) a the coitio (8), so that alyig Proositio, we see (6) lim c f (c) c c M( f ) {}( {}) Let f be a multilicative fuctio efie by (). By (4) a (5) f (), f ( l ), l,, l 3 for rime a iteger l,. Claim 3 the imlies that M( f ) f () f () l f ( l ) l..

14 7 T.K. DUY AD S. TAKAOBU Collectig (5), (6) a this, we have lim E [(Y () ) ] {}( {}). Let us fi the value of a itegral o the right-ha sie. The Fourier easio of a fuctio {}( {}) is as follows: {}( {}) 6 cos cos si. (because È 6) Termwise itegratio yiels that {}( {}) si si y y (y) si y y y si y y y We here ote that from a formula: Ê (si Ú) u Ú u ((u) si(u)) ( u, Ú ). si y y y ( y ) si y y [ y si y] si y y y ( y ) si y cos y y

15 -TH POWER FREE ITEGERS II 7 (because lim y y si y lim y y ((si y)y), lim y y si y lim y si yy ) () si(). Substitutig this ito the above, we have {}( {}) () si() ( ) () () si(). Cosequetly, the assertio of the theorem follows at oce. REMARK. Sice, by the fuctioal equatio (s) ( s) si s () s ( s) of the Riema zeta fuctio, we see si (), ( ) ( )(si(()))() ( ) () () si(()) () () si(()) ( ). The the aearace of Theorem becomes goo as lim E [(Y () ) ] Proof of Theorem. (i) For M ( ). E [(Y () M Y () ) ] E [(Y () M ) ] E [Y () M Y () ] E [(Y () ) ]. The assertio of (i) is obvious from Claim a Theorem. (ii) By Theorem, {Y () } is L -boue, a thus for ay subsequece { i } i {i m } m : subsequece, Y ¾ L (Ç, B, ) s.t. w-limm Y () im Y.

16 7 T.K. DUY AD S. TAKAOBU The lim E [Y () m im Y () i ] E [Y Y () i ], ¾ Æ. But, by Claim E [Y Y () i ] ( ¾ Æ). Lettig yiels that E [Y ]. This imlies that w-lim Y (). 5. Proof of Proositio We ow tae u the roof of Proositio. Suose f ( ) satisfies the coitio (7) or (8). Fi ¾ (, ) a h ¾ C [, ] with h(). We ivie È f ()h({ }) ito three terms as (7) f ()h M( f ) h ( f () M( f ))h f ()h. To fi a limit of each term as, we reset the followig lemma: (i) Lemma. Let a b a ³ ¾ C [a, b]. Give a sequece {a }, set S(t)È t a (t ¾ Ê). The, for a y b a ³() y y S(t)³ (t) t S(y)³(y) S()³(). (ii) For a y b y ³() y ³(t) t {y} ³(y) {} y ³() {t} ³ (t) t. Proof. Let a b, ³ ¾ C [a, b] a a y b.

17 -TH POWER FREE ITEGERS II 73 (i) I case y, otig that a y y b, we have the left-ha sie y y y y a ³() (S() S( ))³() S()³() y y (because S(t) S(t)) y y S()³( ) S()(³() ³( )) S(y)³(y) S()³() S()³ (t) t S(y)³(y) S()³() S(t)³ (t) t S(y)³(y) S()³() S(t)³ (t) t S(y)³(y) S()³() y S(t)³ (t) t S(t)³ (t) t y S(y)³(y) S()³() y S(t)³ (t) t S(y)³(y) S()³() the right-ha sie. I case y, sice y y, the left-ha sie a ³(), y S(t)³ (t) t the right-ha sie S()(³(y) ³()) S()(³(y) ³()). Thus, we obtai the assertio of (i).

18 74 T.K. DUY AD S. TAKAOBU (ii) Let a ( ¾ Æ). I this case, S(t) t (t ), so by (i) y ³() y t³ (t) t y³(y) ³() y t³ (t) t y {t}³ (t) t y³(y) ³() {y}³(y){}³() [t³(t)] y y ³(t) t y {t} ³ (t) t (³(y) ³())[t³(t)]y {y}³(y){}³() ³() y y ³(t) t {y} {t} ³ (t) t. ³(y) {} REMARK 3. This ietity is calle the Euler summatio formula (cf. [3, Theorem. i Chater I]) or the Euler Maclauri summatio formula (cf. [, Lemma.]). Proof of Proositio uer the coitio (7). Æ The first term of (7). Æ - For L ¾ Æ with L, ( (L)) h L l Á l h (ote that ( (L )) L) L l l l h l (whe l, { } l. Also l l l ) L h l ( (l)) ( l) l

19 L l ( l) h ( (l)) ( l) -TH POWER FREE ITEGERS II 75 l t h() t l {t} h l t ( (l)) t t µ (aly Lemma (ii) for ³(t) h( t l) (( (l )) t ( l) )) L l ( l) h ( (l)) ( l) t t h() {t} h ( (l)) t t l t (ote that ( (l )) t ( l) t l) h ( (L)) t t h() L l {t} h t ( (L)) t t L h({}) L h() l µ L h ({}) µ µ l l µ (by chage of variable: t ). Æ - Let a L L() (). The L() () L(), a so L(), (L()) () (). Sice, by Æ - h ( (L())) h h ( (L())) L() L() h ({}) h({}) h() µ L() l ( (L())) h, l µ

20 76 T.K. DUY AD S. TAKAOBU we see L() h h ( (L())) L() µ h() l l L() h ({}) µ L() L() L() h({}) ma h() h() L() ma () h L() ma () h (ote that ma h() ma h ()) This shows that () (()) () ma () h (()) ma h () as. (8) the first term of (7) M( f ) Æ The seco term of (7). For simlicity, set a f () M( f ), S() È a. Æ - First y S(y) y y f () ym( f ) y y h({}) as. f () M( f ) {y} y M( f )

21 -TH POWER FREE ITEGERS II 77 y y as y. f () M( f ) M( f ) y Æ - I the same way as i Æ -, we have that for L ¾ Æ with L a h ( (L)) L a h l l ( (l)) ( l) L l ( l) ( (l)) h t l S(t) t t h()s l (aly Lemma (i) for ³(t) h( t l) (( (l )) t ( l) )) S(t) L h ( (L)) t t t h()s l l L L l h ({})S h()s l l l L h ({})S h()s. Æ -3 Fi L ¾ Æ with L. By Æ - a h ( (L)) L h ({}) S h() S ma h () L ma h () L S(( ) ) ( ) S(( ) ) ( ) S(( ) ) ( ) S(( ) ) ( ) (ote that Lµ Lµ( (L)) ( ) ( ) ) ma () h su y( (L)) S(y) y L

22 78 T.K. DUY AD S. TAKAOBU S(y) L y ma () h su ma h () ma h () y( (L)) su y( (L)) S(y) y su y( (L)) (L) S(y) y (L). Also ( (L)) a h ( (L)) ma h() ma h() ma h() f ()h M( f ) ( (L)) f ()M( f ) ( (L)) ( (L)) M( f ) L ma h() L M( f ) ma h() L ma h() L f () f () ( (L)) h ( (L)) f ()M( f ) ± L f () ( (L)) (L ). f () L

23 Combiig two estimates above, we have -TH POWER FREE ITEGERS II 79 (9) a h ma () h ma h() Ï A su y( (L)) S(y) y su y( (L)) f () S(y) (L ) y (L ) (L ) B(L ). Æ -4 Tae so that B(( )A). By Æ - y S(y) s.t. (y y ). y Let L B(( )A) ¾ Æ. For y B(( )A), B L. y y ( )A Sice ( (L )) y, su y( (L)) S(y)y. Usig this i (9), we have a h A(L ) B(L ) A A A B ( )A B ( )A B ( )A B B B B ( )A B ( )A B ( )A ( ) A B ( ) A B (( ) ( ) )A B.

24 7 T.K. DUY AD S. TAKAOBU Lettig yiels lim This shows that a h (( ) ( ) )A B as ². () the seco term of (7) as. 3 Æ The thir term of (7). 3 Æ - First, we chec the covergece of a series È f ()h( ). Let L, M ¾ Æ, L M. Lemma (i) for ³(t) h( t ) (L t M ) tells us that L M M L f ()h f () t M t M f () h t t h( M) M t M L L f () h( L) L L. Ê Here, otig that sice, tt, lim M M lim L L a sice h(), lim h() h (), we see the covergece of this series. 3 Æ - et, lettig L a M i the above yiels that t f ()h t f () h t t t f () h ( ) f () f () h() h() (by chage of variable: t ).

25 By multilyig both sies by, it turs out that -TH POWER FREE ITEGERS II 7 the thir term of (7) f () h ( ) f () h(). Thus, by the Lebesgue covergece theorem, () the thir term of (7) M( f ) M( f ) M( f ) M( f ) M( f )h ( ) M( f )h() (h ( ) ) h() ( h( )) h() [ h( )] h( ) h( (because h( ) (h( ) )( ) as ) M( f ) (by chage of variable: ) M( f ) h() 4 Æ Collectig (8), () a (), we have lim M( f ) M( f ) h({}) as. f ()h h({}) M( f ) h({}). ) h() h({}) Proof of Proositio uer the coitio (8). The argumet of Æ a 3 Æ i the revious roof is also vali i this case. Thus we have the covergeces (8) a ().

26 7 T.K. DUY AD S. TAKAOBU I the followig, we slightly moify the argumet of Æ i the revious roof. Let a f () M( f ), S() È a. Æ - First Clearly Æ - For L ¾ Æ with L L ( (L)) a h S(y) as y. y h ({})S Æ -3 Fi L ¾ Æ with L. By Æ - a h ( (L)) ma h () a h ( (L)) Combiig these estimates, we have () a h ma () h sua ma Ï A su y( (L)) h()s su y( (L)). S(y) (L ). y ( (L)) a h sua ma h() (L ) sua ma h() h() S(y) y su y( (L)) (L ) S(y) (L ) y (L ) C(L ). (L ).

27 -TH POWER FREE ITEGERS II 73 Æ -4 Tae so that C(( a choose y such that S(y)y (y y ). Let L C(( )A) ¾ Æ. For y C(( )A), su y( (L)) S(y)y. Usig this i () a h A(L ) C(L ) (( ) ( ) )A C. Lettig, a the ², we have the covergece (). Cosequetly, the assertio of Proositio uer the coitio (8) follows. Refereces [] T.K. Duy: O the istributio of -th ower free itegers, Osaa J. Math. 48 (), [] K. Matsumoto: Riema Zeta Fuctio, Asaura, 7 (i Jaaese). [3] W. Schwarz a J. Siler: Arithmetical Fuctios, Loo Mathematical Society Lecture ote Series 84, Cambrige Uiv. Press, Cambrige, 994. [4] H. Sugita a S. Taaobu: The robability of two itegers to be co-rime, revisite o the behavior of CLT-scalig limit, Osaa J. Math. 4 (3), Trih Khah Duy Deartmet of Mathematics Grauate School of Sciece Osaa Uiversity Osaa Jaa hahuy6@gmail.com Satoshi Taaobu Faculty of Mathematics a Physics Istitute of Sciece a Egieerig Kaazawa Uiversity Kaazawa 9-9 Jaa taaob@staff.aazawa-u.ac.j