THE LEAST COMMON MULTIPLE OF RANDOM SETS OF POSITIVE INTEGERS JAVIER CILLERUELO, JUANJO RUÉ, PAULIUS ŠARKA, AND ANA ZUMALACÁRREGUI Abstact. We study the typical behavio of the least coon ultiple of the eleents of a ando subset A {1,..., n}. Fo exaple we pove that lc{a : a A} = 2 n(1+o(1 fo alost all subsets A {1,..., n}. 1. Intoduction The function ψ(n = log lc { : 1 n} was intoduced by Chebyshev in his study on the distibution of the pie nubes. It is a well nown fact that the asyptotic elation ψ(n n is equivalent to the Pie Nube Theoe, which was poved independently by J. Hadaad and C.J. de la Vallée Poussin. In the pesent pape, instead of consideing the whole set {1,..., n}, we study the typical behavio of the quantity ψ(a := log lc{a : a A} fo a ando set A in {1,..., n} when n. We define ψ( = 0. We conside two natual odels. In the fist one, denoted by B(n, δ, each eleent in A is chosen independently at ando in {1,..., n} with pobability δ = δ(n, typically a function of n. Theoe 1.1. If δ = δ(n < 1 and δn then ψ(a n δ log(δ 1 asyptotically alost suely in B(n, δ when n. The case δ = 1 coesponds to the classical Chebyshev function and its asyptotic estiate appeas as the liiting case, as δ tends to 1, in Theoe 1.1, since li δ 1 δ log(δ 1 1 δ = 1. When δ = 1/2 all the subsets A {1,..., n} ae chosen with the sae pobability and Theoe 1.1 gives the following esult. Coollay 1.1. Fo alost all sets A {1,..., n} we have that lc{a : a A} = 2 n(1+o(1. Fo a given positive intege = (n, again typically a function of n, we conside the second odel, whee each subset of eleents is chosen unifoly at ando aong all sets of size in {1,..., n}. We denote this odel by S(n,. When δ = /n the heuistic suggests that both odels ae quite siila. Indeed, this is the stategy we follow to pove Theoe 1.2. 1
2 J. CILLERUELO, J. RUÉ, P. ŠARKA, AND A. ZUMALACÁRREGUI Theoe 1.2. Fo = (n < n and we have ψ(a = log(n/ 1 /n ( 1 + O(e C log alost suely in S(n, when n fo soe positive constant C. The case = n, which coesponds to Chebyshev s function, is also obtained as a liiting case in Theoe 1.2 in the sense that li /n 1 log(n/ 1 /n = 1. This wo has been otivated by a esult of the fist autho about the asyptotic behavio of ψ(a when A = A q,n := {q( : 1 q( n} fo a quadatic polynoial q(x Z[x]. We wondeed if that behavio was typical aong the sets A {1,..., n} of siila size. We analyze this issue in the last section. 2. Chebyshev s function fo ando sets in B(n, δ. Poof of Theoe 1.1 The following lea povides us with an explicit expession fo ψ(a in tes of the Mangoldt function log p if = p fo soe 1 Λ( = 0, othewise. Lea 2.1. Fo any set of positive integes A we have ψ(a = Λ(I A(, whee Λ denotes the classical Von Mangoldt function and { 1 if A {, 2, 3,... }, I A ( = 0 othewise. Poof. We obseve that fo any positive intege l, the nube log l can be witten as log l = p l log p, whee the su is taen ove all the powes of pies. Thus, using that p lc{a : a A} if and only if A {p, 2p, 3p,... }, we get log lc(a : a A = log p = (log pi A (p = Λ(I A (. p p lc(a: a A Note that if A = {1,..., n} then ψ(a = n Λ( is the classical Chebychev function ψ(n. 2.1. Expectation. Fist of all we give an explicit expession fo the expected value of the ando vaiable X = ψ(a whee A is a ando set in B(n, δ. Poposition 2.1. Fo the ando vaiable X = ψ(a in B(n, δ we have E (X = n δ log(δ 1 + δ R ( 1, 1 whee R(x = ψ(x x denotes the eo te in the Pie Nube Theoe.
THE LEAST COMMON MULTIPLE OF RANDOM SETS OF POSITIVE INTEGERS 3 Poof. The abiguous case δ = 1 ust be undestood as the liit as δ 1, which ecoves the equality ψ(n = n + R(n. In the following we assue that δ < 1. By lineaity of the expectation, Lea 2.1 clealy iplies E(X = n Λ(E(I A (. Since E(I A ( = P(A {, 2,... } = = 1 n/ P( A = 1 ( n/, we obtain (1 E(X = ( Λ( 1 ( n/. n n We obseve that n/ = wheneve +1 < n, so we split the su into intevals J = +1, n ], obtaining E (X = (1 ( Λ( J 1 = ( ( 1(1 ψ = δ ψ ( 1 1 = δn 1 = n δ log(δ 1 ( 1 Coollay 2.1. If δ = δ(n < 1 and δn then fo soe constant C > 0. E (X = n δ log(δ 1 ψ + 1 + δ R ( 1. 1 + δ R ( 1. 1 ( ( 1 + O e C log(δn. Poof. We estiate the absolute value of su appeaing in Poposition 2.1. Fo any positive intege T and using that R(y < 2y fo all y > 0 we have R (n/ ( 1 = R (n/ ( 1 + R (n/ ( 1 1 n 1T 1T ( n ax x n/t n log(δ 1 ( T +1 R (n/ ( 1 + 2n ( 1 (n/ T +1 R(x ( 1 + 2n ( 1 x 1T T +1 ( R(x ax + 2n ( T x n/t x T + 1 δ Taing into account that ( T < e δt and the nown estiate ax x>y R(x x e C1 log y
4 J. CILLERUELO, J. RUÉ, P. ŠARKA, AND A. ZUMALACÁRREGUI fo the eo te in the PNT, we have 1 R (n/ ( 1 n log(δ 1 ( e C1 log(n/t + n e δt δt. Thus we have poved that fo any positive intege T we have ( 1 + O (e C1 log(n/t + O E(X = n δ log(δ 1 We tae T δ 1 log(δn to iniize the eo te. ( e δt log(δ 1. δt To estiate the fist eo te we obseve that log(n/t log(δn/ log(δn log(δn, so e C1 log(n/t e C log(δn fo soe constant C. To bound the second eo te we siply obseve that δt > 1 and that 1 δ log(δ 1 1 and we get a siila uppe bound. 2.2. Vaiance. Poposition 2.2. Fo the ando vaiable X = ψ(a in B(n, δ we have V (X δn log 2 n. Poof. By lineaity of expectation we have that V (X = E ( X 2 E 2 (X = Λ(Λ(l (E (I A (I A (l E (I A ( E (I A (l.,ln We obseve that if Λ(Λ(l 0 then l, l o (, l = 1. Let us now study the te E(I A (I A (l in these cases. (i If l then (ii If (l, = 1 then E(I A (I A (l = 1 ( n/. E(I A (I A (l = 1 ( n/ ( n/l + ( n/ + n/l n/l. Both of these elations ae subsued in E(I A (I A (l = 1 ( n/ ( n/l + ( n/ + n/l n(,l/l. Theefoe, it follows fo (1 that fo each te in the su we have Λ(Λ(l (E (I A (I A (l E (I A ( E (I A (l = Λ(Λ(l( n/ + n/l n(,l/l ( 1 ( n(,l/l. Finally, by using the inequality 1 (1 x x we have and theefoe: Λ(Λ(l (E (I A (I A (l E (I A ( E (I A (l δn Λ(l l V (X 2δn 1ln Λ(l l Λ( (, l. Λ( (, l,
THE LEAST COMMON MULTIPLE OF RANDOM SETS OF POSITIVE INTEGERS 5 We now split the su accoding to l o (l, = 1 and estiate each one sepaately. 1ln l Λ(l l 1ln (l,=1 Λ( Λ(l l (, l pn 1i Λ( (, l 1ln log p p Λ(l l log p p i p = pn 1n 1i i log 2 p p i log 2 n, Λ( log 2 n, as we wanted to pove. We finish the poof of Theoe 1.1 by obseving that V (X = o(e(x 2 when δn, so X E(X asyptotically alost suely. 3. Chebyshev s function fo ando sets in S(n,. Poof of Theoe 1.2 Let us conside again the ando vaiable X = ψ(a, but in the odel S(n,. Fo now on E (X and V (X will denote the expected value and the vaiance of X in this pobability space. Clealy, fo s = 1, 2 we have E (X s = V (X = Lea 3.1. Fo s = 1, 2 and 1 < we have that 1 ψ s (A A = 1 (ψ(a E (X 2 A = E (X s E (X s E (X s + ( s s log s n. Poof. In ode to pove the lowe bound it is enough to conside the case = 1. Obseve that the function ψ is onotone with espect to inclusion, i.e. ψ (A {a} ψ(a fo any A, {a} [n]. Using this we get A = 1 ψ s (A 1 n + 1 a [n]\a Inequality then follows fo ( n 1 = n (n +1(. ψ s (A {a} = (n + 1 A = ψ s (A. and any patition into two sets Fo the second inequality we obseve that fo any set A ( [n] A = A A with A =, A = we have that ψ(a ψ(a +ψ(a ψ(a +( log n. Siilaly, ψ 2 (A (ψ(a + ( log n 2 = ψ 2 (A + 2ψ(A ( log n + ( 2 log 2 n ψ 2 (A + 2( log 2 n + ( 2 log 2 n = ψ 2 (A + ( 2 2 log 2 n.
6 J. CILLERUELO, J. RUÉ, P. ŠARKA, AND A. ZUMALACÁRREGUI Thus, fo s = 1, 2 we have Then, A = ψ s (A ψ s (A = = = and the second inequality holds. ( 1 (ψ s (A + ( s s log s n A A A = 1( ( ( 1 A A A = A = A A A = ψ s (A + ( s s log s n. ψ s (A + ( 1 ψ s (A 1 + A = A A A = ( 1 ( n ( n ψ s (A + A = A = ψ s (A + ( n ( s s log s n ( n ( s s log s n ( s s log s n, ( n ( s s log s n Poposition 3.1. Fo s = 1, 2 we have that E (X s = E(X s + O( s 1/2 log s n whee E(X s denotes the expectation of X s in B(n, /n and E (X s the expectation in S(n,. Poof. Obseve that fo s = 1, 2 we have E(X s E (X s = E (X s + = E (X s + = n fo s = 1, 2. Using Lea 3.1 we get ( n (2 E (X s E(X s log s n n n ( 1 n n ( ( 1 n ψ s (A n n A = ( ( 1 n n n E (X s n ( n (E (X s E (X s, ( ( 1 n ( n s s. The su in (2 fo s = 1 is E( Y E(Y, whee Y Bin(n, /n is the binoial distibution of paaetes n and /n. Cauchy Schwaz inequality fo the expectation iplies that this quantity is bounded by the standad deviation of the binoial distibution. n ( ( (3 1 n ( n n(/n(1 /n, which poves Poposition 3.1 fo s = 1.
THE LEAST COMMON MULTIPLE OF RANDOM SETS OF POSITIVE INTEGERS 7 To estiate the su in (2 fo s = 2, we split the expession in two tes: the su indexed by 2 and the one with > 2. We use (3 to get 2 ( ( 1 n ( n 2 2 3 n 3 3/2. ( ( 1 n ( n On the othe hand, >2 ( ( 1 n ( n 2 2 (l + 1 2 2 l 2 l<(l+1 (l + 1 2 2 P(Y > l l 2 ( ( 1 n ( n whee, once again, Y Bin(n, /n. Chenoff s Theoe iplies that fo any ɛ > 0 we have Applying this inequality to P(Y > l we get P(Y > (1 + ɛ e ɛ2 /3. >2 ( ( 1 n ( n 2 2 (l + 1 2 2 e (l 12/3 2 e /3 3/2. l 2 The next coollay poves the fist pat of Theoe 1.2. Coollay 3.1. If = (n < n and then E (X = log(n/ 1 /n ( 1 + O (e C log Poof. Poposition 3.1 fo s = 1 and Coollay 2.1 iply that E (X = log(n/ 1 /n ( ( ( 1 + O e C log + O 1/2 log n 1 /n. log(n/ It is clea that 1/2 (1 /n ( ( log n log(n/ = O 1/2 log = o e C log, when, which concludes the poof.
8 J. CILLERUELO, J. RUÉ, P. ŠARKA, AND A. ZUMALACÁRREGUI To conclude the poof of Theoe 1.2 we cobine Poposition 2.2 and Poposition 3.1 to estiate the vaiance V (X in S(n, : V (X = E (X 2 E 2 (X = V (X + ( E (X 2 E(X 2 + (E(X E (X (E(X + E (X ( log 2 n + 1/2 log n ( log n 3/2 log 2 n. The second assetion of Theoe 1.2 is a consequence of the estiate V (X = o ( E 2 (X when. 3.1. The case when is constant. The case when is constant and n is not elevant fo ou oiginal otivation but we give a bief analysis fo the sae of copleteness. In this case Fenández and Fenández [3] have poved that E (ψ(a = log n + C + o(1 whee C = + ( =2 ( 1 ζ ( ζ(. Actually, they conside the pobabilistic odel with independent choices in {1,..., n}, but when is fixed it does not ae big diffeences because the pobability of a epetition between the choices is tiny. It is easy to pove that with pobability 1 o(1 we have that ψ(a log n. To see this we obseve that a 1 a (a i, a 1 lc(a 1,..., a a 1 a n, i< so i=1 log a i i< log(a i, a ψ(a log n. Now, let us note that P(a i n/ log n fo soe i = 1,..., / log n and that P((a i, a log n d>log n P(d a i, d a d>log n 1 d < 2 2 log n. These obsevations iply that with pobability at least 1 +2( 2 log n = 1 2 log n we have that log n (1 O (log log n/ log n ψ(a log n. The analysis in the odel B(n, δ when δn c can be done using again Poposition 2.1. E (ψ(a = n δ log(δ 1 + δ <n/ log n R ( 1 + δ n/ log nn R ( 1 We use the estiate R(x x/ log x in the fist su and the estiate R(x x in the second one. We have E (ψ(a = c log n(1 + o(1 + O c ( 1 ( + O c 1 log log n < n log n ( c log δ = c log n + O + O (c log log n log log n = c log n(1 + o(1. n log n n Of couse in this odel we cannot expect concentation aound the expectation because fo exaple the pobability that A is the epty set tends to a positive constant, P(A = e c, and then P(ψ(A = 0 e c.
THE LEAST COMMON MULTIPLE OF RANDOM SETS OF POSITIVE INTEGERS 9 4. The least coon ultiple of the values of a polynoial Chebyshev s function could be also genealized to ψ q (n = log lc {q( : 1, 1 q( n} fo a given polynoial q(x Z[x] and it is natual to ty to obtain the asyptotic behavio fo ψ q (n. Soe pogess has been ade in this diection. While the Pie Nube Theoe is equivalent to the asyptotic ψ q (n n fo q(x = x, Paul Batean noticed that the Pie Nube Theoe fo aithetic pogessions could be exploited to obtain the asyptotic estiate when q(x = a 1 x + a 0 is a linea polynoial and poposed it as a poble [1] in the Aeican Matheatical Monthly: ψ q (n n a 1 φ( 1l (l,=1 whee = a 1 /(a 1, a 0. The fist autho [2] has extended this esult to quadatic polynoials. Fo a given ieducible quadatic polynoial q(x = a 2 x 2 + a 1 x + a 0 with a 2 > 0 the following asyptotic estiate holds: (4 ψ q (n = 1 2 (n/a 2 1/2 log (n/a 2 + B q (n/a 2 1/2 + o(n 1/2, whee the constant B q depends only on q. In the paticula case of q(x = x 2 + 1, he got ψ q (n = 1 2 n1/2 log n + B q n 1/2 + o(n 1/2 with B q = γ 1 log 2 2 1 l, ( 1 p 1 2 log p, p 1 p 2 whee γ is the Eule constant and the su is consideed ove( all odd pie nubes. It has been poved [4] that the eo te in (4 fo q(x = x 2 + 1 is O n 1/2 (log n 4/9+ɛ fo each ɛ > 0. When q(x is a educible polynoial the behavio is, howeve, diffeent. In this case it is nown (see Theoe 3 in [2] that: ψ q (n cn 1/2 whee c is an explicit constant depending only on q. Fo exaple fo q(x = x 2 1 the constant is c = 1. The asyptotic behavio of ψ q (n eains unnown fo ieducible polynoials of highe degee. Conectue 1 (Cilleuelo [2]. Let q(x be an ieducible polynoial of degee d 3. Then (5 ψ q (n (1 1/d (n/a d 1/d log (n/a d, whee a d > 0 is the coefficient of x d in q(x. Fo exaple, this conectue would iply ψ q (n 2 3 n1/3 log n fo q(x = x 3 + 2. We obseve that ψ q (n = ψ(a q,n whee A q,n := {q( : 1, 1 q( n}. It is natual to wonde whethe fo a given polynoial q(x the asyptotic E (X ψ q (n holds, when n, whee = A q,n and X = ψ(a fo a ando set A of eleents in {1,..., n}.
10 J. CILLERUELO, J. RUÉ, P. ŠARKA, AND A. ZUMALACÁRREGUI Howeve, conside fo exaple the polynoials q(x = x 2 1 and q(x = x 2 + 1. In both cases A q,n n but the asyptotic behavios of ψ q (n ae distinct: n when q(x = x 2 1 ψ q (n 1 2 n log n when q(x = x 2 + 1. So, what is the typical behavio of ψ(a when A n? Is it lie in the educible case o lie in the ieducible one? Maybe neithe of the epesent the typical behavio of a ando set. This question was the oiginal otivation of this wo. Theoe 1.2 with = A q,n = n/a2 + O(1 gives E (X = log(n/ 1 /n ( ( 1 + O e C log = 1 ( 2 (n/a 2 1/2 log(n/a 2 + o n 1/2. This shows that, when q(x is an ieducible quadatic polynoial, the asyptotic behavio of ψ q (n coincides with ψ(a, fo alost all sets of size A q,n. Theoe 1.2 also suppots Conectue 1 fo any q(x = a d x d + + a 0 ieducible polynoial of degee d 3. Nevetheless, thee ae soe diffeences in the second te. Fo exaple, if q(x = x 2 + 1, we have ψ q (n = 1 2 n1/2 log n + B q n 1/2 + o(n 1/2, fo B = 0.06627563... On the othe hand, Theoe 1.2 iplies that in coesponding odel S(n, with = A q,n = n 1 we have that ψ(a = 1 2 n1/2 log n + o(n 1/2 alost suely. In othe wods, when q(x is an ieducible quadatic polynoial, the asyptotic behavio of ψ q (n is the sae that ψ(a in the coesponding odel S(n,. But, the second te is not typical unless B q = 0. Pobably B q 0 fo any ieducible quadatic polynoial q(x but we have not found a poof. Acnowledgents: This wo was suppoted by gants MTM 2011-22851 of MICINN, ICMAT Seveo Ochoa poect SEV-2011-0087 and Reseach Council of Lithuania Gant No. MIP- 068/2013/LSS-110000-740. Pat of this wo was ade duing thid autho s visit at Univesidad Autónoa de Madid, he would lie to than the people of the Matheatics Depatent and especially Javie Cilleuelo fo thei wa hospitality. The authos ae also gateful to Suya Raana and Javie Cácao fo useful coents, and especially to Ben Geen whose suggestions ipoved the claity and exposition of the esults. Refeences [1] P. Batean, J. Kalb and A. Stenge, A liit involving Least Coon Multiples, Aeican Matheatical Monthly 10797, 109 (2002, no. 4, 393-394. [2] J. Cilleuelo, The least coon ultiple of a quadatic sequence, Copositio Matheatica 147 (2011 no. 4, 1129-1150. [3] J.L. Fenández and P. Fenández, On the pobability distibution of the gcd and lc of -tuples of integes. Pepint. [4] J. Rué, P. Šaa, A. Zualacáegui, On the eo te of the logaith of the lc of a quadatic sequence, Jounal de Théoie des Nobes de Bodeaux 25(2013, no. 2, p. 457-470.
THE LEAST COMMON MULTIPLE OF RANDOM SETS OF POSITIVE INTEGERS 11 J. Cilleuelo: Instituto de Ciencias Mateáticas (CSIC-UAM-UC3M-UCM and Depataento de Mateáticas, Univesidad Autónoa de Madid, 28049 Madid, Spain E-ail addess: fanciscoavie.cilleuelo@ua.es J. Rué: Institut fü Matheati, Feie Univesität Belin, Aniallee 3-5, D-14195 Belin, Geany E-ail addess: ue@zedat.fu-belin.de P. Šaa: Institute of Matheatics and Infoatics, Aadeios 4, Vilnius LT-08663, Lithuania and Depatent of Matheatics and Infoatics, Vilnius Univesity, Naugaduo 24, Vilnius LT-03225, Lithuania E-ail addess: paulius.saa@gail.co A. Zualacáegui: Instituto de Ciencias Mateáticas (CSIC-UAM-UC3M-UCM and Depataento de Mateáticas, Univesidad Autónoa de Madid, 28049 Madid, Spain E-ail addess: ana.zualacaegui@ua.es