ON THE AVERAGE ASYMPTOTIC BEHAVIOR OF A CERTAIN TYPE OF SEQUENCE OF INTEGERS

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1 #A42 ITEGERS 9 (2009), O THE AVERAGE ASYMPTOTIC BEHAVIOR OF A CERTAI TYPE OF SEQUECE OF ITEGERS Bakir Farhi Institut des Hautes Études Scientifiques, Le bois-marie, Bures-sur-Yvette, France bakir.farhi@gmail.com Received: 0/2/08, Revised: 4/4/09, Accepted: 5/8/09, Published: /9/09 Abstract In this paper, we prove the following result: Let A be an infinite set of positive integers. For all positive integer n, let τ n denote the smallest element of A which doesn t divide n. Then we have τ n n n0 lcm{a A a n}. In the two particular cases when A is the set of all positive integers and when A is the set of the prime numbers, we give a more precise result for the average asymptotic behavior of (τ n ) n. Furthermore, we discuss the irrationality of the it of τ n (in the average sense) by applying a result of Erdős.. Introduction and Results In umber Theory, it is frequent that a sequence of positive integers does not have a regular asymptotic behavior but has a simple and regular asymptotic average behavior. As eamples, we can cite the following: (i) The sequence (d n ) n, where d n denotes the number of divisors of n. (ii) The sequence (σ(n)) n, where σ(n) denotes the sum of divisors of n. (iii) The Euler totient function (ϕ(n)) n, where ϕ(n) denotes the number of positive integers, not eceeding n, that are relatively prime to n. We refer the reader to [3] for many other eamples. In this paper, we give another type of sequence which we describe as follows: Let a < a 2 <... be an increasing sequence of positive integers which we denote by A. For all positive integers n, let τ n denote the smallest element of A which doesn t divide n. Then, we shall prove the following theorem.

2 ITEGERS: 9 (2009) 556 Theorem We have: τ n n n0 lcm{a A a n} in both cases when the series on the right-hand side converges or diverges. In the particular cases when A is the sequence of all positive integers and when it is the sequence of the prime numbers, we refine the proof of Theorem to obtain the following more precise results: Corollary 2 For all positive integers n, let e n denote the smallest positive integer which doesn t divide n. Then, we have n ( ) (log ) 2 e n l O, log log where l : n lcm(, 2,..., n) <. Corollary 3 For all positive integers n, let q n denote the smallest prime number which doesn t divide n. Then, we have where n ( ) (log ) 2 q n l 2 O, log log l 2 : n p prime, p n p <. Further, by applying a result of Erdős [], we derive a sufficient condition for the average it of (τ n ) n to be an irrational number. We have the following. Proposition 4 Let d(a) denote the lower asymptotic density of A; that is, d(a) : inf. a A a Suppose that d(a) > log 2. Then the average it of (τ n ) n is an irrational number. In particular, the numbers l and l 2, appearing respectively in Corollaries 2 and 3, are irrational.

3 ITEGERS: 9 (2009) The Proofs 2.. Some Preparations and Preinary Results Throughout this paper, we let denote the set \ {0} of all positive integers. For a given real number, we let and denote respectively the integer part and the fractional part of. Further, we adopt the natural convention that the least common multiple and the product of the elements of an empty set are equal to. We fi an increasing sequence of positive integers a < a 2 <, which we denote by A, and for all positive integer n, we let τ n denote the smallest element of A which doesn t divide n. For all α A, we let L(α) denote the positive integer defined by L(α) : We then let B denote the subset of A defined by lcm{a A a α} lcm{a A a < α}. () B : {a A L(a) > }. (2) We shall see later that B is just the set of the values of the sequence (τ n ) n. We begin with the following lemma. Lemma 5 For all positive integer n, we have: lcm {a A a n} lcm {b B b n}. Proof. Let n and let a,..., a k be the elements of A not eceeding n. By using the following well-known property of the least common multiple: lcm(a,..., a i,..., a k ) lcm(lcm(a,..., a i ), a i,..., a k ) (for i, 2,..., k), we remark that when a i B, we have L(a i ) and then lcm(a,..., a i ) lcm(a,..., a i ). So, each a i not belonging to B can be einated from the list a,..., a k without changing the value of the least common multiple of that list. The lemma follows. have From Lemma 5, we derive another formula for L(α) (α A). For all α A we L(α) lcm{b B b α} lcm{b B b < α}. (3) The net lemma gives a useful characterization for the terms of the sequence (τ n ) n. Lemma 6 For all positive integers n and all α A, we have: τ n α k, k 0 mod L(α) such that n k lcm{b B b < α}.

4 ITEGERS: 9 (2009) 558 Proof. Let n and α A. By the definition of the sequence (τ n ) n, the equality τ n α amounts to saying that n is a multiple of each element a A satisfying a < α and that n is not a multiple of α. Equivalently, τ n α if and only if n is a multiple of lcm{a A a < α} without being a multiple of lcm{a A a α}. So, it suffices to set k : n lcm{a A a<α} to obtain the equivalence: τ n α k, k 0 mod L(α) such that n k lcm{a A a < α}. The lemma then follows from Lemma 5. Corollary 7 The sequence (τ n ) n takes its values in the set B. Besides, any element of B is taken by (τ n ) n infinitely often. Proof. Let α A. If α B, then we have L(α) and thus there is no k such that k 0 mod (L(α)). It follows, according to Lemma 6, that α cannot be a value of (τ n ) n. et, if α B, then L(α) 2 and thus there are infinitely many k such that k 0 mod (L(α)). This implies (according to Lemma 6) that there are infinitely many n satisfying τ n α. The corollary is proved. Actually, given α B, Lemma 6 even gives an estimation for the number of solutions of the equation τ n α in an interval [, ] ( R ). For > 0 and α B, define Then, we have the following: ϕ(α; ) : #{n, n τ n α}. Corollary 8 Let α B and > 0. Then we have ϕ(α; ) L(α) lcm{b B b α}. c α,, where c α, <. Furthermore, if lcm{b B b < α} >, then we have ϕ(α; ) 0. Proof. Let α B and > 0. By Lemma 6, we have { } ϕ(α; ) # k k and k 0 mod (L(α)) lcm{b B b < α} lcm{b B b < α} L(α).lcm{b B b < α} lcm{b B b < α} L(α).lcm{b B b < α} c α,, where c α, :. So, it is clear that c α, <. lcm{b B b<α} L(α).lcm{b B b<α}

5 ITEGERS: 9 (2009) 559 et, we have lcm{b B b < α} L(α).lcm{b B b < α} L(α) L(α).lcm{b B b < α}. by using (3). This confirms the first part of the corollary. L(α) lcm{b B b α}., ow, if lcm{b B b < α} >, then none of the integers of the range [, ] is a multiple of lcm{b B b < α}. It follows, according to Lemma 6, that the equation τ n α doesn t have any solution in the range [, ]. Hence ϕ(α, ) 0. This confirms the second part of the corollary and ends this proof. ow, let b < b 2 <... be the elements of B. To prove our main result, we will need some properties of the sequence (b n ) n. For simplicity, for all n, let Then, by (3), we have: which gives L n : L(b n ) 2. L n lcm{b B b b n} lcm{b B b < b n } lcm(b, b 2,..., b n ) lcm(b, b 2,..., b n ), lcm(b, b 2,..., b n ) L n lcm(b, b 2,..., b n ) ( n ). By iteration, we obtain for all n : lcm(b, b 2,..., b n ) L L 2 L n 2 n. (4) For the simplicity of some formulas in what follows, it is useful to set b 0 : 0 and L 0 :. ote that b 0 B. We have the two following lemmas: Lemma 9 We have n lcm{a A a n}. L L 2 L k Proof. According to Lemma 5, we have: n lcm{a A a n} n lcm{b B b n} lcm{b B b n} n<

6 ITEGERS: 9 (2009) 560 lcm(b, b 2,..., ) L L 2 L k (according to (4)). The lemma is proved. Lemma 0 Let r be a positive integer and be an integer such that Then, we have where lcm(b, b 2,..., ) < lcm(b, b 2,..., ). τ n S () S 2 ()f(, r), n S () : L L 2 L r L L 2 L k, S 2 () : and f(, r) is a function of and r, satisfying f(, r) <. Proof. According to Corollaries 7 and 8, and to relation (4), we have τ n n α B n,τ nα αϕ(α; ) α B α αϕ(α; ) α B lcm{b B b<α} ϕ( ; ) ( ) Lk c k, L L 2 L k ( L L 2 L k L L 2 L k ) (where c k, < ) c k,.

7 ITEGERS: 9 (2009) 56 Then, the lemma follows by remarking that: ( ) L L 2 L k L L 2 L k ( ) L L 2 L k L L 2 L k L L 2 L k L L 2 L r L L 2 L k and by defining f(, r) : r c k, r b, k which satisfies f(, r) < (since c k, < for all k ). This completes the proof. We will finally need two lemmas on the convergence of sequences and series. Lemma Let ( n ) n be a real non-increasing sequence. Suppose that the series n n converges. Then, we have n o ( ) n (as n tends to infinity). Lemma 2 Let (θ n ) n be a sequence of real numbers. Suppose that θ n tends to 0 as n tends to infinity. Then the sequence with general term given by also tends to 0 as n tends to infinity. 2 n n 2 k θ k Remark. ote that Lemma 2 is in fact a particular case of a more general theorem in summability theory, called Silverman-Toeplitz theorem (see e.g., [2]) Proofs of the Main Results Proof of Theorem. Let be a positive integer. Since (according to (4)) the sequence (lcm(b, b 2,..., b n )) n increases and tends to infinity with n, must lie somewhere between two consecutive terms of this sequence. So, let r such that lcm(b, b 2,..., ) < lcm(b, b 2,..., ).

8 ITEGERS: 9 (2009) 562 Then, by using Lemma 0, we have τ n S () S 2 ()f(), (5) n where S () : S 2 () : L L 2 L r L L 2 L k, (6) (7) and f() <. et, set S : n0 lcm{a A a n}. To prove Theorem, we distinguish two cases according to whether S converges or diverges. st case: S <. In this case, since the sequence (/lcm{a A a < n}) n is clearly non-increasing, then by Corollary, we have n n lcm{a A a < n} 0. By specializing in this it n to the integers (k ), we obtain (according to Lemma 5 and Formula (4)) that k L L 2 L k 0. (8) On the one hand, according to (8) and to Lemma 9, we have (because r tends to infinity with ) S () and, on the other hand, we have S 2 () : 2 r lcm(b,..., ) L L 2 L k S (9) L L 2 L r L L 2 L k 2 k r (since L i 2 for all i ) 2 k L L 2 L k.

9 ITEGERS: 9 (2009) 563 But by applying Lemma 2 for θ k : to 0 as k tends to infinity, we have r 2 r L L 2 L k 2 k L L 2 L k 0. So, it follows (because r tends to infinity with ) that Finally, by inserting (9) and (0) into (5), we get as required. which is seen (from (8)) to tend S 2() 0. (0) τ n S, n 2 nd case: S. In this case, by using (5), we are going to bound from below n τ n by an epression tending to infinity with. On the one hand, we have S () : L L 2 L r L L 2 L r L L 2 L k r L L 2 L r L L 2 L k r 2 () L L 2 L r L L 2 L k (because we obviously have b i lcm(b,..., b i ) L L 2 L i, for all i ). On the other hand, we have S 2 () : < L L 2 L r r L L 2 L k (since b i L L 2 L i, for all i) L L 2 L r L L 2 L r r L L 2 L r L k L k2 L r r L L 2 L r 2 r k (since L i 2 for all i) L L 2 L r 2. (2)

10 ITEGERS: 9 (2009) 564 It follows, by inserting () and (2) into (5), that τ n S () S 2 ()f() n S () S 2 () (since f() < ) r L L 2 L k 4. But since (according to Lemma 9) L L 2 L k S, we conclude that τ n. n This completes the proof of the theorem. Proof of Corollary 2. In the situation of Corollary 2, A is the set of all positive integers and then B is the set of the powers of prime numbers. We must repeat the proof of Theorem and give more precision to the two quantities r. First let us show that L L 2 L r and log (as tends to infinity). (3) By the definition of r, recall that lcm(b, b 2,..., ) < lcm(b, b 2,..., ). (4) et, by Lemma 5, we have (since A ): lcm(b, b 2,..., ) lcm(, 2,..., ), lcm(b, b 2,..., ) lcm(, 2,..., ) and by the prime number theorem (see, e.g., [3]), we have, on the one hand, log lcm(, 2,..., n) n (as n tends to infinity) and, on the other hand (because B is the set of the powers of prime numbers), b n b n (as n tends to infinity). Taking into account all these facts, we derive from (4) that effectively log, confirming (3).

11 ITEGERS: 9 (2009) 565 ow, we are going to be precise the order of magnitude of We have L L 2 L r and r. L L 2 L r lcm(b, b 2,..., ) < (according to (4)). It follows, according to (3), that L L 2 L r O ( log ). (5) et, because B is the set of the powers of the prime numbers, we have e,p prime p e p prime ( p log br e log p ) e p prime p p ( ) log br p log p. p e p p Since for any prime number p, we have it follows that p p 2 and p log br/ log p p log br/ log p, 2 π( ), where π denotes the prime-counting function. But, by using again the prime number theorem and (3), we have π( ) O( b2 r log ) O( (log )2 log log ). Hence ( ) (log ) 2 O. (6) log log It finally remains to insert (5) and (6) into (6) and (7) respectively to obtain (according to (5) and to Lemma 9) that n τ n n ( ) (log ) 2 lcm(, 2,..., n) O. log log The corollary is proved. Proof of Corollary 3. In the situation of Corollary 3, A is the set of the prime numbers and then B A. So, for all n, we have a n b n L n p n, where p n denotes the n th prime number. Consequently, we have (in the contet of the proof of Theorem ) p p 2 p r < p p 2 p r. (7) So, by the prime number theorem (see, e.g., [3]), we have p r log (as tends to infinity).

12 ITEGERS: 9 (2009) 566 From this last, it follows that p r < p r L L 2 L r p p 2 p r log (as tends to infinity), which gives and that p k ( log O L L 2 L r p2 r 2 log p r (log )2 2 log log ) (as tends to infinity), (8) which gives ( ) (log ) 2 O. (9) log log To conclude, it suffices to insert (8) and (9) into (6) and (7) respectively and use (5) and Lemma 9. The result of the corollary follows. ow, we are going to prove Proposition 4. To do so, we need the following result of Erdős []. Theorem (Erdős []) Let u < u 2 < be an infinite sequence of positive integers. Set U : {u, u 2,... } and suppose that d(u) > log Then, the real positive number is irrational. n lcm{u U u n} Proof of Proposition 4. The first part of Proposition 4 which concerns a general set A is clearly an immediate consequence of the Main Theorem and the above theorem of Erdős. et, since the set of all positive integers has asymptotic density > log 2, the irrationality of the constant l of Corollary 2 is a direct application of the first part of the proposition. ow, let us prove the irrationality of the constant l 2 appearing in Corollary 3. We must notice that this is not a direct application of the first part of the proposition, because the set of the prime numbers has asymptotic density 0 < log 2.

13 ITEGERS: 9 (2009) 567 Let F denote the set of square-free numbers, that is the set of all positive integers which are a product of pairwise distinct prime numbers. It is known that F has asymptotic density 6 π > log 2 (see, e.g., [3]). So, it follows by Erdős result that 2 the number l 3 : lcm{f F f n} n is irrational. But we remark that for all n, we have: lcm{f F f n} p prime p n p lcm{p prime p n}, which shows that actually l 3 l 2. Consequently l 2 is an irrational number. This completes the proof of the proposition. Remark. The irrationality of the constants l and l 2 appearing in Corollaries 2 and 3 respectively can be shown by a more elementary way than that presented in Erdős paper for the general case. Acknowledgments I would like to thank the referee for his/her comments and suggestions which certainly improved the readability of this paper. References [] P. Erdős, On the irrationality of certain series: problems and results, in ew Advances in Transcendence Theory (Durham, 986), p , Cambridge Univ. Press, Cambridge, 988. [2] G.H. Hardy, Divergent Series, Oford Univ. Press, London, 949. [3] G.H. Hardy and E.M. Wright, The Theory of umbers, 5th ed., Oford Univ. Press, London, 979.