Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) ON ADDITIVE FUNCTIONS SATISFYING CONGRUENCE PROPERTIES

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1 Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae ) ON ADDITIVE FUNCTIONS SATISFYING CONGRUENCE PROPERTIES Bui Minh Phong Budapest, Hungary) Dedicated to the memory of Professor Péter Kiss Abstract. In this paper, we consider those integer-valued additive functions f 1 f 2 for which the congruence f 1an+b) f 2cn)+d mod n) is satisfied for all positive integers n for some fixed integers a 1, b 1, c 1 d. Our result improve some earlier results of K. Kovács, I. Joó, I. Joó & B. M. Phong P. V. Chung concerning the above congruence. 1. Introduction The problem concerning the characterization of some arithmetical functions by congruence properties initiated by Subbarao [10] was studied later by several authors. M. V. Subbarao proved that if an integer-valued multiplicative function gn) satisfies the congruence gn + m) gm) mod n) for all positive integers n m, then there is a non-negative integer α such that gn) = n α holds for all positive integers n. Recently some authors generalized improved this result in a variety of ways. A. Iványi [3] obtained that the same result holds when m is a fixed positive integer g is an integer-valued completely multiplicative function. For further results generalizations of this problem we refer to the works of B. M. Phong [7] [8], B. M. Phong & J. Fehér [9], I. Joó [4] I. Joó & B. M. Phong [5]. For example, it follows from [8] that if an integer-valued multiplicative function gn) satisfies the congruence gan + B) C mod n) for all positive integers n for some fixed integers A 1, B 1 C 0 with A, B) = 1, then there are a non-negative integer α a real-valued Dirichlet character χ A mod A) such that gn) = χ A n)n α It was financially supported by OTKA T

2 124 Bui Minh Phong holds for all positive integers n which are prime to A. In the following let A A denote the set of all integer-valued additive completely additive functions, respectively. Let IN denote the set of all positive integers. A similar problem concerning the characterization of a zero-function as an integer-valued additive function satisfying a congruence property have been studied by K. Kovács [6], P. V. Chung [1] [2], I. Joó [4] I. Joó & B. M. Phong [5]. It was proved by K. Kovács [6] that if f A satisfies the congruence fan + B) C mod n) for some integers A 1, B 1, C for all n IN, then fn) = 0 holds for all n IN which are prime to A. This result was extended in [1], [2], [4] [5] for integer-valued additive functions f. It follows from the results of [2] [4] that for integers A 1, B 1, C functions f 1 A, f 2 A the congruence f 1 An + B) f 2 n) + C mod n) n IN) implies that f 2 n) = 0 for all n IN f 1 n) = 0 for all n IN which are prime to A. Our purpose in this paper is to improve the above results by showing the following Theorem 1. Assume that a 1, b 1, c 1 d are fixed integers the functions f 1, f 2 are additive. Then the congruence 1) f 1 an + b) f 2 cn) + d mod n) is satisfied for all n IN if only if the equation 2) f 1 an + b) = f 2 cn) + d holds for all n IN. Theorem 2. Assume that a 1, b 1, c 1 d are fixed integers. Let a 1 = a a, b), b 1 = b a, b) µ := { 1 if 2 a1 b 1 2 if 2 a 1 b 1. If the additive functions f 1 f 2 satisfy the equation 2) for all n IN, then f 1 n) = 0 for all n IN, n, µab 1 ) = 1

3 On additive functions satisfying congruence properties 125 f 2 n) = 0 for all n IN, n, µcb 1 ) = Lemmas Lemma 1. Assume that f A satisfies the congruence f An + B) f n) + D mod n) for some fixed integers A 1, B 1 D. Then f n) = 0 holds for all n IN. Proof. Lemma 1 follows from Theorem 2 of [4]. Lemma 2. Assume that f A satisfies the congruence fan + B) D mod n) for some fixed integers A 1, B 1 D. Then fn) = 0 holds for all n IN which are prime to A. Proof. This is the result of [1]. Lemma 3. Assume that f 1, f A satisfy the congruence 3) f 1 An + 1) fcn) + D mod n) holds for all n IN with some integers A 1, C 1 D. Then fn) = f[n, 6C 2 )] for all n IN f 1 m) = 0 holds for all m IN, which are prime to 6AC. Here x, y) denotes the greatest common divisor of the integers x y. Proof. In the following we shall denote by n the product of all distinct prime divisors of positive integer n. For each positive integer M let P = PM) be a positive integer for which 4) M 2 1) ACP. It is obvious from 4) that ACMM + 1)Pn + 1, ACM + 1)Pn + 1) = 1, C 2 M + 1) 2 Pn, ACMPn + 1 ) = 1

4 126 Bui Minh Phong ACMM + 1)Pn + 1)ACM + 1)Pn + 1) = ACM + 1) 2 Pn[ACMPn + 1] + 1 hold for all n IN. Using these relations appealing to the additive nature of the functions f 1 f, we can deduce from 3) that 5) facmpn + 1) f C 2 M + 1) 2 Pn ) + f C 2 MM + 1)Pn ) + f C 2 M + 1)Pn ) + D mod n) is satisfied for all n, M IN, where P = PM) satisfies the condition 4). Let M = 2, P2) = 3 M = 3, P3) = 2. In these cases 4) is true so it follows from 5) that 6) f6acn + 1) f27c 2 n) + f18c 2 n) + f9c 2 n) + D mod n) 7) f6acn + 1) f32c 2 n) + f24c 2 n) + f8c 2 n) + D mod n) are satisfied for all n IN. Let N n be positive integers with the condition 8) NN + 1), 6ACn + 1) = 1. By using the relation that 6ACn + 1)6 2 A 2 C 2 Nn 2 + 1) = 6ACn [6ACNn6ACn + 1) + 1] + 1 6ACn + 1, 6 2 A 2 C 2 Nn ) = 6ACn + 1, N + 1) = 1, 6ACNn, 6ACn + 1) = 6ACn + 1, N) = 1, it follows from 6) 7) that 9) f 162AC 3 Nn 2) + f 108AC 3 Nn 2) + f 54AC 3 Nn 2) f 27C 2 Nn ) +f 18C 2 Nn ) + f 9C 2 Nn ) f 27C 2 n ) + f 18C 2 n ) + f 9C 2 n ) + D mod n) 10) f 192AC 3 Nn 2) + f 144AC 3 Nn 2) + f 48AC 3 Nn 2) f 32C 2 Nn ) +f 24C 2 Nn ) + f 8C 2 Nn ) f 32C 2 n ) + f 24C 2 n ) + f 8C 2 n ) + D mod n) hold for all n, N IN satisfying 8).

5 On additive functions satisfying congruence properties 127 Let Q be a fixed positive integer. First we apply 9) when N = 1, n = Qm, m, Q) = 1 m. It is obvious that 8) holds, so by 9) we have 11) f Q 2) = 2fQ) for Q IN, Q, 6AC) = 1. Now let N = Q n = Q k 6CQm + 1) with k, m IN. It is obvious that 8) holds for infinity many integers m, because 36AC 2 Q k+1, 6ACQ k + 1 ) = 1. These with 9) show that 12) f Q 2k+1) = f Q k) + f Q k+1) for all Q IN, Q, 6AC) = 1. From 11) 12) we obtain that 13) f Q k) = kfq) for all Q IN, Q, 6AC) = 1. Thus, by using the additivity of f it follows from 8) 13) that 9) 10) hold for all N, n IN, they with n = Qm, m, 6ACNQ) = 1, m imply that f 162AC 3 NQ 2) + f 108AC 3 NQ 2) + f 54AC 3 NQ 2) = f 27C 2 NQ ) +f 18C 2 NQ ) + f 9C 2 NQ ) f 27C 2 Q ) + f 18C 2 Q ) + f 9C 2 Q ) D f 192AC 3 NQ 2) + f 144AC 3 NQ 2) + f 48AC 3 NQ 2) = f 32C 2 NQ ) +f 24C 2 NQ ) + f 8C 2 NQ ) f 32C 2 Q ) + f 24C 2 Q ) + f 8C 2 Q ) + D hold for all N, Q IN. Consequently 14) f 27C 2 NQ ) +f 18C 2 NQ ) +f 9C 2 NQ ) f 27C 2 Q ) +f 18C 2 Q ) +f 9C 2 Q ) f 27C 2 NQ 2) + f 18C 2 NQ 2) + f 9C 2 NQ 2) f 27C 2) + f 18C 2) + f 9C 2) 15) f 32C 2 NQ ) +f 24C 2 NQ ) +f 8C 2 NQ ) f 32C 2 Q ) +f 24C 2 Q ) +f 8C 2 Q ) = f 32C 2 NQ 2) +f 24C 2 NQ 2) +f 8C 2 NQ 2) f 32C 2) +f 24C 2) +f 8C 2) are satisfied for all N, Q IN. For each prime p let e = ep) be a non-negative integer for which p e C 2.

6 128 Bui Minh Phong First we consider the case when p, 6) = 1. By applying 14) with Q = p, N = p l l 0), we have f p l+ep)+2) f p l+ep)+1) = f p ep)+1) f p ep)) for all l 0, which shows that for all integers β ep) 16) f p β+1) f p β) = f p ep)+1) f p ep)). Now we consider the case p = 2. Applying 14) with Q = 2 n = 2 l, l 0) one can check as above that 17) f 2 β+1) f 2 β) = f 2 e2)+2) f 2 e2)+1). Finally, we consider the case p = 3. Applying 15) with Q = 3 N = 3 l, l 0 we also get 18) f 3 β+1) f 3 β) = f Now we write 3 e3)+2) f 3 e3)+1). fn) = f n) + Fn), where f is a completely additive function defined as follows: f p ep)+1) f p ep)) for p, 6) = 1 19) f p) := f p ep)+2) f p ep)+1). for p = 2 or p = 3 Then, from 16)-19) it follows that Thus, we have proved that F p k) = F [ p k, 6C 2 ) ] for k = 0, 1,...). 20) Fn) = F [ n, 6C 2 ) ] is satisfied for all n IN. We shall prove that f n) = 0 for all n IN f 1 m) = 0 for all m IN which are prime to 6AC. We note that, by considering n = 2m taking into account 6), we have f12acm + 1) f54c 2 m) + f36c 2 m) + f18c 2 m) + D mod m)

7 On additive functions satisfying congruence properties 129 Since f = f + F, from the last relation 20) we get f 12ACm + 1) f m) + [f 12C 2 ) + F6C 2 ) + D] mod m), which with Lemma 1 shows that f n) = 0 for all n IN. This shows that f F, i.e. fn) = f[n, 6C 2 )] holds for all n IN. Now, by applying 3) with n = 6Cm using the last relation Lemma 2, we have that f 1 n) = 0 holds for all n IN which are prime to 6AC. The proof of Lemma 3 is completed. 3. Proof of Theorem 1 It is obvious that 1) follows from 2). We shall prove that if 1) is true, then 2) holds. Assume that the functions f 1 f 2 A satisfy the congruence 1) for some integers a 1, b 1, c 1 d. It is obvious that 1) implies the fulfilment of for all n IN. By Lemma 3, f 1 abn + 1) f 2 b 2 cn) + d f 1 b) mod n) 21) f 2 n) = f 2 [n, 6b 4 c 2 )] for all n IN 22) f 1 n) = 0 for all n IN which are prime to 6abc. We shall prove that 23) f 1 an + b) = f 2 cn) + d is true for all n IN. Let K be a positive integer. By 21) 22), we have f 1 6ab 4 ct + 1) = 0, f 2 [6b 4 c 2 ak + b)t + ck] = f 2 ck)

8 130 Bui Minh Phong hold for all positive integers t, consequently f 1 ak + b) f 2 ck) d = f 1 ak + b) + f 1 6ab 4 ct + 1) f 2 ck) d = f 1 [a6b 4 cak + b)t + K) + b] f 2 [6b 4 c 2 ak + b)t + ck] d holds for every positive integer t. Thus, by applying 1) with n = 6b 4 cak+b)t+k, the last relation proves that 23) holds for n = K. This completes the proof of Theorem Proof of Theorem 2 As we have shown in the proof of Theorem 1, if the functions f 1, f 2 A satisfy 2), then 21) 22) imply 24) f 1 m) = 0 for all m IN, m, 6abc) = 1 25) f 2 n) = 0 for all n IN, m, 6bc) = 1. Let D = a, b), a 1 = a D, b 1 = b D. It is clear that for each positive integer M, M, a 1 ) = 1 there are m 0, n 0 IN such that 26) Mm 0 = a 1 n 0 + b 1, m 0, a 1 ) = 1 M, n 0 ) = M, b 1 ). Let 27) um) := { 1, if 2 a1 M 2, if 2 a 1 M b 1 M, b 1) M, b, 1) b 1 M, b 1) M, b. 1) By applying the Chinese Remainder Theorem using 26) 27), we can choose a positive integer t 1 such that m 1 = a 1 t 1 + m 0, n 1 = Mt 1 + n 0 satisfy the following conditions: Mm 1 = a 1 n 1 + b 1, n 1 um)m, b 1 ) is an integer, ) n 1 m 1, 6abc) = um)m, b 1 ), 6bc = 1. Hence, we infer from 2) 24)-25) that f 1 DM) = f 1 DMm 1 ) = f 1 an 1 + b) = f 2 cn 1 ) + d = f 2 [cum)m, b 1 )] + d,

9 On additive functions satisfying congruence properties 131 consequently 28) f 1 [DM] = f 2 [cum)m, b 1 )] + d hold for all M IN, M, a 1 ) = 1. This implies that 29) f 1 n) = 0 for all n IN, n, µab 1 ) = 1, where µ {1, 2} such that 2 µa 1 b 1. Now we prove that 30) f 2 n) = 0 for all n IN, n, µcb 1 ) = 1. For each positive integer n, let Mn) := a 1 n + b 1 Un) := ua 1 n + b 1 ). Since Mn), b 1 ) = n, b 1 ) a 1 [ ] Mn) b 1 Mn), b 1 ) Mn), b 1 ) a b 1 n 1 n, b 1 ) n, b 1 ) + 1 mod 2), we have [ b 1, if 2 a 1 n 1 n, b 1) n, b 1) ], Un) := [ + 1 b 2, if 2 a 1 n 1 n, b 1) n, b + 1 1) ]. Hence, 2) 28) show that f 2 cn) = f 1 an + b) d = f 1 [DMn)] d = f 2 [cun)n, b 1 )] is satisfied for all n IN, which implies 29). Thus, 29) is proved. By 29) 30), the proof of Theorem 2 is completed. References [1] Chung, P. V., Note on additive functions satisfying some congruence properties I. II., Studia Math. Hungar., ), , [2] Chung, P. V., Note on additive functions satisfying some congruence properties III., Ann. Univ. Sci. Budapest. Eötvös, Sect. Math., ), [3] Iványi, A., On multiplicative functions with congruence property, Ann. Univ. Sci. Budapest. Eötvös, Sect. Math., ), [4] Joó, I., Arithmetical functions satisfying a congruence property, Acta Math. Hungar., ), [5] Joó, I. Phong, B. M., Arithmetical functions with congruence properties, Ann. Univ. Sci. Budapest. Eötvös, Sect. Math., ), [6] Kovács, K., On additive functions satisfying some congruence properties, Periodica Math. Hungar., 23 3) 1991),

10 132 Bui Minh Phong [7] Phong, B. M., Multiplicative functions satisfying a congruence property, Studia Sci. Math. Hungar., ), [8] Phong, B. M., Multiplicative functions satisfying a congruence property V., Acta Math. Hungar., )1993), [9] Phong, B. M. Fehér, J., Note on multiplicative functions satisfying congruence property, Ann. Univ. Sci. Budapest. Eötvös, Sec. Math., ), [10] Subbarao, M. V., Arithmetic functions satisfying congruence property, Canad. Math. Bull., ), Bui Minh Phong Eötvös Loránd University Department of Computer Algebra Pázmány Péter sét. I/C H-1117 Budapest Hungary bui@compalg.inf.elte.hu