The least common multile of a quadatic sequence Javie Cilleuelo Abstact Fo any ieducible quadatic olynomial fx in Z[x] we obtain the estimate log l.c.m. f1,..., fn n log n Bn on whee B is a constant deending on f. 1. Intoduction The oblem of estimating the least common multile of the fist n ositive integes was fist investigated by Chebyshev [Che52]. He intoduced the function Ψn m n log log l.c.m.1,..., n in his study on the distibution of ime numbes. The ime numbe theoem claims that Ψn n, so the asymtotic estimate log l.c.m.1,..., n n is equivalent to the ime numbe theoem. Indeed the Riemann Hiothesis is equivalent to say that log l.c.m.1,..., n n n 1/2ɛ fo all ɛ > 0. The analogous asymtotic estimate fo aithmetic ogessions is also known [Bat02] and it is a consequence of the ime numbe theoem fo aithmetic ogessions: log l.c.m.a b,..., an b n q 1 φq k, 1 whee q a/a, b. 1 k q k,q1 We addess hee the oblem of estimating log l.c.m.f1,..., fn when f is an ieducible quadatic olynomial in Z[x]. The same oblem fo educible quadatic olynomials is solved in section 4 with consideably less effot than the ieducible case. We state ou main theoem. Theoem 1. Fo any ieducible quadatic olynomial fx ax 2 bx c in Z[x] we have whee B B f is defined by the fomula log l.c.m. f1,..., fn n log n Bn on B f γ 1 2 log 2 log a 2aD d/ log 1 1 φq 1 q,q1 1 d/ log sf, k 1 k. k 1 log 1 q In this fomula γ is the Eule constant, D b 2 4ac dl 2, whee d is a fundamental disciminant, d/ is the Konecke symbol, q a/a, b and sf, k is the numbe of solutions of fx 0 mod k, which can be easily calculated using Lemma 2. 2 2000 Mathematics Subject Classification 2000 Mathematics Subject Classification: 11N37. Keywods: least common multile, quadatic sequences, equidistibution of oots of quadatic conguences This wok was suoted by Gant MTM 2008-03880 of MYCIT Sain
Javie Cilleuelo Fo the simlest case, fx x 2 1, the constant B f in Theoem 1 can be witten as B f γ 1 log 2 2 2 1/ log, 3 1 whee 1/ is the Konecke symbol o Legende symbol since is odd defined by 1/ 1 1 2 when is odd. In section 3 we give an altenative exession fo the constant B f, which is moe convenient fo numeical comutations. As an examle we will see that the constant B f in 3 can be witten as B f γ 1 log 2 2 k1 ζ 2 k ζ2 k k0 0.066275634213060706383563177025... L 2 k, χ 4 L2 k, χ 4 k1 log 2 2 2k 1 It would be inteesting to extend ou estimates to ieducible olynomials of highe degee, but we have found a seious obstuction in ou agument. Some heuistic aguments and comutations allow us to conjectue that the asymtotic estimate log l.c.m. f1,..., fn degf 1n log n holds fo any ieducible olynomial f in Z[x] of degee 2. To obtain the asymtotic estimate log l.c.m. f1,..., fn n log n fo ieducible quadatic olynomials is not difficult. The main idea is to comae this quantity with log n i1 fi, which we can estimate easily. See subsection 2.2 fo moe details. Howeve to obtain the linea tem in Theoem 1 we need a moe involved agument. An imotant ingedient in this at of the oof is a dee esult about the distibution of the solutions of the quadatic conguences fx 0 mod when uns ove all the imes. It was oved by Duke, Fiedlande and Iwaniec [DFI95] fo D < 0, and by Toth fo D > 0. Actually we will need a moe geneal statement of this esult, due to Toth. Theoem 2 [Tot00]. Fo any ieducible quadatic olynomial f in Z[x], the sequence {ν/, 0 ν < x, S, fν 0 mod } is well distibuted in [0, 1 as x tends to infinity fo any aithmetic ogession S containing infinitely many imes fo which the conguence fx 0 mod has solutions. Acknowledgment. We thank to Aad Toth fo claifying the statement of Theoem 1.2 in [Tot00], to Guoyou Qian fo detecting a mistake in a fome vesion of Lemma 2, to Adolfo Quiós fo convesations on some algebaic asects of the oblem, to Enique González Jiménez fo the calculations of some constants and to Fenando Chamizo fo some suggestions and a caefully eading of the ae. 2. Poof of Theoem 1 2.1 Peliminaies Fo fx ax 2 bx c we define D b 2 4ac and L n f l.c.m.f1,..., fn. Since L n f L n f we can assume that a > 0. Also we can assume that b and c ae nonnegative integes. If this is not the case, we conside a olynomial f k x fk x fo a k such that f k x has nonnegative coefficients. Then we obseve that L n f L n f k O k log n and that this eo tem is negligible fo the statement of Theoem 1. We define the numbes β n by the fomula L n f βn 4 2
The least common multile of a quadatic sequence whee the oduct uns ove all the imes. The imes involved in this oduct ae those fo which the conguence fx 0 mod has some solution. Excet fo some secial imes those such that 2aD the conguence fx 0 mod has 0 o 2 solutions. We will discus this in detail in Lemma 2. We denote by P f the set of non secial imes fo which the conguence fx 0 mod has exactly two solutions. Moe concetely P f { : 2aD, D/ 1} whee D/ is the Konecke symbol. This symbol is just the Legende symbol when is an odd ime. The quadatic eciocity law shows that the set P f is the set of the imes lying in exactly ϕ4d/2 of the ϕ4d aithmetic ogessions modulo 4D, coimes with 4D. As a consequence of the ime numbe theoem fo aithmetic ogessions we have #{ x : P f } x 2 log x o equivalently, 0 ν< x fν 0 mod 1 x log x. Let C 2a b. We classify the imes involved in 4 in Secial imes: those such that 2aD. Small imes : < n 2/3. P f : Medium imes: n 2/3 < Cn : Lage imes: Cn fn. We will use diffeent stategies to deal with each class. 2.2 Lage imes We conside P n f and the numbes α n defined by n P n f { bad imes: 2 fi fo some i n. good imes: 2 fi fo any i n. i1 fi Next lemma allow us to analyze the lage imes involved in 4. Lemma 1. If 2an b then α n β n. α n. 5 Poof. If β n 0 then α n 0. If α n > β n 1 then thee exist i < j n such that fi and fj. It imlies that fj fi j iaj i b. Thus j i o aj i b, which is not ossible because 2an b. Since C 2a b we can wite log L n f log P n f <Cn β n α n log. 6 Indeed we can take C to be any constant geate than 2a b. As we will see, the final estimate of log L n f will not deend on C. The estimate of log P n f is easy: log P n f log n fk log k1 n log a logn! 2 n k1 ak 2 1 b ka c k 2 a n k1 2n log n nlog a 2 Olog n log 1 b ka c k 2 a 7 3
Javie Cilleuelo and we obtain log L n f 2n log n nlog a 2 β n α n log Olog n. 8 <Cn 2.3 The numbe of solutions of fx 0 mod k and the secial imes The numbe of solutions of the conguence fx 0 mod k will lay an imotant ole in the oof of Theoem 1. We wite sf, k to denote this quantity. Lemma below esumes all the casuistic fo sf, k. We obseve that excet fo a finite numbe of imes, those dividing 2aD, we have that sf; k 2 o 0 deending on D/ 1 o 1. Lemma 2. Let fx ax 2 bx c be an ieducible olynomial and D b 2 4ac. i If 2a, D l D and D, 1 then k/2, k l sf, k 0, k > l, l odd o D / 1 2 l/2, k > l, l even D / 1. { ii If a, 2 then sf, k 0, if b 1, if b. 1 if a is even iii If b is odd then, fo all k 2, sf, 2 k sf, 2 0 if a is odd and c is odd 2 if a is odd and c is even. iv If b is even and a is even then sf, 2 k 0 fo any k 1. v If b is even and a is odd, let D 4 l D, D 0 mod 4. a If k 2l 1, sf; 2 k { 2 k/2 b If k 2l, sf; 2 k 2 l, D 1 mod 4 0, D 1 mod 4. { c If k 2l 1, sf; 2 k 2 l1, D 1 mod 8 0, D 1 mod 8. Poof. The oof is a consequence of elementay maniulations and Hensel s lemma. Coollay 1. If 2aD then sf, k 1 D/. Poof. In this case, l 0 and D D in Lemma 2. Thus sf, k 0 1 D/ if D/ 1 and sf, k 2 1 D/ if D/ 1. Lemma 3. α n n sf, k log n k O. 9 log k 1 whee sf; k denotes the numbe of solutions of fx 0 mod k, 0 x < k. Poof. We obseve that the maximum exonent α,i such that α,i fi can be witten as α,i k 1, k fi 1. Thus α n α,i 1 1. 10 i n i n k 1 k 1 The tivial estimate [ ] n sf; k k i n, k fi k fi 1 sf; k i n k fi [ ] n k 1 4
gives The least common multile of a quadatic sequence i n k fi 1 n sf; k k Osf; k. 11 Putting 11 in 10 and obseving that k log fn/ log and that sf, k 1, we get α n n sf, k log n k O. log k 1 Since β n fn we have always the tivial estimate β n log fn/ log log n/ log. 12 Now we substitute 12 and 9 in 8 fo the secial imes obtaining log L n f 2n log n n log a 2 sf, k log k 13 <Cn, 2aD 2.4 Small imes Lemma 3 has an easie fomulation fo small imes. Lemma 4. Fo any 2aD we have α n n 1 D/ 1 Poof. It is a consequence of Lemma 3 and Coollay 1. 2aD k 1 β n α n log Olog n. log n O log. 14 By substituting 14 and 12 in 13 fo small imes we obtain log L n f 2n log n n log a 2 sf, k log k 15 <n 2/3 2aD 1 D/ log 1 2aD k 1 n 2/3 <Cn P f β n α n log On 2/3. 2.5 Medium imes These imes can be also classified in bad and good imes. Bad imes ae those such that 2 fi fo some i n. Good imes ae those ae not bad imes. As we have seen in the evious section, fo any ime P f the conguence fx 0 mod has exactly two solutions, say 0 ν,1, ν,2 <. If is a good ime, we have that α n is just the numbe of integes i n such that fi. All these integes have this fom [ ] n ν,1 ν,1 k, 0 k 16 [ ] n ν,2 ν,2 k, 0 k. 17 Also it is clea that if is a good ime then β n 1. These obsevations motivate the following definition: 5
Definition 1. Fo any P f we define [ n αn ν,1 Lemma 5. Fo any P f we have β n i α n αn 2n 1 Olog n/ log Javie Cilleuelo ] { 1, if β n 1 0, othewise. [ n ν,2 ii α n α n and β n β n if 2 fi fo any i n. ] 2 18 Poof. i Lemma 4 imlies that α n 2n 1 Olog n/ log when P f. On the othe hand we have that αn 2n O1. Thus, α n αn 2n 1 Olog n/ log. ii The fist assetion has been exlained at the beginning of the subsection. Fo the second, if fi fo any i n then β n βn 0. And if fi fo some i n we have that βn β n 1 since 2 fi. 19 Now we slit the last sum in 15 in β n α n log n 2/3 <Cn P f n 2/3 <Cn P f β <Cn P f β n β n α n α n log 20 n log n 2/3 <Cn P f S 1 n S 2 n S 3 n On 2/3. α n log On 2/3 To estimate S 1 n we obseve that Lemma 5 ii imlies that β n β n α nα n 0 fo any good ime. On the othe hand, Lemma 5 i and 12 imlies that β n β n α nα n log n/ log. Thus, S 1 n log n { : n 2/3 < < Cn, bad}. 21 Lemma 6. The numbe of bad imes D, Q < 2Q is n 2 /Q 2. Poof. Let P the set of all imes such that fi ai 2 bi c 2 fo some i n. Fo P we have 2ai b 2 4a 2 D and then, 2aib 2 a 1 1 2 Q. We obseve that all the factions 2 2aib, 1 i n, Q < 2Q ae aiwise diffeent. Othewise 2ai b 2ai b and then 2ai b. But it would imly that 2ai b 2 4a 2 D, which is not ossible. On the othe hand, 2aib 2ai b 1 1 Q. Thus, the numbe of imes P 2 lying in [Q, 2Q] is 1. We finish the oof by obseving that fn/q 2 n 2 /Q 2. Now, if we slit the inteval [n 2/3, Cn] in dyadic intevals and aly lemma above to each inteval we obtain S 1 n n 2/3 log n. To estimate S 3 n n 2/3 <<Cn, P f αn we stat by witing [ ] [ ] n αn ν,1 n ν,2 2 2n 1 2 ν,1 1 2 ν,2 1 { n 2 ν,1 } 1 { } n 2 ν,2. 6
The least common multile of a quadatic sequence Thus S 3 n n n 2/3 <<Cn n 2/3 <<Cn 0 ν< fν 0 mod n n 2/3 <<Cn 0 ν<<cn fν 0 mod 1 D/ log 1 2 ν log 1 D/ log 1 1 2 ν log n 2/3 <<Cn 0 ν< fν 0 mod 22 1 { n ν } 2 log 23 On 2/3 24 0 ν<<cn fν 0 mod 1 { n ν } 2 log 25 Substituting this in 20 and then in 15 we obtain log L n f 2n log n n log a 2 <Cn 2aD 1 D/ log 1 2aD k 1 sf, k log k 26 S 2 n T 1 n T 2 n On 2/3 log n whee S 2 n T 1 n T 2 n β <Cn P f n log 27 0 ν<<cn fν 0 mod 0 ν<<cn fν 0 mod 1 2 ν log 28 1 2 { n ν } log. 29 Sums T 1 n and T 2 n will be on as a consequence of Theoem 2. But this is not comletely obvious and we will ovide a detailed oof in the next subsection. Fist we will obtain in the next lemma a simlified exession fo 26. Lemma 7. log L n f n log n cn S 2 n T 1 n T 2 n On 2/3 log n, 30 whee c log a log C 2 γ 2aD d/ log 1 2aD 1 log 1 sf, k k k 1 and S 2 n, T 1 n and T 2 n ae as in 27, 28 and 29. Poof. Let D l 2 d whee d is a fundamental disciminant. Fist we obseve that D/ l/ 2 d/ and that if D then D/ d/. As a consequence of the ime numbe theoem on aithmetic ogessions we know that the sum is convegent. On the othe hand, the well known estimate d/ log 1 7
Javie Cilleuelo log x 1 log x γ o1 whee γ is the Eule constant, imlies that <Cn 2aD 1 D/ log 1 log n log C γ 2aD d/ log 1 2aD o1. log 1 31 Finally we substitute 31 in 26. 2.6 Equidistibution of the oots mod of a quadatic olynomial Now we develo a method to ove that T 1 n, T 2 n and othe simila sums which will aea in the estimate of S 2 n ae all on. These sums ae all of the fom 0 ν< x, S fν 0 mod fo some function aν,, x 1. By atial summation we also get easily that 0 ν< x, S fν 0 mod aν,, x log log x log x 0 ν< x, S fν 0 mod 0 ν< x, S fν 0 mod Hence, to ove that the sums 32 ae ox we must show that 0 ν< x, S fν 0 mod aν,, x log 32 x 1 aν,, x 1 t 0 ν< t, S fν 0 mod aν,, x 33 aν,, x ox/ log x. 34 aν,, x ox/ log x. Theoem 2 imlies, in aticula, that fo any aithmetic ogession S and fo any iecewise continuos function g in [0, 1] such that 1 g 0 we have that 0 gν/ ox/ log x. 35 0 ν< x, S fν 0 mod Lemma 8. Let f be an ieducible olynomial in Z[x]. We have that the sums T 1 n and T 2 n defined in 28 and 29 ae both on. Poof. To ove that T 1 n on we aly 35 to the function gx x 1/2. To ove that T 2 n on, the stategy is slitting the ange of the imes in small intevals such that n/ ae almost constant in each inteval. We take H a lage, but fixed numbe and we divide the inteval [1, Cn] in H intevals L h h 1 H Cn, h H Cn], h 1,..., H. Now we wite 0 ν<<n fν 0 mod { n ν } 1 Σ 31 Σ 32 Σ 33 On/H 1/3 log n 36 2 8
whee The least common multile of a quadatic sequence Σ 31 Σ 32 Σ 33 H 2/3 h H 0 ν< L h fν 0 mod H 2/3 h H 0 ν< L h fν 0 mod ν [ H h, H h 1 ] H 2/3 h H 0 ν< L h fν 0 mod ν [ H h, H h 1 ] To estimate Σ 31 we aly 35 with the function { H h ν } 1 2 { n ν } { H h ν } { n ν } { H h ν }. { H h x } 1 2 in each L h and we obtain Σ 31 ohn/ log n on log n 37 since H is a constant. To bound Σ 32 we obseve that if L h and ν [ H h, H h 1 ], then { n 0 ν } { H h ν } n H h H hh 1. Thus Σ 32 H 2/3 h<h L h To bound Σ 33 fist we obseve that Σ 33 H h 2 H 2/3 h<h H 2/3 h<h 0 ν< L h fν 0 mod ν [ H h, H h 1 ] H 2/3 h<h 0 ν< L h fν 0 mod H 2/3 h<h 0 ν< L h fν 0 mod 1 1 πn H1/3 H n 1/3 H 1/3 log n. 38 L h χ [H/h,H/h 1] ν/ H hh 1, H hh 1 whee, hee and late, χ [a,b] x denotes the chaacteistic function of the inteval [a, b]. Thus, Theoem 2 imlies that 0 ν< L h fν 0 mod Σ 33 H 2/3 h<h χ [H/h,H/h 1] ν/ n o log n H on/ log n. hh 1 H 2/3 h<h 0 ν< L h fν 0 mod on/ log n πn H 1/3 on/ log n On/H1/3 log n. 1 H 1/3 39 Estimates 37, 38 and 39 imly Σ 3 on/ log n n/h 1/3 log n. Since H can be chosen abitaily lage, we have that Σ 3 on/ log n which finishes the oof. 9
Javie Cilleuelo To esent Lemma 10 we need some evious consideations. Fo imes P f the conguence fx 0 mod has exactly two solutions, say 0 ν,1, ν,2 <. In some ats of the oof of Theoem 1 we will need to estimate some quantities deending on minν,1, ν,2. Fo this eason it is convenient to know how they ae elated. If fx ax 2 bx c and P f then ν,1 ν,2 b/a mod. Next lemma will give moe infomation when the ime belongs to some aticula aithmetic ogession. Lemma 9. Let q a/a, b, l b/a, b. Fo any,, q 1 and fo any ime l 1 mod q and P f we have ν,1 ν,2 mod 1. 40 q Poof. To avoid confusions we denote by q and q the inveses of q mod and mod q esectively. Fom the obvious conguence qq q 1 mod q we deduce that q mod q we obtain q 2q 1 q l q q mod 1. Thus ν,1 ν,2 lq l 1 q l q q q q q mod 1. [ Since the two oots ae symmetic esect to 2 2q, necessaily one of then lies in mod 1 and the othe in the comlementay set. l 2q 1 q Z. Since l q 2q l 2q, 1 2 Definition 2. Fo, q 1, 1 q, l 1 mod q and P f we define ν,1 the oot of fx mod such that ν [,1 T 2 2q, 1 2 2 mod 1, 2q and we define ν,2 the oot of fx 0 mod such that ν,2 [0, 1 \ T. Lemma 10. Assume the notation above. Let α 1, α 2, β 1, β 2, c 1, c 2 be constants and g 1 x, g 2 x two linea functions satisfying that [ n J n g 1 c 1 n, g 2 c ] 2 T fo any ime K n [α 1 n β 1, α 2 n β 2 ]. We have K n P f l 1 mod q χ Jn ν,1 whee χ I is the chaacteistic function of the set I. 2 J n Poof. Since J n T then ν 2 / J n and we can wite ν,1 χ Jn log and Thus, K n P f l 1 mod q K n P f l 1 mod q K n P f l 1 mod q χ Jn ν,1 2 J n log 2 J n log 1 ν K n, fν 0 mod l 1 mod q 1 ν K n, fν 0 mod l 1 mod q 1 ν K n, fν 0 mod l 1 mod q log on 41 χ Jn ν log J n log. ν χ Jn J n log. 10
The least common multile of a quadatic sequence The oof will be accomlished by showing that ν χ Jn J n on/ log n. 42 1 ν K n, fν 0 mod l 1 mod q We slit K n in intevals L h h 1 H n, h H n] of length n/h and two exta intevals I, F the initial and the final intevals of length n/h. Hee h uns ove a suitable set of consecutive integes H of cadinality α 2 α 1 H. Let I h denote the inteval [g 1 H/h c 1 H/nh, g 2 H/h c 2 H/nh]. We wite 1 ν K n, fν 0 mod l 1 mod q ν χ Jn J n Σ 1 Σ 2 Σ 3 Σ 4 43 whee Σ 1 h H ν χ Ih I h Σ 2 h H Σ 3 h H Σ 4 0 ν< L h fν 0 mod l 1 mod q 0 ν< L h fν 0 mod l 1 mod q 0 ν< L h fν 0 mod l 1 mod q 0 ν< I F fν 0 mod l 1 mod q χ Ih ν I h J n χ Jn ν ν χ Ih J n. The inne sum in Σ 1 can be estimated as we did in Lemma 8, with the function gx χ I x I instead of gx x 1/2, and we get again that Σ 1 on/ log n. To estimate Σ 2 and Σ 3 we obseve that if L h then J n and I h ae almost equal. Actually, comaing the end oints of both intevals and because g is a linea function, we have that and χ Jn x χ Ih x excet fo an inteval o union of two intevals E h of measue E h min1, H/h 2. In aticula, the estimate J n I h min1, H/h 2 holds. Thus, we have Σ 2 min1, H/h 2 h H L n h H 2/3 L h H 2/3 <h H πn/h 1/3 1 H 1/3 πα 1n α 2 n/h 1/3 log n. 1 H 1/3 L h 11
Javie Cilleuelo To bound Σ 3 fist we obseve that Σ 3 h H χ Eh ν/ h H 0 ν< L h fν 0 mod l 1 mod q χ Eh ν/ E h h H E h. Theoem 2 imlies that 0 ν< L h fν 0 mod l 1 mod q 0 ν< L h fν 0 mod l 1 mod q χ Eh ν/ E h on/ log n. On the othe hand, 0 ν< L h fν 0 mod l 1 mod q h H 0 ν< L h fν 0 mod l 1 mod q Thus, Σ 3 on/ log n n/h 1/3 log n. Finally we estimate Σ 4. We obseve that Σ 4 I E h h H 2/3 L h 1 H 2/3 <h H L h πn/h 1/3 1 H πα 1n α 1/3 2 n H 1/3 log n. 1 F 1 n/h log n H h 2 as a consequence of the ime numbe theoem. Then Σ 1 Σ 2 Σ 3 Σ 4 On/H 1/3 log n On/H log n on/ log n finishing the oof because we can take H abitaily lage. 2.7 Estimate of S 2 n and end of the oof Lemma 11. S 2 n n 1 log C log 4 1 φq,q1 Poof. Following the notation of Lemma 9 we slit S 2 n S 2 n βn log l whee,q1 1 q S 2 n l< Cn l 1 mod q log1 q on 44,q1 1 q S 2 n O1 β n log. 45 Since l 1 mod q, Lemma 9 imlies that ν,1 since > l we have that 0 < q 1. ν,2 q mod 1. We obseve also that, 12
Now we will check that The least common multile of a quadatic sequence n 1, if 1 2 2q χ βn [ 2q l 2q, n ]ν,1/, if q < n < 1 2 2q χ [ 2q l 2q, ]ν,1/, if q 2 2q n q χ [ q n, ]ν n,1/ if q < 2 2q We obseve that βn 1 if and only if ν,1 n o ν,2 n. We emind that l 2q l 2q 2q l 2q ν,1 < 1 2 2q l 2q 46 Also we obseve that Lemma 9 imlies that ν,2 { q ν,1 if ν,1 q ν,1 1 if ν,1 q Assume n 1 2 2 2q. Then ν 1,1 < 2 2 2q < n, so βn 1 Assume q < n < 1 2 2q If χ [ If χ [ l 2q. 2 2q, n ]ν,1/ 1 then ν,1 n, so βn 1. 2 2q, n ]ν,1/ 0 then ν,1 > q. 47 > n > q. Relations 46 and 47 imly that ν,2 1 q ν,1 > 1 2 2 2q > n. Since ν,1 > n and ν,2 > n we get β n 0. Assume 2 2q n q. If χ [ 2 2q, ]ν,1/ 1 then 47 imly that 0 < ν,2 q 2 2q, which imlies that ν,2 n, so βn 1. If χ [ 2 2q, ]ν,1/ 0 then ν,1 q > q n and elation 47 imly that ν,2 q ν,1 Assume n < 2 2q. If χ [ ν,1 1 > q n. Since ν,1 > n and ν,2 > n we get β n 0. q n, ]ν,1/ 1 then ν,1 q q q n n, so β n 1 If χ [ q n, q ]ν,1/ 0 we distinguish two cases: If 2 2q ν,1 q ν,1 > < q n q q and elation 47 imlies that ν,2 q ν1, then q n If q < ν,1 < 1 2 2 2q then ν,1 q ν,1 1 > q 1 2 2 2q that β n 0. 2 2q > n, and also we have that ν,2 n. Thus β n 0 > 1 2 q > n. On the othe hand, ν,2 1 1 2 2 2q > n. Thus, again we have 13
Javie Cilleuelo Now we slit S 2 n 4 i1 S 2in accoding the anges of the imes involved in lemma above. S 21 n log S 22 n S 23 n S 24 n l< nl/2q 1/2/2q l 1 mod q P f χ [ 2 nl/2q nl/q << 1/2/2q /q l 1 mod q P f χ [ 2 2q, q n l 2q q n l q l 1 mod q P f 2q n l 2q <<Cn l 1 mod q P f 2q, n ]ν,1/ log q ]ν,1/ log χ [ q n, q ]ν,1/ log. Since q, D 1 and the imes ae odd numbes, the imes l 1 mod q, P f lie in a set of φ4qd/2φq aithmetic ogessions modulo 4qD. The ime numbe theoem fo aithmetic ogessions imlies that and l 1 x mod q, P f ax< bx l 1 mod q, P f log log x 2φq logb/a 2φq 48 o1 49 We will use these estimates and lema 10 to estimate S 2i n, i 1, 2, 3, 4. By 48 we have To estimate S 52 we wite S 22 n nl/2q nl/q << 1/2/2q /q l 1 mod q nl/2q nl/q << 1/2/2q /q l 1 mod q nl/2q nl/q << 1/2/2q /q l 1 mod q S 21 n n q on. 50 φq q χ [ 2 2q, n ]ν,1/ log 2n q l log q χ [ 2 2q, n ]ν,1/ 2 n 2q l log 2q 14
The least common multile of a quadatic sequence Lemma 10 imlies that the last sum is on. Thus, 2n S 52 log on q nl/2q nl/q << 1/2/2q /q l 1 mod q P f 2n by 48 and 49. To estimate S 23 n we wite S 23 n nl/2q nl/q << 1/2/2q /q l 1 mod q P f n φq log 1 2 q 2 q n l 2q q n l q l 1 mod q P f q n l 2q q n l q l 1 mod q P f n 2φq on by 48 and lema 10. To estimate S 24 n we wite S 24 n by 49 and Lemma 10. Thus 2q n l 2q <<Cn l 1 mod q P f 2q n l 2q <<Cn l 1 mod q P f n φq log q n 1 φq 2 q q χ [ 2n 2l q nl/2q nl/q << 1/2/2q /q l 1 mod q P f log 2 2q, log χ [ q n, log C log2q/ on on log on ]ν,1/ q q q ]ν,1/ log 2n 2l log q S 2 n S 21 n S 22 n S 23 n S 24 n O1 n q φq q on n 1 φq log 2 q n 1 2 φq 2 on q n 2φq on n log C log2q/ on φq n φq 1 log C log 4 log1 /q on. 15
Javie Cilleuelo Now sum in all q,, q 1 to finish the estimate of S 2 n. Finally we substitute 44 in 30 to conclude the oof of Theoem 1. d/ log 3. Comutation of the constant B f The sum 1, aeaing in the fomula of the constant B f conveges vey slowly. Next lemma gives an altenative exession fo this sum, moe convenient in ode to obtain a fast comutation. Lemma 12. d/ log 1 k1 ζ 2 k ζ2 k k0 L 2 k, χ d L2 k, χ d s. 51 d whee s k1 log 2k 1. Poof. Fo s > 1 we conside the function G d s d/. 1 1 Taking the deivative of the logaithm s of G d s we obtain that G d s G d s d/ log s. 52 1 Since Ls, χ d whee T s d 1 d/ 1 1 s. s 1 we have The deivative of the logaithm gives G d sls, χ d 1 1 d/ s 1 d/ s 1 1 1 2s d/ 1 1 1 2s d/ 1 2 d 1 1 1 2s 1/2 53 54 55 G 1/2 d 2sζ 1/2 2sT 1/2 2s 56 G d s G d s G d 2s G d 2s ζ 2s ζ2s T d 2s T d 2s L s, χ d Ls, χ d. Thus G d s G d s G d 2m m 1 s G d 2 m s G d 2 k s G d 2 k s G d 2k1 s G d 2 k1 s k0 m k1 ζ 2 k s m ζ2 k s k1 T d 2k m 1 s T d 2 k s k0 57 L 2 k s, χ d L2 k s, χ d. 58 16
By 52 we have that fo s 2, The least common multile of a quadatic sequence ζ s ζs n 2 Λn log 2 n s 1 2 s 1 n 3 4 log 2 3 2 s 9 8 n 3 4 log 2 3 2 s 9 log 2 8 1 20 log 2 8 2 s s 1 9 log n n s 1 log n n s 4 log 2 3 2 s 9 8 2 s 1 s 1 1 2 s 1 s 1 2 5 2 2 s s 1. 2 log x x s dx Thus, ζ 2 k ζ2 k 5 2 k 2 2 2 k 1. The same estimate holds fo G d 2k G d 2 k, T d 2k T d 2 k and L 2 k,χ d L2 k,χ d. When m and then s 1 we get d/ log 1 k1 ζ 2 k ζ2 k k0 L 2 k, χ d L2 k, χ d Finally we obseve that T d 2k T d 2 k log d, so T d 2k 2k 1 k1 T d 2 k d s. k1 T d 2k T d 2 k. 59 The advantage of the lemma above is that the seies involved convege vey fast. Fo examle, with Eo 10 40. k0 L 2 k, χ d 6 L2 k, χ d k0 L 2 k, χ d L2 k, χ d Eo Hence we can wite B f C 0 C d Cf whee C 0 is an univesal constant, C d deends only on d, and Cf deends on f. Moe ecisely, C 0 γ 1 2 log 2 C d k0 Cf 1 φq 1 q,q1 k1 ζ 2 k ζ2 k 1.1725471674190148508587521528364 L 2 k, χ d L2 k, χ d s d log 1 log a 1 d/ log sf, k q 1 k. 2aD k 1 The values of s and k 0 L 2 k, χ d /L2 k, χ d, can be calculated with MAGMA with high ecision. We include some of the values of C d and Cf: C 4 0.346538435736895987549 s 2 0.066550762366036180349... C 8 0.076694093066485311184 s 2 0.356681766437345118384... C 3 0.586272400297149523649 s 3 0.435045713698422447292... C 7 0.070022837990444988815 s 7 0.111373766208260107471... C 15 0.486320692903261758405 s 3 s 5 0.707190640126000030028... 17
Javie Cilleuelo Cx 2 1 3 log 2/2 1.039720770839917964125... Cx 2 2 3 log 2/2 1.039720770839917964125... Cx 2 x 1 log 2 log 3/6 0.876249228671296924649... Cx 2 x 2 log 2 log 7/42 0.739478374585071816681... C2x 2 1 3 log 2 2.079441541679835928251... C2x 2 x 1 2 log 2 log 3 log 7/42 1.838090663253181508076... C2x 2 x 2 log 2 7 log 36 log 5/20 2.055333412961111634775... C2x 2 2x 1 3 log 2 2.079441541679835928251... Table below contains the constant B B f fo all ieducible quadatic olynomial fx ax 2 bxc with 0 a, b, c 2. When f 1, f 2 ae ieducible quadatic olynomials such that f 1 x f 2 x k fo some k, we only include one of them since L n f 1 L n f 2 Olog n. fx d q B f x 2 1-4 1 0.06627563421306070638... x 2 2-8 1 0.48950816301644200511... x 2 x 1-3 1 0.13874777495070452108... x 2 x 2-7 1 0.54444255904220314164... 2x 2 1-8 1 0.55021260782347595900... 2x 2 x 1-7 2 0.55416972962590654974... 2x 2 x 2-15 2 0.17559560541609675388... 2x 2 2x 1-4 1 0.97344513662685725774... Table below shows the eo tem E f n log L n f n log n B f n fo the olynomials above and some values of n. fx E f 10 2 E f 10 3 E f 10 4 E f 10 5 E f 10 6 E f 10 7 x 2 1 18 6 111 34 2634 1557 x 2 2 36 11 263 761 1462 8457 x 2 x 1 6 9 17 654 2528 1685 x 2 x 2 9 20 218 2120 687 686 2x 2 1 15 1 301 251 1084 14821 2x 2 x 1 1 6 18 1289 235 2553 2x 2 x 2 34 4 295 27 1169 1958 2x 2 2x 1 9 89 9 232 2876 10624 4. Quadatic educible olynomials To comlete the oblem of estimating the least common multile of quadatic olynomials we will study hee the case of educible quadatic olynomials. Being this case much easie than the ieducible case, we will give a comlete descition fo the sake of the comleteness. If fx ax 2 bx c with g a, b, c > 1, it is easy to check that log L n f log L n f O1 whee f x a x 2 b x c with a a/g, b b/g, c c/g. 18
The least common multile of a quadatic sequence If fx ax b 2 with a, b 1 then, since m 2, n 2 m, n 2, we have that L n ax b 2 L 2 nax b and we can aly 1 to get log l.c.m.{a b 2,..., an b 2 } 2n a 1 φa k. 60 1 k a k,a1 Now we conside the moe geneal case fx ax bcx d, a, b c, d 1. Theoem 3. Let fx ax bcx d with a, b c, d 1 and ad bd. Let q ac/a, c. We have log L n f n a c max,. 61 ϕq b a d c 1 q,,q1 Poof. Suose 2 L n f. It imlies that 2 ai bci d fo some i. If ai b and ci d then ad bci. If ad bc then i and consequently b and d. Thus, if ad bcbd and 2 aibcid then 2 aib o 2 cid. In these cases M n max an b, cn d, ad bdbd. Thus we wite L n f ɛn ɛn, 62 >M n M n βn M n βn ɛn whee ɛ n 1 if fi fo some i n and ɛ n 0 othewise. Since β n fn we have that β n log n/ log and then M n β n ɛ n log log nπm n n. 63 Thus, log L n f fi fo some i n log O n. 64 Let q ac/a, c. Suose that 1 mod q,, q 1. Let k b a the least ositive intege such that k b mod a. Then ai b fo some i n if and only if k an b. Similaly, let j d c be the least ositive intege such that j d mod c. Again, ci d fo some i n if j cn d. Thus, the imes 1 mod ac counted in the sum above ae those such that max anb k, cnd j. The ime numbe theoem fo aithmetic ogessions imlies that thee ae n ϕq max a k, c j of such imes. We finish the oof summing u in all 1 q,, q 1. Refeences Bat02 P. Bateman, A limit involving Least Common Multiles: 10797, Ameican Mathematical Monthly 109 2002, no. 4, 393-394. Che52 P.L. Chebyshev, Memoie su les nombes emies. J. Math. ues et al. 17 1852, 366-390. DFI95 W. Duke, J. Fiedlande and H. Iwaniec, Equidistibution of oots of a quadatic conguence to ime moduli, Ann. of Math. 141 1995, no. 2, 423 441. Tot00 A. Toth, Root of quadatic conguences, Intenat. Math. Res. Notices 14 2000, 719 739. Javie Cilleuelo fanciscojavie.cilleuelo@uam.es Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Deatamento de Matemáticas, Univesidad Autónoma de Madid, 28049 Madid, Sain 19