Counting carefree couples

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1 Counting carefree coules arxiv:math/050003v2 [math.nt] 4 Jul 204 Pieter Moree Abstract A air of natural numbers a,b such that a is both squarefree an corime to b is calle a carefree coule. A result conjecture by Manfre Schroeer in his book Number theory in science an communication on carefree coules an a variant of it are establishe using stanar arguments from elementary analytic number theory. Also a relate conjecture of Schroeer on triles of integers that are airwise corime is rove. Introuction It is well known that the robability that an integer is squarefree is 6/π 2. Also the robability that two given integers are corime is 6/π 2. More generally the robability that n ositive integers chosen arbitrarily an ineenently are corime is well-known [7, 22, 27] to be /ζn, where ζ is Riemann s zeta function. For some generalizations see e.g. [3, 4, 2, 23, 25]. One can woner how statistically ineenent squarefreeness an corimality are. To this en one coul for examle consier the robability that of two ranom natural numbers a an b, a is both squarefree an corime to b. Let us call such a coule a,b carefree. If b is also squarefree, we say that a,b is a strongly carefree coule. Let us enote by C x the number of carefree coules a,b with both a x an b x an, similarly, let C 2 x enote the number of strongly carefree coules a,b with both a x an b x. The urose of this note is to establish the following result, art of which was conjecture, on the basis of heuristic arguments, by Manfre Schroeer [26,. 54]. In it an in the rest of the aer the mathematical symbol is exclusively use to enote rimes. Theorem We have C x = x2 ζ2 +Oxlogx, + an C 2 x = x2 ζ2 2 +Ox 3/

2 The interretation of Theorem is that the robability for a coule to be carefree is K := ζ2 + an to be strongly carefree is K 2 := ζ Using the ientity ζn = n vali for n > we can alternatively write K 2 = 2 = ζ2 + For m 3 an 0 k m we ut Z k m = + k m k m. 6 Note that Z 2 3 = K an Z 3 3 = K 2. The constants K an K 2 we coul call the carefree, resectively strongly carefree constant, cf. [0, Section 2.5]. Assuming ineenence of squarefreeness an corimality we woul exect that K = ζ2 2 an K 2 = ζ2 3. Now note that K = ζ2 2 +, K = + ζ We have ζ2 2 K.5876 an ζ2 3 K Thus, there is a ositive correlation between squarefreeness an corimality. Let I 3 x enote the number of triles a,b,c with a x, b x,c x such that a,b = a,c = b,c =. Schroeer [26, Section 4.4] claims that I 3 x K 2 x 3. Inee, in Section 2.2 we will rove the following result. Theorem 2 We have I 3 x = K 2 x 3 +Ox 2 log 2 x. The work escribe in this note was carrie out in 2000 an with some imrovement in the error terms was oste on the arxiv in Setember of 2005 [2], with the remark that it was not intene for ublication in a research journal as the methos use involve only rather elementary an stanar analytic number theory. Over the years various authors referre to [2], an this inuce me to try to ublish it in a mathematical newsletter. For ublications in this area after 2005 see, e.g, [, 6, 7, 8, 9, 4, 5, 6, 30, 3]. In [2] there was a mistake in the roof of 2 leaing to an error term of Oxlog 3 x, rather than Ox 3/2. Excet for this, the resent version has essentially the same mathematical content as the earlier one, but is written in a less carefree way an with the mathematical etails more selle out. 2

3 2 Proofs As usual we let µ enote the Möbius function an ϕ Euler s totient function. Note that n is squarefree if an only if µn 2 =. We will reeately make use of the basic ientities { if n = ; µ = 7 0 otherwise, an n ϕn n = n µ = n. 8 Wewill also use several times that if s is a comlex number anf a multilicative function such that ν fν νs <, then n= fn n s = ν f ν νs. 9 For a roof see, e.g., Tenenbaum [28,. 07]. In the roof of Theorem we will make use of the following lemma. Lemma Let be arbitrary. Put We have S x = S x = n x,n= µn 2. x ζ2 + +O2ω x, 0 where ω enotes the number of istinct rime ivisors of. Proof. Let T x enote the number of natural numbers n x that are corime to. Using 7 an 8 an [x] = x+o we euce that T x = n x n,= = n x µα = α α n α By the rincile of inclusion an exclusion we fin that µα[ x α ] = ϕ x+o2ω. Hence, on invoking, we fin S x = µmt x m 2. m x,m= S x = x ϕ m x,m= µm m 2 +O2 ω x. 3

4 an hence, on comleting the sum, S x = x ϕ m=,m= µm m 2 +O2 ω x Note that m=,m= Using this an 8 the roof is comlete. µm = m 2 2 = ζ2 /2. Let n enote the number of ivisors of n. We have 2 ωn n with equality iff n is squarefree. The estimates below also hol with 2 ωn relace by n. Lemma 2 We have x 2 ω = O, 3/2 x 2 ω = O xlogx, x 4 ω = Olog 3 x. Proof. Using the convergence of 3/2 we fin by 9 that = 2ω 3/2 = O. The remaining estimates follow on invoking Theorem at. 20 of Tenenbaum s book [28] together with artial integration. 2. Proof of Theorem Note that C x = a x µa 2 b x a, bµ = xµ a x a µa 2 b x/ after swaing the summation orer. Using [x/] = x/+o, we then obtain C x = x x µ µa 2 +Oxlogx. a x a, On noting that µa 2 = µ 2 a x a n x/,n= µn 2 = µ 2 S x 2 an µ = µ 3, we fin C x = x x On using Lemma we obtain the estimate C x = x2 ζ2 x µ µ S x +Oxlogx. 2 +/ +O x x 4 2 ω +Oxlogx.

5 On comleting the latter sum an noting that we obtain = C x = x2 ζ2 µ 2 +/ =, + +O x 2 ω +Oxlogx. + x Estimate now follows on invoking Lemma 2. The roof of 2 is very similar to the roof of. We start by noting that C 2 x = µa 2 µb 2 µ. a x b x On swaing the summation orer, we obtain C 2 x = x µ a x a a, b µa 2 b x b On noting that µ = µ 5 an invoking 2 we obtain µb 2. 3 C 2 x = xµs x 2. 4 On using Lemma we obtain the estimate C 2 x = x2 µ ζ Ox3/2 x +/2 x 2 ω +Ox 4 ω 3/2. x On comleting the first sum an noting that µ 2 = +/2 =, + 2 we fin C 2 x = x2 ζ2 2 +Ox 3/2 2 ω + 2 +Ox 4 ω 3/2. x x On invoking Lemma 2 estimate 2 is then establishe. 2.2 Proof of Theorem 2 We write [n,m] for the least common multile of n an m, an n,m for the greatest common ivisor. Recall that n, m[n, m] = nm. Note that I 3 x = a,b,c x a b µ 2 a 2 c 5 µ 2 3 b 3 c µ 3,

6 which can be rewritten as I 3 x = Now ut J x = [, 2 ] x [, 3 ] x [ 2, 3 ] x [, 2 ] x [, 3 ] x [ 2, 3 ] x x µ µ 2 µ 3 [ [, 2 ] ][ x [, 3 ] ][ x [ 2, 3 ] ]. µ µ 2 µ 3 [, 2 ][, 3 ][ 2, 3 ], J 2x = J 3 x = [, 2 ] x [, 3 ] x [ 2, 3 ] x Using that [x] = x+o we fin that We will show first that J x = [, 2 ] x [, 3 ] x [ 2, 3 ] x [, 2 ] an J 4x = [, 2 ] x [, 3 ] x [ 2, 3 ] x [, 2 ][, 3 ], I 3 x = x 3 J x+ox 2 J 2 x+oxj 3 x+oj 4 x. 5 = 2 = 3 = [, 2 ]>x 3. µ µ 2 µ 3 logx [, 2 ][, 3 ][ 2, 3 ] +O. x To this en it is enough, by symmetry of the argument of the sum, to show that. 6 logx [, 2 ][, 3 ][ 2, 3 ] = O x Put, 2 = α,, 3 = β an 2, 3 = γ. Since α an β, we can write = [α,β]δ for some integer δ, an similarly 2 = [α,γ]δ 2, 3 = [β,γ]δ 3. Note that any trile, 2, 3 corresons to a uniqe 6-tule,δ,δ 2,δ 3. Since αδ,δ 2 ivies [α,β]δ,[α,γ]δ 2 on the one han an [α,β]δ,[α,γ]δ 2 =, 2 = α on the other, it follows that δ,δ 2 = an likewise δ,δ 3 = δ 2,δ 3 =. Write u = αβγ/ 2. On noting that, 2, 2, 3 =, 2, 3 =, 2,, 3, 2, 3 we infer that α,β = α,γ = β,γ = an hence we fin that [, 2 ] = uδ δ 2, [, 3 ] = uδ δ 3 an [ 2, 3 ] = uδ 2 δ 3. Now [, 2 ]>x 3 [, 2 ][, 3 ][ 2, 3 ] u 3 δ δ 2 >x/uδ 3 δ δ 2 δ 3 2, where the trile sum is over all 6-tules,δ,δ 2,δ 3 an is of orer O u 3 δ δ 2 >x/u n logx = O = O δ δ 2 2 u 3 n 2 x n>x/u where we use the well-known estimate n>x nn 2 = Ologx/x. Now, u 2 u = 4 = O = O, 7 2 αβγ 2 2 α β γ 2 = α,β,γ 6

7 where we have written =, α = α, β = β an γ = γ. Thus we have establishe equation 6. In the same vein J 2 x can be estimate to be J 2 x = O = O δ δ 2 x/u δ δ 3 x/u δ 2 δ 3 x/u u 2 = O [, 2 ][, 3 ] δ 2 δ 3 x/u 3/2 δ 2 δ 3 = O u 2 u 2 δ δ 2 δ 3 x/u 3/2 n n n x/u 3/2 δ 2 δ 2δ 3 Using the classical estimate n x n/n = Olog2 x an 7, one obtains J 2 x = Olog 2 x. Note that 0 J 4 x xj 3 x x 2 J 2 x. Using 5 we see that it remains to evaluate the trile infinite sum, which we rewrite as = 2 = 3 = which can be rewritten as µ 2 = 2 = µ µ 2 µ 3, 2, 3 2, , µ 2, = µ 3, 3 2, Note that the argument of the inner sum is multilicative in 3. By Euler s rouct ientity 9 it is zero if, 2 > an ζ2 2 +/ otherwise. Thus the latter trile sum is seen to yiel ζ2 = µ 2 +/ 2 =, 2 = 2 2 µ 2 2 +/, the argument of the inner sum is multilicative in 2 an roceeing as before we obtain that it equals ζ2 + = 2 µ +, + which is seen to equal ζ2 which by equation 5 equals K 2. 2, + 3 Numerical asects Direct evaluation of the constants K an K 2 through 3, resectively 4 yiels only about five ecimal igits of recision. By exressing K an K 2 as infinite roucts involving ζk for k 2, they can be comute with high recision. To 7

8 this en Theorem of [20] can be use. The error analysis can be ealt with using Theorem 2 of [20]. Using [20, Theorem ] it is inferre that K = k 2ζk e k k, where e k = b µ k Z, k with the sequence {b k } k=0 efine by b 0 = 2 an b = an b k+2 = b k+ +b k. Using the same theorem, it is seen that K 2 = {ζk 2 k } f k k, where f k = 2 µ k Z. 2 k k 2 Tyically in analytic number theory constants of the form f/ with f rational arise as ensities. Their numerical evaluation was consiere by the author in [20]. By similar methos any constant of the form f/ with f an analytic function on the unit isc satisfying f0 = an f 0 = 0 can be evaluate [9]. 4 Relate roblems Let us call a coule a,b with a, b x, a an b corime an either a or b squarefree, weakly carefree. A little thought reveals that C 3 x = 2C x C 2 x. By Theorem it then follows that the robability K 3 that a coule is weakly carefree equals K 3 = 2K K The roblem of estimating I 3 x has the following natural generalisation. Let k 2 be an integer an let I k x be the number of k-tules a,...,a k with a i x for i k such that a i,a j = for every i j k. The number of k-tules such that none of the gc s is ivisible by some fixe rime is easily seen to be x k Thus, it seems lausible that I k x x k k k + k = x k k k + k + k., x. 8 For k = 2 an k = 3 by Theorem 2 an equation 5 this is true. In 2000 I i not see how to rove this for arbitrary k, however the conjecture 8 was establishe soon afterwars in 2002 by L. Tóth [29], who rove that for k 2 we have I k x = x k k + k +Ox k log k x. 9 Let I u k x enote the number of k-tules a,...,a k with a i x that are airwise corime an moreover satisfy a i,u = for i k. It is easy to 8

9 see that I u k+ n = n j= j,u= I ju k n. Note that I u n = T u n can be estimate by. Then by recursion with resect to k an estimate for I u k n can be establishe that imlies 9. In [3] Havas an Majewski consiere the roblem of counting the number of n-tules of natural numbers that are airwise not corime. They suggeste that the ensity δ n of these tules shoul be δ n = n ζ2 The robability that a air of integers is not corime is /ζ2. Since there are n airs of integers in an n-tule, one might naively esect the robability 2 for this roblem to be as given by 20. T. Freiberg [] stuie this roblem for n = 3 using my aroach to estimate I 3 x it seems that the recursion metho of Tóth cannot be alie here. Freiberg showe that the ensity of triles a,b,c with a,b >, a,c > an b,c > equals F 3 = 3 ζ2 +3K K , whereas /ζ Thus the guess of Havas an Majewski for n = 3 is false. Inee, it is easy to see as Peter Pleasants ointe out to the author [24] that for every n 3 their guess is false. Since all n-tules of even numbers are airwise not corime, δ n, if it exists, satisfies δ n 2 n. Since n n 2 an /ζ2 < 0.4 the reicte ensity by Havas an Majewski [3] satisfies δ n < 2 n for n 3 an so must be false. In 2006 the author learne [8] that the result of Freiberg is imlicit in the PhD thesis of R.N. Buttsworth [2] an inee can be foun there in more general form. Buttsworth showe that the ensity of relatively rime m-tules for which k rescribe m -tules have gc equals Z k m given in 6. Consequently by inclusion an exclusion the set of relatively rime m-tules such that every m -tule fails to be relatively rime has ensity m m k Z k m. k k=0 Form = 3thisyiels/ζ3 3/ζ2+3K K 2. Sotheensityofrelativelyrime 3-tules such that at least one 2-tule is relatively rime, is equal to 3/ζ2 3K + K 2. However this is also equal to the ensity of 3-tules such that at least one 2-tule is relatively rime. Hence the ensity of 3-tules such that all 2-tules are not relatively rime is 3/ζ2 + 3K K 2, which is Freiberg s formula. To close this iscussion, we like to remark that Freiberg establishe his result with error term Ox 2 log 2 x an that Buttsworth s result gives only a ensity. Some relate oen roblems are as follows: 9

10 Problem a To comute the ensity of n-tules such that at least k airs are corime. b To comute the ensity of n-tules such that exactly k airs are corime. Problem 2 To comute the ensity of n-tules such that all airs are not corime. Remark. Recently Jerry Hu [6] announce that he solve Problem. 5 Conclusion In stark constrast to what exerience from aily life suggests, strongly carefree coules are quite common... Acknowlegement. The author likes to thank Steven Finch for bringing Schroeer s conjecture to his attention an his instignation to write own these results. Also Finch aneweger ointe out thatone has n x kn = ζ2k x 2 /2+Ox 3/2, where kn = n, an that in [2] the K was inavertently roe. For a roof of this formula see Eckfor Cohen [5, Theorem 5.2]. The authorlikes tothank TristanFreiberg anjerry Huforointing out some references an helful comments. Keith Matthews rovie me kinly with very helful information concerning the relevant results of his former PhD stuent Buttsworth. In articular he ointe out how Freiberg s result follows from that of Buttsworth. References [] J. Arias e Reyna an R. Heyman, Counting tules restrice by corimality conitions, arxiv: [2] R.N. Buttsworth, A general theory of inclusion-exclusion with alications to the least rimitive root roblem an other ensity questions, PhD thesis, University of Queenslan, 983. [3] J. Chiambaraswamy an R. Sitaramachanra Rao, On the robability that the values of m olynomials have a given g.c.., J. Number Theory 26987, [4] E. Cohen, Arithmetical functions associate with arbitrary sets of integers, Acta Arith , [5] E. Cohen, Arithmetical functions associate with the unitary ivisors of an integer, Math. Z , [6] J.L. Fernánez an P. Fernánez, Asymtotic normality an greatest common ivisors, arxiv: [7] J.L. Fernánez an P. Fernánez, On the robability istribution of the gc an lcm of r-tules of integers, arxiv: [8] J.L. Fernánez an P. Fernánez, Equiistribution an corimality, arxiv:

11 [9] J.L. Fernánez an P. Fernánez, Ranom inex of coivisibility, arxiv: [0] S.R. Finch, Mathematical constants, Encycloeia of Mathematics an its Alications 94, Cambrige University Press, Cambrige, [] T. Freiberg, The robability that 3 ositive integers are airwise norime, unublishe manuscrit, [2] J.L. Hafner, P. Sarnak an K. McCurley, Relatively rime values of olynomials. In: A tribute to Emil Grosswal: number theory an relate analysis, , Contem. Math. 43, Amer. Math. Soc., Provience, RI, 993. [3] G. Havas an B.S. Majewski, A har roblem that is almost always easy. Algorithms an comutations Cairns, 995, Lecture Notes in Comut. Science , [4] R. Heyman, Pairwise non-corimality of triles, arxiv: [5] J. Hu, The robability that ranom ositive integers are k-wise relatively rime, Int. J. Number Theory 9 203, [6] J. Hu, Pairwise relative rimality of ositive integers, arxiv: [7] D.N. Lehmer, An asymtotic evaluation of certain totient sums, Amer. J. Math , [8] K. Matthews, to author, July 6, [9] M. Mazur, B.V. Petrenko, Reresentations of analytic functions as infinite roucts an their alication to numerical comutations, htt://arxiv.org/abs/ [20] P. Moree, Aroximation of singular series an automata, Manuscrita Mathematica , [2] P. Moree, Counting carefree coules, arxiv contribution 2005, htt://front.math.ucavis.eu/ [22] J.E. Nymann, On the robability that k ositive integers are relatively rime, J. Number Theory 4 972, [23] J.E. Nymann, On the robability that k ositive integers are relatively rime. II, J. Number Theory 7 975, [24] P.A.B. Pleasants, to author, May 7, [25] S. Porubský, On the robability that k generalize integers are relatively H-rime, Colloq. Math , [26] M.R. Schroeer, Number theory in science an communication, 2n eition, Sringer Series in Information Sciences 7, Sringer-Verlag, Berlin-New York, 986. [27] J.J. Sylvester, The collecte aers of James Joseh Sylvester, vol. III, Cambrige Univ. Press, Lonon, New York, 909. [28] G. Tenenbaum, Introuction to analytic an robabilistic number theory. Cambrige Stuies in Avance Mathematics, 46. Cambrige University Press, Cambrige, 995.

12 [29] L. Tóth, The robability that k ositive integers are airwise relatively rime, Fibonacci Quart , 3 8. [30] L.Tóth, Onthenumberofcertainrelatively rimesubsetsof{,2,...,n}, Integers 0 200, A35, [3] L. Tóth, Multilicative arithmetic functions of several variables: a survey, arxiv: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53 Bonn, Germany. moree@mim-bonn.mg.e 2