THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE
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1 THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe f(x = ax 2 +c we rove the more recise estimate, log l.c.m. {f(1,..., f(} = log + B + o( for a suitable costat B = B(a, c. 1. Itroductio It is well kow that log l.c.m.{1,..., }. Actually it is equivalet to the rime umber theorem. The aalogous for arithmetic rogressios is also kow [2], log l.c.m.{a + b,..., a + b} a 1 ϕ(a k. 1 k a (k,a=1 We address here the roblem of estimate L (f = l.c.m.{f(1,..., f(}, where f is a irreducible quadratic olyomial i Z. The case f(x = x has bee cosidered i [1] where the estimate log L (f = log l.c.m.{f(1,..., f(} A+B was obtaied i this case for exlicit costats A, B > 0. I sectio 2 we give a asymtotic estimate for a geeral irreducible quadratic olyomial f(x = ax 2 + bx + c. Theorem 1.1. For ay irreducible quadratic olyomial we have log l.c.m. {f(1,..., f(} log. I sectio 3, which is the mai art of this work, we obtai a more recise estimate i some articular cases. Theorem 1.2. For ay irreducible olyomial f(x = ax 2 + c we have log l.c.m. {f(1,..., f(} = log + B + o(, for a exlicit costat B = B(a, c. It icludes the simlest case f(x = x with B(1, 1 = 1 3 log 2 2 lim t t 1 (mod 4 2 log 1 log t Mathematics Subject Classificatio Mathematics Subject Classificatio: 11A05. Key words ad hrases. least commo multile, quadratic sequeces. This work was suorted by Grat MTM of MYCIT (Sai. 1
2 2 JAVIER CILLERUELO To rove theorem 1.2 we eed to use a dee result about the distributio of the solutios of the quadratic cogrueces f(x 0 (mod due to Duke, Friedlader ad Iwaiec (for D > 0, ad Toth (for D < 0, where D = b 2 4ac is the discrimiat of f. Theorem 1.3 ([3], [4]. For ay irreducible quadratic olyomial f, the sequece {ν/, 0 ν < x, f(ν 0 (mod } is well distributed i [0, 1 whe x. It would be iterestig to exted these estimates to irreducible olyomials of higher degree, but we have foud a serious obstructio i our argumet. Some heuristic argumets allow us to cojecture that the asymtotic log l.c.m. {f(1,..., f(} (deg(f 1 log holds for ay irreducible olyomial f. 2. Prelimiaries ad the geeral case For ay irreducible f(x = ax 2 + bx + c we write P (f = i=1 f(i = α ad L (f = l.c.m.{f(1,..., f(} = β. Lemma 2.1. With the otatio above we have that α = β for 2a + b. Proof. Notice that α β if ad oly if there exist i, j, i < j such that f(i ad f(j. I that case we have f(j f(i, so (j i(a(j + i + b ad the < 2a + b. For short we write L = L (f ad P = P (f. The lemma above allow us to write (2.1 log L = log P + (β α log. <2a+b We have log P = i log(ai2 + bi + c = log a + i 2 log i + O( i 1/i, so (2.2 log P = 2 log + (log a 2 + o(. Notice also that if β f(i for some i, the β log f(/ log = O(log / log. Thus <2a+b β log = O(log π(2a + b = O( ad the (2.3 log L = 2 log α log + O(. 2a+b We observe that we ca write α = x ν (f(x, where ν (m deotes the maximum l such that l m. As ν (m = { k χ k(m, with χ k(m = 1, k m, 0, otherwise we have (2.4 α = χ k(f(x = 1 k x k ad we obtai the trivial estimate (2.5 s(f; k [/ k ] x k f(x x k f(x 1 s(f; k ( [/ k ] + 1,
3 THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE 3 where s(f; k deotes the umber of solutios of f(x 0 (mod k, 0 x < k. Sice k log f(/ log we have (2.6 α = s(f, k k + O s(f; k. k log(f(/ log Lemma belove resumes all the casuistic for s(f, k. k log f(/ log Lemma 2.2. Let f(x = ax 2 +bx+c be a irreducible olyomial ad D = b 2 4ac. (1 If 2a, D = l D, (D, = 1,the k k/2, k l s(f, k = 0, k > l, l odd or (D / = 1. 2 l/2, k > l, l eve (D / = 1 { (2 If a, 2 the s(f, k 0, b = 1, b (3 If b is odd the s(f, 2 k = s(f, 2 for ay k 2. (4 If b is eve, let D = 4 l D, D 0 (mod { 4. (a If D 2, 3 (mod 4, s(f; 2 k 2 [k/2], k 2l 1 = 0, k 2l { (b If D 1 (mod 8, s(f; 2 k 2 [k/2], k 2l = 2 l+1, k 2l + 1 { (c If D 5 (mod 8, s(f; 2 k 2 [k/2], k 2l = 0, k 2l + 1 Proof. The roof is a cosequece of elemetary maiulatios ad Hesel s lemma. Sice for ay rime s(f; k is bouded we have (2.7 α = k 1 s(f, k k + O (log / log which gives the trivial estimate α = O(/ for ay rime. If 2aD the s(f; k = 2 or 0 accordig with (D/ = 1 or 1. The, for these rimes we have { 0, (D/ = 1 (2.8 α = ( O log log, (D/ = 1. Now we ut together the estimate α = O(/ for the rimes 2aD, the estimate π(x x/ log x ad the formulas (2.3 ad (2.8 to obtai (2.9 log L = 2 log 2 log 1 + O(. 2a+b (D/=1 The quadratic recirocity law allow us to slit the residues d, (d, D = 1 i two sets D 1, D 1 of the same size, ϕ(d/2, such that (D/ = 1 d (mod D
4 4 JAVIER CILLERUELO with d D 1. The rime umber theorem for arithmetic rogressios says that (2.10 log = t ϕ(d + O(t/ log2 t for d D 1. The (2.11 t d (mod D t (D/=1 t (D/=1 log = t 2 + O(t/ log2 t. Actually a better error term is kow i (2.10 ad (2.11, but this oe is eough to deduce, by artial summatio, that log ( = log t + A D + o(1 2 for a costat A D. We fiish the roof of theorem 1.1 by erformig a substitutio i (2.9 with the formula above. 3. A more recise estimate for f(x = ax 2 + c We ca stregth lemma 2.1 whe b = 0. Lemma 3.1. If f(x = ax 2 + c we have that α = β for 2. Proof. If α > β the there exists i < j such that ai 2 + c ad aj 2 + c. But it imlies that a(i j(i + j, so i j, a or i + j. I ay case < 2. The we ca write (3.1 log L = log P + (β α log. Defie Q = {, 2c} ad P = { < 2, 2ac, ( ac/ = 1}. Lemma 2.2 imlies that if P Q the β = α = 0. We defie also the bad ad the good rimes as (3.2 P bad = { P, 2 ai 2 + c for some i } ad P good = P \ P bad. Lemma 3.2. Suose that P good. The i α = 2 +z where z = 1 { +x of ax 2 + c 0 (mod such that 0 < x ( 1/2 ii β = 1 for < 2. Proof. i Sice is a good rime we have } { x }, where x deotes the solutio α = #{l, x + (l 1 } + #{l, x + (l 1 }. So α = [ x ] [ +x ] = 2 + z where z = 1 { +x } { x }. ii Sice is good we have always that β 1. O the other had, sice x ( 1/2 < ad ax 2 + c we have that β 1. Lemma 3.3. For ay J 1 we have #{, /J < 2, P bad } J 3.
5 THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE 5 Proof. If > /J is bad, the there exists i such that ai 2 + c = 2 r for some 2 r aj 2. For each r, 2 r aj 2, cosider P r = { > /J, ai 2 + c = 2 r, for some i }. If P r we have that r a i c 2 ra c J 2 2 ad the ra all i / lie o a iterval of legth 2 c J 2 2 ra. O the other had, i i 1 1 4, 2 so P r 8 c J 2 ra + 1 ad r J P 2 r J 3. We fix a large iteger J ad use the lemmas above to write β log = β log + log + (β 1 log + 2c /J O(log + log + /J /J< O(log + By (2.11 ad lemma 3.2 we have ( (3.3 β log = + o( + O J log(/j log + O(J 3 log, whe. Now we write (3.4 α log + (3.5 (3.6 (3.7 where α log = 2c ( 2 + z C(a, c + O(log log ( 1 + (3.8 C(a, c = 2c (3.9 We use (2.12 ad the estimates (3.10 z log = log </J log + O ( 2 log + O(log + 1 log 1 + O log k 1 bad ( log ( 1 = O log = J log(/j log z log + O(J 4 log, s(f; k k. ( log(/j /J ( z + O log J, log (β 1 log = /J< bad O(log.
6 6 JAVIER CILLERUELO to obtai (3.11 (3.12 α log = log + (C(a, c + log 2 + 2A D + log z + o( + O(/J whe. Notice that most of the error terms have bee icluded i o(. To estimate z we first slit the rimes /J < < 2 i short itervals [/J, H/J] ad I j = ( j 1 J (3.13 z = (3.14 /J H/J H<j 2J, j J ], H < j < 2J to write z + H<j 2J I j z = H z + O( J log(h/j I j where H is a iteger which will be chose later. For I j, j > H we ca write = t j ɛ j ( where t j = J j 1 ad ɛ j( = J (j 1 (j 1. Notice that 0 ɛ j ( J (j 1 J 2 H. The we have 2 z = 1 {t j + x / + ɛ j (} {t j x / + ɛ j (}. We deote by E j the set of the rimes I j such that (3.15 {t j } x / {t j } + J H 2 or 1 {t j } x / 1 {t j } + J H 2. If I j \ E j we have that z = 1 {t j + x /} {t j x /} 2ɛ j (, so (3.16 z = 1 {t j + x /} {t j x /} + O(J/H 2 for these rimes. For rimes E j it is useful to write (3.17 z = 1 {t j + x /} {t j x /} + O(1. Theorem 1.3 imlies that the sequeces {t j + x /} ad {t j x /} are well distributed o M + j ad M j resectively, where M± j = t j ±[0, 1/2 (mod 1. Observe also that M + j M j = [0, 1. The we have that (3.18 {t j + x /} = 2 sds π(p; y + o(π(p; y, y where π(p; y = y 1. For the same reaso we have that (3.19 {t j x /} = 2 sds π(p; y + o(π(p; y y M + j M j
7 THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE 7 ad the (3.20 (3.21 ( 1 2 (1 {t j + x /} {t j x /} = y, [0,1 sds π(p; y + o(π(p; y = o(y/ log y. I articular we have that (3.22 (1 {t j + x /} {t j x /} = o(/ log. I j P So, if j H, by (3.16, (3.17 ad (3.22 we obtai (3.23 z = o(/ log + O J H 2 + O( E j ad the, (3.24 I j P H j 2J I j P ( J z = o(j/ log + O H 2 Sice x / is well distributed we have that Hece (3.25 I j 1 log + O( H j 2J E j. E j = 2J (π(p, j/j π(p, (j 1/J + o(π(p, j/j. H2 H<j 2J E j = 2J (π(p; 2 π(p; H/J + o(j/ log. H2 If we take H = [J 2/3 ], formulas (3.25, (3.24, (3.13 ad (3.14 give ( (3.26 z = O + o(/ log. J 1/3 log This ad (3.11 yield (3.27 α log = log + (C(a, c + log 2 + 2A D + O(/J 1/3 + o(. Puttig (3.27, (3.3, (2.1 ad (2.2, we have fially where B(a, c = log a 1 log 2 2c log L = log + B(a, c + o( + O(/J 1/3 log k 1 s(ax 2 + c; k k lim 2 log t 1 log t ad we fiish the roof of theorem 1.2 observig that we ca choose J arbitrary large.
8 8 JAVIER CILLERUELO Refereces [1] B. Farhi, No trivial lower bouds for the least commo multile of some fiite sequeces of itegers. Joural of Number theory (2007,doi: /j.jt [2] P. Batema, A limit ivolvig Least Commo Multiles: 10797, America Mathematical Mothly 109 (2002, o. 4, [3] W. Duke, J. Friedlader ad H.Iwaiec, Equidistributio of roots of a quadratic cogruece to rime moduli, A. of Math. 141 (1995, o. 2, [4] T. Arad, Root of quadratic cogrueces, Iterat. Math. Res. Notices 14 (2000, Deartameto de Matemáticas, Uiversidad Autóoma de Madrid, Madrid, Sai address: fraciscojavier.cilleruelo@uam.es
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