On Erdős and Sárközy s sequences with Property P

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1 Monatsh Math : DOI /s On Erdős and Sárközy s sequences with Proerty P Christian Elsholtz 1 Stefan Planitzer 1 Received: 7 November 015 / Acceted: 7 October 016 / Published online: 18 October 016 The Authors 016. This article is ublished with oen access at Sringerlink.com Abstract A sequence A of ositive integers having the roerty that no element a i A divides the sum a j + a k of two larger elements is said to have Proerty P. We construct an infinite set S N having Proerty P with counting function S log log log. This imroves on an eamle given by Erdős and log log log Sárközy with a lower bound on the counting function of order log. Keywords Sequences with Proerty P Sums of two squares Primes in arithmetic rogressions Distribution of integers with given rime factorization Mathematics Subject Classification 11B83 11N13 1 Introduction Erdős and Sárközy [9] define a monotonically increasing sequence A ={a 1 < a <...} of ositive integers to have Proerty P if a i a j +a k for i < j k. They roved Communicated by J. Schoißengeier. The authors are suorted by the Austrian Science Fund FWF: W130, Doctoral Program Discrete Mathematics. B Stefan Planitzer lanitzer@math.tugraz.at Christian Elsholtz elsholtz@math.tugraz.at 1 Institute of Analysis and Number Theory, Graz University of Technology, Koernikusgasse 4/II, 8010 Graz, Austria 13

2 566 C. Elsholtz, S. Planitzer that any infinite sequence of integers with Proerty P has density 0. Schoen [15] showed that if an infinite sequence A has Proerty P and any two elements in A are corime then the counting function A = a i < 1 is bounded from above by A < 3 and Baier [1] imroved this to A <3+ɛ 3 log 1 for any ɛ>0. Concerning finite sequences with Proerty P, Erdős and Sárközy [9] get the lower bound ma A 3 +1 by just taking A to be the set A ={, 1,..., 3 } for N. Erdős and Sárközy also thought about large sets with Proerty P with resect to the size of the counting function cf. [9,. 98]. They observed that the set A ={qi : q i the i-th rime with q i 3mod4} has Proerty P. This uses the fact that the square of a rime 3 mod 4 has only the trivial reresentation = + 0 as the sum of two squares. With this set A they get A log. Erdős has asked reeatedly to imrove this see e.g. [6,. 185], [7,. 535] and in articular, Erdős [7,8] asked if one can do better than a n n log n. He wanted to know if it is ossible to have a n < n. We will not quite achieve this but we go a considerable ste in this direction. First, we observe that a set of squares of integers consisting of recisely k rime factors 3 mod 4 also has Proerty P. As for any fied k this would only lead to a moderate imrovement, our net idea is to try to choose k increasing with. In order to do so, we actually use a union of several sets S i with Proerty P. Together, this union will have a good counting function throughout all ranges of. However, in order to ensure that this union of sets with Proerty P still has Proerty P, we emloy a third idea, namely to equi all members a S i with a secial indicator factor. This seems to be the first imrovement going well beyond the eamle given by Erdős and Sárközy since Our main result will be the following theorem. Theorem The set S N constructed elicitly below has Proerty P and counting function S log log log log log log. We achieve this imrovement by not only considering squares of rimes 3mod4 but roducts of squares of such rimes. More formally we set Here the sets S i are defined by 13 S i := S = S i. 1 i=1 { n N : n = q 4 i ν},

3 On Erdős and Sárközy s sequences with Proerty P 567 where ν is the roduct of eactly i distinct rimes 3 mod 4 and we recall that q i is the i-th rime in the residue class 3 mod 4. The rôle of the q i is an indicator which uniquely identifies the set S i a given integer n S belongs to. Results from robabilistic number theory like the Theorem of Erdős-Kac suggest that for varying different sets S i will yield the main contribution to the counting function S. In articular for given > 0 the main contribution comes from the sets S i with log log log log i log log + log log. The study of sequences with Proerty P is closely related to the study of rimitive sequences, i.e. sequences where no element divides any other and there is a rich literature on this toic cf. [10, Chater V]. Indeed a similar idea as the one described above was used by Martin and Pomerance [13] to construct a large rimitive set. While Besicovitch [3] roved that there eist infinite rimitive sequences with ositive uer density, Erdős [4] showed that the lower density of these sequences is always 0. In his roof Erdős used the fact that for a rimitive sequence of ositive integers the sum 1 i=1 a i log a i converges. In more recent work Banks and Martin [] make some rogress towards a conjecture of Erdős which states that in the case of a rimitive sequence i=1 1 a i log a i 1 log holds. Erdős [5] studied a variant of the Proerty P roblem, also in its multilicative form. Notation Before we go into details concerning the roof of the Theorem we need to fi some notation. Throughout this aer P denotes the set of rimes and the letter with or without inde will always denote a rime number. We write log k for the k-fold iterated logarithm. The functions ω and count, as usual, the rime divisors of a ositive integer n without resectively with multilicity. For two functions f, g : R R + the binary relation f g and analogously f g denotes that there eists a constant c > 0 such that for sufficiently large f cg f cg resectively. Deendence of the imlied constant on certain arameters is indicated by subscrits. The same convention is used for the Landau symbol O where f = Og is equivalent to f g. We write f = og if lim f g = 0. 3 The set S has Proerty P In this section we verify that any union of sets S i defined in has Proerty P. 13

4 568 C. Elsholtz, S. Planitzer Lemma 1 Let n 1, n and n 3 be ositive integers. If there eists a rime 3mod4 with n 1 and gcdn, n 3, then n 1 n + n 3. Proof We rove the Lemma by contradiction. Suose that n 1 n +n 3. By our assumtion there eists a rime 3 mod 4 such that n 1 and gcdn, n 3. Hence, w.l.o.g. n.wehave n + n 3 0mod and since does not divide n, we get that n is invertible mod. Hence n3 1mod n a contradiction since 1 is a quadratic non-residue mod. Lemma Any union of sets S i defined in has Proerty P. Proof Suose by contradiction that there eist a i S i, a j S j and a k S k with a i < a j a k and a i a j + a k. First suose that either S i = S j or S i = S k. Define l {0, } to be the largest eonent such that q l i gcda i, a j, a k where we again recall that q i was defined as the i-th rime in the residue class 3 mod 4. Then a i qi l a j q l i + a k qi l. By construction of the sets S i, S j and S k we have that q i a i q l i and w.l.o.g. q i a j.an qi l alication of Lemma 1 finishes this case. If S i = S j = S k then a i = a j = a k. If there is some rime with a i qi 4 we may again use Lemma 1. If no such eists, then a i a j and and a j q 4 i or a k q 4 i a i a k trivially holds. With the restriction on the number of rime factors we get that a i = a j = a k. 4 Products of k distinct rimes In order to establish a lower bound for the counting functions of the sets S i in we need to count square-free integers containing eactly k distinct rime factors 3 mod 4, but no others, where k N is fied. For k and π k := #{n : ωn = n = k} Landau [11] roved the following asymtotic formula: 13 π k log k 1 k 1! log.

5 On Erdős and Sárközy s sequences with Proerty P 569 We will need a lower bound of similar asymtotic growth as the formula above for the quantity π k ; 4, 3 := #{n : n 3mod4,ωn = n = k}. Very recently Meng [14] used tools from analytic number theory to rove a generalization of this result to square-free integers having k rime factors in rescribed residue classes. The following is contained as a secial case in [14, Lemma 9]: Lemma A For any A > 0, uniformly for k A log log, we have log log k 1 log k 1! π k ; 4, 3 = 1 k 1+ k 1 k 1k C3, 4+ log log log log h k 3 +O A 3loglog k log log 3, where C3, 4 = γ + log λ, γ is the Euler-Mascheroni constant, λ is the indicator function of rimes in the residue class 3mod4and 1 h = Ɣ / 1 + λ. We will show that Lemma A with some etra work imlies the following Corollary. log log log log Corollary 1 Uniformly for 1 k + log log we have π k ; 4, 3 1 k log log k 1. k 1! log log Proof In view of Lemma A and with k we see that it suffices to check that, indeendent of the choice of k and for sufficiently large, there eists a constant c > 0 such that C3, k 3 h c. 3 3loglog Note that the left hand side of the above inequality is eactly the coefficient of the main term 1 log k 1 k log k 1! for k in the range given in the Corollary. The constant C3, 4 does not deend on k. Using Mertens Formula cf. [16,. 19: Theorem 1.1] in the form log 1 1 = γ log log + o1 13

6 570 C. Elsholtz, S. Planitzer we get C3, 4 = γ + log λ = M3, 4, where M3, 4 is the constant aearing in λ = log log 1 + M3, 4 + O, log which was studied by Languasco and Zaccagnini in [1]. 1 The comutational results of Languasco and Zaccagnini imly that < M3, 4 < and hence allow for the following lower bound for C3, 4: C3, 4 = M3, 4 > It remains to get a lower bound for h k 3 3loglog, where the function h is defined as in Lemma A. A straight forward calculation yields that h = 1 1 / 1 + λ Ɣ log 1 + λ +λ 1 Ɣ + 1 Ɣ + 1 and h = f 1 1 / 1 + λ, where f = 1 log λ +λ Ɣ + 1 Ɣ log 1 1 Ɣ + 1 Ɣ + 1 4Ɣ λ +λ + Ɣ + 1 Ɣ λ +λ Ɣ Note that our constant M3, 4 corresonds to the constant M4, 3 in the work of Languasco and Zaccagnini. 13

7 On Erdős and Sárközy s sequences with Proerty P 571 log log log log Note that for and 1 k + log log the term k 3 3loglog gets arbitrarily close to Hence we may suose that 300 k 3 3loglog and it suffices to find a lower bound for h where For in this range Mathematica rovides the following bounds on the Gamma function and its derivatives Ɣ , Ɣ , Ɣ Furthermore we have Later we will use that λ + < λ < 10 4 d < =10 4 = log λ = 1 + > log 1 log λ + n> λ λ = γ + M3, 4 λ 1 n λ + λ > 0.905, and 1 log λ < 1 + log λ = γ + M3, 4 < Finally, using log1 +, we get / 1 + λ = e γ + M3, 4 < e e 1 log < λ 13

8 57 C. Elsholtz, S. Planitzer Alying the elicit bounds calculated above, for we obtain: f This imlies for sufficiently large : > k 3 h > loglog Together with 4 this leads to an admissible choice of c = 0.80 in 3. 5 The counting function S Proof of Theorem As in 1weset S = i=1 where the sets S i are defined as in. The set S has Proerty P by Lemma and it remains to work out a lower bound for the size of the counting function S. For sufficiently large there eists a uniquely determined integer k N such that e ek < e ek+1 hence k log S i < k It deends on the size of, which S i makes the largest contribution. For a given log we take several sets S k+, S k+3,...,s k+l, l =, as the number of rime factors 3 mod 4 of a tyical integer less than isin [ log log, log ] log +. Using Corollary 1 as well as the fact that the i-th rime in the residue class 3 mod 4 is asymtotically of size i log i for given j l we get 16k+ j 4 log 4 k+ j S k+ j log 16k+ j 4 log 4 k+ j }{{} F 1 13 log k+ j 1 16k+ j 4 log 4 k+ j } k+ j k + j 1! {{ } F. 6

9 On Erdős and Sárközy s sequences with Proerty P 573 We deal with the fractions F 1 and F on the right hand side of 6 searately. With the given range of j and 5 we have that F 1 log log log 3. It remains to deal with F. Using the given range of k and j we have that k + j log and, again for sufficiently large, for the numerator of F we get k+ j 1 log 16k + j 4 log 4 k + j loglog log 4 log 3 log4 k+ j 1 loglog 5log 3 k+ j 1 = log + log 1 5log 3 k+ j 1 log 10 log log 3 k+ j 1 log 1 10 log 3 log log k+ j 1 log. log + log 1 k+ j 1 log Here we used that log log lim 1 10 log 3 log log + 1 = 1 and that for 0 y 1 we certainly have that log1 y y. To deal with the denominator of F we aly Stirling s Formula and get k + j 1 k+ j 1 k + j 1! k + j 1 e k+ j 1 log + j 1 log e log + j 1 k+ j 1 log k+ j 1 e log + j log + j 1 k+ j 1 log k+ j 1 e j log. 13

10 574 C. Elsholtz, S. Planitzer Altogether we get k+ j 1 log F log e j log log + j 1 log log log e j log log + j 1 + j 1. 7 Since log log + j 1 log 1 e j 1 it suffices to check that for any > 0 and for our choices of j there eists a fied constant c > 0 such that 1 + j 1 1 j c. 8 log 1 j j 1 For j we have that 1 + log is monotonically decreasing in j and get log j 1 1 j log log log = log log 1 e. Therefore for j the constant c in 8 may be chosen as c = 1 e large. Together with 7 this imlies for sufficiently F log log. Altogether for the counting function of any of the sets S i with log + i log log + we have S i log log 5 log 3. Summing these contributions u we finally get 13 S log log log 3.

11 On Erdős and Sárközy s sequences with Proerty P 575 Acknowledgements Oen access funding rovided by Graz University of Technology. Parts of this research work were done when the first author was visiting the FIM at ETH Zürich, and the second author was visiting the Institut Élie Cartan de Lorraine of the University of Lorraine. The authors thank these institutions for their hositality. The authors are also grateful to the referee for suggestions on the manuscrit and would like to thank Xianchang Meng for some discussion on his recent aer [14]. Oen Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License htt://creativecommons.org/licenses/by/4.0/, which ermits unrestricted use, distribution, and reroduction in any medium, rovided you give aroriate credit to the original authors and the source, rovide a link to the Creative Commons license, and indicate if changes were made. References 1. Baier, S.: A note on P-sets. Integers 4A13, Banks, W. D., Martin, G.: Otimal rimitive sets with restricted rimes. Integers 13, A69, Besicovitch, A.S.: On the density of certain sequences of integers. Math. Ann. 1101, Erdős, P.: Note on Sequences of integers no one of which is divisible by any other. J. London Math. Soc. 101, Erdős, P.: On sequences of integers no one of which divides the roduct of two others and on some related roblems. Mitt. Forsch.-Inst. Math. Mech. Univ. Tomsk, Erdős, P.: Some old and new roblems on additive and combinatorial number theory. In: Combinatorial Mathematics: Proceedings of the Third International Conference New York, 1985, Ann. New York Acad. Sci., vol. 555, New York Acad. Sci., New York Erdős, P.: Some of my favourite unsolved roblems. Math. Jaon. 463, Erdős, P.: Some of my new and almost new roblems and results in combinatorial number theory. In: Number theory Eger, 1996, de Gruyter, Berlin Erdős, P., Sárközi, A.: On the divisibility roerties of sequences of integers. Proc. London Math. Soc. 1, Halberstam, H., Roth, K.F.: Sequences, nd edn. Sringer-Verlag, New York-Berlin Landau, E.: Sur quelques roblèmes relatifs à la distribution des nombres remiers. Bull. Soc. Math. France 8, Languasco, A., Zaccagnini, A.: Comuting the Mertens and Meissel-Mertens constants for sums over arithmetic rogressions. Eeriment. Math. 193, With an aendi by Karl K. Norton, comutational results available online: htt:// Mqa/Msumfinalresults.df URL last checked: Martin, G., Pomerance, C.: Primitive sets with large counting functions. Publ. Math. Debrecen 793 4, Meng, X.: Large bias for integers with rime factors in arithmetic rogressions. ArXiv e-rints, available at Schoen, T.: On a roblem of Erdős and Sárközy. J. Combin. Theory Ser. A 941, Tenenbaum, G.: Introduction to analytic and robabilistic number theory, Graduate Studies in Mathematics, vol. 163, third edn. American Mathematical Society, Providence, RI