Extremal Orders of Certain Functions Associated with Regular Integers (mod n)

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1 Journal of Integer Sequences, Vol ), Article Extremal Orders of Certain Functions Associated with Regular Integers mod n) Brăduţ Apostol Spiru Haret Pedagogical High School 1 Timotei Cipariu St. RO Focşani Romania apo brad@yahoo.com Lucian Petrescu Henri Coă Technical College 2 Tineretului St. RO Tulcea Romania petrescureea@yahoo.com Abstract Let Vn) denote the number of positive regular integers mod n) that are n, let V k n) be a multidimensional generalization of the arithmetic function Vn). We find the Dirichlet series of V k n) give the extremal orders of some totients involving arithmetic functions which generalize the sum-of-divisors function the Dedekind function. We also give the extremal orders of other totients regarding arithmetic functions which generalize the sum of the unitary divisors of n the unitary function corresponding to φn), the Euler function. Finally, we study extremal orders of some compositions, involving the functions mentioned previously. 1

2 1 Introduction Letn > 1beaninteger. Anintegeraiscalledregularmodn)ifthereexistsanintegerxsuch that a 2 x a mod n) sequence A in Sloane s Encyclopedia of Integer Sequences). Several authors investigated properties of regular integers mod n). Alkam Osba [1], using ring-theoretic considerations, rediscovered some of the statements proved by Morgado [6, 7], who showed that a > 1 is regular mod n) if only if gcda,n) is a unitary divisor of n. Tóth [15] gave direct proofs of some properties, because the proofs of [6, 7] were lengthy those of [1] were ring-theoretical. Let Reg n = {a : 1 a n a is regular mod n)}, Vn) = #Reg n. The function V is multiplicative Vp α ) = φp α )+1 = p α p α 1 +1, where φ is the Euler function. Consequently, Vn) = d nφd), for every n 1, where d n means that d is a unitary divisor of n, that is, d n gcdd, n ) = 1. Also φn) < Vn) n, for every n > 1, d Vn) = nifonlyifnisasquarefree; see[7,15,1]. Thus, thefunctionvn)isananalogue of the Euler function φn). The function φn) is the sequence A in Sloane s On-Line Encyclopedia of Integer Sequences. Also, the function Vn) is the sequence A055653; see [12]. Apostol Tóth [4] considered the multidimensional generalization of the function Vn), V k n), where k 1 is a fixed integer. The function V k n) is multiplicative V k p α ) = φ k p α ) + 1 = p αk p α 1)k + 1, where φ k is the Jordan function of order k. Consequently, V k n) = d n φ kd), for every n 1. Also φ k n) < V k n) n k, for every n > 1 V k n) = n k if only if n is squarefree; see [4]. Tóth[15] proved results concerning the minimal maximal orders of the functions Vn) Vn)/φn). Alkam Osba [1] investigated the minimal order of Vn). Sándor Tóth [10] Apostol [2] studied the extremal orders of compositions of certain functions. In Section 2 we present some notation results involving arithmetical functions. Section 3 is devoted to the study of the Dirichlet series of V k n). Extremal orders of the function V k n) in connection with σ k n) ψ k n) are given in Section 4. In Section 5 we prove some results regarding V k n) unitary analogues of the functions σ k n) φ k n). In Section 6 we give the extremal orders of some compositions of functions from above. Section 7 provides other its of compositions of arithmetical functions. We also present some open problems regarding extremal orders of these compositions. 2 Preinaries In what follows let n = p α 1 1 p αr r > 1 be a positive integer let k 1 be an integer. Throughout the paper we will use the following notation: p 1,p 2,... the sequence of the primes; σ k n) the generalization of σn), defined by σ k n) = r i=1 2 p α i +1)k i 1 ; p k i 1

3 ψ k n) the generalization of ψn), defined by ψ k n) = n k p n 1+ 1 ) ; p k ζs) the Riemann zeta function, ζs) = ) 1 p 1 1, s = σ +it C σ > 1; p s φn) the Euler function, φn) = n p n ) 1 1 ; p φ k n) the Jordan function of order k, φ k n) = n k p n 1 1 p k ) ; γ the Euler constant, γ = n logn); φ n) the unitary analogue of φn), φ n) = k i=1 pα i i 1); σ n) the unitary analogue of σn), σ n) = k i=1 pα i i +1). For other arithmetic functions defined by regular integers modulo n we refer to the papers [5, 14]. Let fn) be a nonnegative real-valued multiplicative arithmetic function. Let L = Lf) := sup fn) loglogn ρp) = ρp,f) := supfp α ) α 0 for primes p, consider the product R = Rf) := p 1 1 p )ρp). In order to prove the properties below we apply the following results: Lemma 1. Tóth Wirsing [16, Corollary 1]). If f is a nonnegative real-valued multiplicative arithmetic functiouch that for each prime p, 1 i) ρp) = sup α 0 fp α )) 1 p) 1, ii) there is an exponent e p = p o1) N satisfying fp ep ) 1+ 1, p ) fn) then sup = eγ loglogn p 1 1 ρp). p Lemma 2. Tóth Wirsing [16, Theorem 1]). Suppose that ρp) < for all primes p the product R converges unconditionally i.e., irrespectively of order), improper its being allowed, then L e γ R. Lemma 3. Tóth Wirsing [16, Theorem 3]). Suppose that ρp) < for all primes p, that for each prime p there is an exponent e p = p o1) N such that p fpep )ρp) 1 > 0 that the product R converges, improper its being allowed. Then L e γ R. 3

4 3 Dirichlet series of V k n) Apostol Petrescu [3] studied the Dirichlet series of V 1 n) := Vn). In what follows we give the Dirichlet series of V k n) for k 2 some results involving the Möbius function. Proposition 4. For every s = σ +it C with σ > k +1, n 1 V k n) = ζs k)ζs) p 1 p 2s k +p s p s k p 3s k ). Proof. Let fn) = V kn). We have so the series n 1 n 1 V k n) V k n) the claim follows. fn) 1 <, nσ k n 1 n 1 converges absolutely for σ > k +1. Since V k is multiplicative, = p Corollary 5. Let s = σ +it C, σ > k +1. Then n p 1 1 2s k +p s p s k ) 1, 1 1 p p s k p p 3s k s p µn)v k n) = p 1 1 ) p s k = 1 ζs k). Also, n 1 µn) V k n) = p 1+ 1 ) = ζs k) p s k ζ2s 2k). Proof. Forfn) = µn)v kn) n 1 the series n 1 fn) converges absolutely, so µn)v k n) = p For the second assertion take fn) = µn) V kn). 1 1 ) = p s k 1 ζs k). 4

5 4 Extremal orders concerning classical generalized arithmetic functions For the quotient σ kn) V k n), we notice that σ kn) V k n) we get hence the minimal order of σ kn) V k n) Since for every n 1 σ k p) p for every prime p, it is immediate that Thus, the minimal order of ψ kn) V k n) V k p) = p inf 1 for every n 1. Since σ k n) V k n) = 1; p k +1 p k = 1, is 1. Now consider the quotient sup sup ψ k n) V k n). ψ k n) V k n) 1 ψ k p) V k p) = pk +1 p k inf ψ k n) V k n) = 1. is 1. It is known that σn) Vn)loglogn) 2 = e2γ ψn) Vn)loglogn) 2 = 6 π 2e2γ ; see [2]. Proposition 6 shows that the maximal order of σ kn) V k n) ψ kn) V k n) is 6 π 2 e 2γ loglogn) 2. Proposition 6. For k 2, sup σ k n) V k n)loglogn) 2 = sup ψ k n) V k n)loglogn) 2 = 6 π 2e2γ. 5

6 Proof. Take fn) = σk n). Then V k n) fp α ) = p α+1)k 1 p k 1)p αk p α 1)k +1) p+1 p 1 = ρp) < 1 1 ) 1 p fp 2 p ) = 3k 1 p k 1)p 2k p k +1) 1+ 1 p for every prime p, so ii) in Lemma 1 is satisfied. We obtain σk n) sup Vk n)loglogn = 1 1 = p 2eγ p so sup σ k n) V k n)loglogn) 2 = 6 π 2e2γ. 6 π 2eγ, Since ψ k n) σ k n) for the primes p we have ψ k p) = σ k p) = p k + 1, the result for ψ k n) V k n)loglogn) 2 follows from the previous one. 5 Extremal orders concerning unitary analogues of σ k φ k In what follows we consider the functions σk n) φ k n), representing the generalizations for the sum of the unitary divisors of n the unitary analogue Euler function, respectively. Let k 1 be a fixed integer. We have σk n) = d n dk σk pα ) = p αk +1. Also, φ kn) := hence φ k pα ) = p αk 1. Note that gcda 1,...,a k ) {1,2,...,n} k gcdgcda 1,a 2,...,a k ),n) =1 1 = d n gcda,b) = max{d : d a,d b} d k µ n d ), µ n) is the unitary analogue of the Möbius function, given by µ n) = 1) ωn), where ωn) is the number of distinct prime factors of n. The functions σk n) φ k n) are multiplicative. Let n = p α 1 1 p αr r be the prime factorisation of n > 1. We obtain φ kn) = p α 1k 1 1) pr αrk 1) σkn) = p α 1k 1 +1) p αrk r +1). 6

7 Observe that σ k n) = σ kn) φ k n) = φ kn) for all squarefree n. Furthermore, for every n 1, φ k n) φ kn) n k σ kn) σ k n). Recall that an integer n > 1 is called powerful if it is divisible by the square of each of its prime factors. A powerful integer is also called a squarefull integer. We give extremal orders for the quotients σ k n) φ k n), the minimal order of φ k n) V k n) V k n) V k being studied for powerful n) numbers. Since σ k n) 1 for every n 1 σk p) V k n) p = V k p) p pk +1 = 1 for prime p k σk numbers p, it follows that inf n) = 1. V k n) If n is powerful, it is easy to see that φ k n) 1, taking into account that φ k pα ) 1 for V k n) V k p α ) α 2. For prime numbers p, we notice that which implies that φ k p2 ) p V k p 2 ) = p 2k 1 p p 2k p k +1 = 1, inf φ k n) V k n) = 1, so the minimal order of φ k n) V k is 1. n) For the maximal orders of these quotients we have Proposition 7. For k 1, sup σ k n) V k n)loglogn = sup φ k n) V k n)loglogn = eγ. Proof. Take fn) = σ k n) V k in Lemma 2, which is a nonnegative real-valued multiplicative n) arithmetic function. We have fp α p αk +1 ) = p αk p α 1)k +1 1 p) 1 1 = ρp) < R = 1, so Now let gn) = φ k n) V k n). Here gp α ) = sup σ k n) V k n)loglogn eγ. p αk 1 p αk p α 1)k +1 1 p) 1 1 = ρp), R = p gp 1 )ρp)) 1 = p p k +1) p 1 p > 0. 7

8 Hence, by Lemma 3 we have sup φ k n) V k n)loglogn eγ. It is obvious that φ k n) σ k n) for every n 1. We obtain which shows that as desired. e γ sup sup φ k n) V k n)loglogn sup σ k n) V k n)loglogn = sup σ k n) V k n)loglogn eγ, φ k n) V k n)loglogn = eγ, Corollary 8. The maximal order of both σ k n) V k n) φ k n) V k n) is eγ loglogn. 6 Extremal orders regarding compositions of arithmetical functions We now move to the study of extremal orders of some composite arithmetic functions. We start with V k V k n)) φ k V k n)). We know that V k n) n k for every n 1, so V k V k p)) p p k2 V k V k n)) n k2 V kn)) k n k2 V k p k ) = p p k2 nk ) k n k2 = 1 p k2 p k 1)k +1 = p = 1, p k2 so the maximal order of V k V k n)) is n k2. Since φ k n) n k V k n) n k for any n 1, we have φ kv k n)) n k2 V kn)) k n k2 φ 1. But p k V k p)) p k2 = p p k 2 p k 1)k p k2 = 1, so the maximal order of φ k V k n)) is n k2. The maximal order of Vφn)) was investigated in [2]. Using the general idea of that proof, we show Proposition 9. The maximal order of V k φ k n)) is n k2. Proof. We will use Linnik s theorem which states that if gcdt,l) = 1, then there exists a prime p such that p l mod t) p t c, where c is a constant one can take c 11). 8

9 Let A = k<p x p. Since gcda 2,A + 1) = 1, by Linnik s theorem there is a prime number q such that q A + 1 mod A 2 ) q A 2 ) c = A 2c, where c satisfies c 11. Also, q k ka+1 mod A 2 ). Let q be the least prime satisfying the above condition. We have φ k q) = q k 1 = AB, where B = k+sa, for some s. Thus gcda,b) = 1, so B is free of prime factors x > k. Since V k n) is multiplicative, we have V k φ k q)) q k2 = V kab) AB +1) = V ka) VkB) AB) k k A k B k AB +1) k. 1) Here AB) k 1 as x, so it is sufficient to study V ka) AB+1) k A k V kb) B k. Clearly, V k A) A k = 1. 2) We have A = k<p x p p x p = eox). Since B A 10 we obtain B e Ox), so logb x. 3) If B = r i=1 qb i i is the prime factorization of B, we obtain, taking into account that k 1 is a fixed integer, that logb = r i=1 b ilogq i > logx) r i=1 b i for sufficiently large x. But r i=1 b i r, so logb > klogx, implying that r < logb x by 3)). Since logx logx V k B) B k > r 1 1 ) qi k i=1 r ) 1 1qi > i=1 1 x) 1 r 1 1 ) O x logx ), x We obtain ) V k B) 1 > 1+O. 4) B k logx By 1), 2), 4) AB)k AB+1) k 1 as x, we obtain By relation 5), since V kφ k n)) n k2 ) V k φ k q)) 1 > 1+O. 5) q k2 logx φ kn)) k n k2 1, the claim follows. The maximal order of Vφ n)) is n see [2]). For the maximal order of V k φ n)) we give Proposition 10. sup V k φ n)) n k = 1. 9

10 Proof. We apply the following lemma: If a is an integer, a > 1, p is a prime number fn) is an arithmetical function satisfying φn) fn) σn), one has fna,p)) p Na,p) = 1, 6) where Na,p) = ap 1 see, e.g., Suryanarayana [13]). a 1 Since φ n) n, it follows that V k φ n)) φ n)) k n k, so k Vk φ n)) 1. 7) n k Obviously, V k n) meets the conditions of the lemma. We have k Vk φ 2 p k )) Vk 2 = p k 1) Vk N2,p)) 2 p p = 2 p 1 p = 1. 8) N2,p) p Now 7) 8) imply sup k V k φ n)) n Apostol [2] proved that sup sup σφ n)) Vφ n))loglogn) 2 = sup ψφ n)) Vφ n))loglogn) 2 sup The maximal orders of σ kφ n)) ψ kφ n)) V k φ n)) V k φ n)) Proposition 11. For k 2 we have i) sup ii) sup Proof. i) Let l 1 := sup = 1, we are done. σφ n)) Vφ n))loglogφ n)) 2 = e2γ ψφ n)) Vφ n))loglogφ n)) 2 = 6 π 2e2γ. are given by σ k φ n)) σ V k = sup k φ n)) φ n))loglogn) 2 V k = 6 e 2γ, φ n))loglogφ n)) 2 π 2 ψ k φ n)) ψ V k = sup k φ n)) φ n))loglogn) 2 V k = 6 e 2γ. φ n))loglogφ n)) 2 π 2 σ k φ n)) V k φ n))loglogn) 2 l 2 := sup Since φ n) n for every n 1, we have l 1 = sup l 2 = sup sup m σ k φ n)) V k φ n))loglogn) 2 σ k φ n)) V k φ n))loglogφ n)) 2 σ k m) V k m)loglogm) = 6, 2 π 2e2γ 10 σ k φ n)) V k φ n))loglogφ n)) 2.

11 by Proposition 6. Since gcdn,1) = 1, by Linnik s theorem, there exists a prime number p such that p 1 mod n) p n c. Let p n be the least prime such that p n 1 mod n), for every n. Then n p n 1 p n n c, so loglogp n loglogn. Observe that a b implies σ ka) V k σ kb). If a) V k b) pβ p α β α), it is easy to see that σ k p β ) σ kp α ). The general case follows, taking into account that σ kn) V k p β ) V k p α ) V k is multiplicative. So, n) σ k φ p n )) V k φ p n ))loglogp n ) = σ k p n 1) 2 V k p n 1)loglogp n ) σ k p n 1) 2 V k p n 1)loglogn) 2. On the other h, Therefore, sup σ k p n 1) V k p n 1)loglogn) 2 σ k φ n)) V k φ n))loglogn) 2 sup sup σ k n) V k n)loglogn) 2. σ k n) V k n)loglogn) 2 = 6 π 2e2γ. We obtain 6 π 2 e 2γ l 1 l 2 6 π 2 e 2γ, hence l 1 = l 2 = 6 π 2 e 2γ. σ k φ p n )) V k φ p n ))loglogp n ) 2 ii) The proof is similar to the proof of i), taking into account that a b implies ψ ka) sup ψ k n) V k n)loglogn) 2 = 6 π 2 e 2γ, by Proposition 6. So, the maximal orders of both σ kφ n)) In a similar manner, since sup V k φ n)) ψ kφ n)) σ k n) V k n)loglogn = sup using Proposition 7), the fact that a b implies σ k a) it can be shown that sup σ k φ n)) V k φ n))loglogn = sup are 6 V k e 2γ loglogn) 2. φ n)) π 2 φ k n) V k n)loglogn = eγ σ k b) φ k a) φ k b) V k a) V k b) V k a) σ k φ n)) V k φ n))loglogφ n) = eγ ψ kb) V k a) V k b) V k b), respectively, sup φ k φ n)) V k φ n))loglogn = sup φ k φ n)) V k φ n))loglogφ n) = eγ. 11

12 7 Open Problems Open Problem 12. Note that inf V k φn)) n k = inf V k φ n)) n k = inf φ k Vn)) n k = 0. For n k = p 1 p r the product of the first r primes), we have V k φn r )) n k r = V kp 1 1) p r 1)) p k 1 p k r p 1 1) k p r 1) k p k 1 p k r = 1 1 p 1 ) 1 1 p r ) ) k, so V k φn r )) r n k r = r 1 1 p 1 ) 1 1 p r ) ) k = 0, similarly the other relations. What are the minimal orders for the V k φn)), V k φ n)), φ k Vn))? Open Problem 13. Taking n r = p 1 p r the product of the first r primes), σ k Vn r)) n k r = σ k p 1 p r ) p k 1 p k r = pk 1 +1) p k r +1) p k 1 p k r = 1+ 1 p 1 ) 1+ 1 p r ) ) k as r, so sup σ k Vn)) n k =. What is the maximal order for σ k Vn))? 8 Acknowledgments The authors would like to thank the referees for helpful suggestions. References [1] O. Alkam E. Osba, On the regular elements in Z n, Turk. J. Math ), 1 9. [2] B. Apostol, Extremal orders of some functions connected to regular integers modulo n, An. Ştiinţ. Univ. Ovidius Constanţa, Ser. Mat ), to appear. [3] B. Apostol L. Petrescu, On the number of regular integers mod n), J. Algebra Number Theory Acad ), [4] B. Apostol L. Tóth, Some remarks on regular integers modulo n, preprint, [5] P. Haukkanen L. Tóth, An analogue of Ramanujan s sum with respect to regular integers mod r), Ramanujan J ),

13 [6] J. Morgado, A property of the Euler φ-function concerning the integers which are regular modulo n, Portugal. Math ), [7] J. Morgado, Inteiros regulares módulo n, Gazeta de Matematica ), 1 5. [8] J. Sándor, Geometric Theorems, Diophantine Equations, Arithmetic Functions, American Research Press, [9] J. Sándor B. Crstici, Hbook of Number Theory II, Kluwer Academic Publishers, [10] J. Sándor L. Tóth, Extremal orders of compositions of certain arithmetical functions, Integers ), # A34. [11] R. Sivaramakrishnan, Classical Theory of Arithmetic Functions, Marcel Dekker, [12] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, [13] D. Suryanarayana, On a class of sequences of integers, Amer. Math. Monthly ), [14] L. Tóth, A gcd-sum function over regular integers modulo n, J. Integer Seq ), Article [15] L. Tóth, Regular integersmod n), Annales Univ. Sci. Budapest., Sect Comp ), [16] L. Tóth E. Wirsing, The maximal order of a class of multiplicative arithmetical functions, Annales Univ. Sci. Budapest., Sect Comp ), Mathematics Subject Classification: Primary 11A25; Secondary 11N37. Keywords: arithmetical function, composition, regular integer mod n), extremal order. Concerned with sequences A000010, A000203, A055653, A ) Received April ; revised versions received July ; July ; August Published in Journal of Integer Sequences, August Return to Journal of Integer Sequences home page. 13