NNTDM 17 (2011), 2, 18-30 New results on some multiplicative functions Mladen Vassilev - Missana1, Peter Vassilev2 1 5 V. Hugo Str., Sofia–1124, Bulgaria e-mail:[email protected] 2 Institute of Biophysics and Biomedical Engineering e-mail:[email protected] Abstract The paper is a continuation of [1]. The considerations are over the class of mul- tiplicative functions with strictly positive values and more precisely, over the pairs (f; g) of such functions, which have a special property, called in the paper property S: For every such pair (f; g) and for every composite number n > 1; the problem of (︀ n )︀ finding the maximum and minimum of the numbers f(d)g d ; when d runs over all proper divisors of n; is completely solved. Since some classical multiplicative func- tions like Euler's totient function '; Dedekind's function ; the sum of all divisors of m; i.e. 휎(m); the number of all divisors of m; i.e. 휏(m); and 2!(m) (where !(m) is the number of all prime divisors of m) form pairs having property S; we apply our results to these functions and also resolve the questions of finding the maximum (︀ n )︀ (︀ n )︀ (︀ n )︀ !(d) (︀ n )︀ and minimum of the numbers '(d)휎 d ;'(d) d ; 휏(d)휎 d ; 2 휎 d ; where d runs over all proper divisors of n: In addition some corollaries from the obtained results, concerning unitary proper divisors, are made. Since many other pairs of multiplicative functions (except the considered in the paper) have property S; they may be investigated in similar manner in a future research. Keywords: multiplicative functions, divisors, proper divisors, unitary divisors, proper unitary divisors, prime numbers, composite number AMS Subject Classification: 11A25 Used Denotations: Z+ - the set of all non-negative integers; N - the set of all positive * integers; P - the set of all prime numbers; for a given n 2 N Dn denotes the set of all ** proper divisors of n; i.e. different than 1 and n; Dn - the set of all proper unitary divisors of n (i.e. different than 1 and n)(see [2]); for n > 1 !(n) denotes the number of all prime def divisors of n (!(1) = 0); for a; b 2 N gcd(a; b) denotes the greatest common divisor of a k and b; for p 2 P ordpn denotes the largest exponent k for which p is a divisor of n: 1 Introduction The present paper is a continuation of the research from [1]. We remind that an artihmetic function F is said to be multiplicative if for every a; b 2 N such that gcd(a; b) = 1 it is 18 fulfilled F (ab) = F (a)F (b) Therefore, F (1) = 1 if F ̸≡ 0: Some classical examples of multiplicative functions that have an importnat meaning in Number Theory are Euler's totient function (the function '), Dedekind's function (the function ); sum of all divisors of a positive integer (the function 휎) and the number of all divisors of a positive integer (the function 휏): When n > 1 these functions admit the following multiplicative representations: Y (︂ 1)︂ '(n) = n 1 − ; p pjn Y (︂ 1)︂ (n) = n 1 + ; p pjn Y p1+ordpn − 1 휎(n) = ; p − 1 pjn Y 휏(n) = (1 + ordpn) ; pjn where p runs over all prime divisors of n: Below we shall consider only the class M of all multiplicative functions with strictly positive values. Our investigation is based on some pairs of multiplicative functions from the class M which have a special property (called in the paper property S). For such pairs we completely decide the question about finding the max {︀f(d)g (︀ n )︀}︀ and min {︀f(d)g (︀ n )︀}︀ * d * d d2Dn d2Dn when n > 1 is a composite number. Since some pairs of classical multiplicative functions (like ('; 휎); ('; ); (휏; 휎);(2!(m); 휎) ) have property S; we apply our theorems to them to obtain some new results. 2 Preliminary results Definition. Let f; g 2 M: We say that the ordered pair (f; g) has the property S when one of the following two cases is fulfilled: (i) 8p 2 P & 8m 2 Z+ f;g def k m−k Hp;m(k) = f(p )g(p ) (1) is an increasing function (not necessarily strictly) with respect to k 2 [0; m] \ Z+ + f;g (ii) 8p 2 P & 8m 2 Z the function Hp;m from (1) is a decreasing function (not neces- sarily strictly) with respect to k 2 [0; m] \ Z+ In [1] the following result is established (see [1, Theorem 5]): 19 Theorem 1. Let f; g 2 M and the pair (f; g) has the property S (satisfying (ii)). If n > 1 is a composite number, then: n (︁n)︁}︁ {︂ (︂n)︂}︂ max f(d)g = max f(p)g (2) * d2Dn d p p n (︁n)︁}︁ {︂ (︂n)︂}︂ min f(d)g = min g(p)f (3) * d2Dn d p p where p runs over all prime divisors of n: Remark 1. If f; g 2 M and the pair (f; g) has the property S (satisfying (i)), then the pair (g; f) has the property S (satisfying (ii)). So we are able to apply Theorem 1 by exchanging the places of f and g. According to Remark 1 the following result is true (see also [1, Theorem 5]): Theorem 2. Let f; g 2 M and the pair (f; g) has the property S (satisfying (i)). If n > 1 is a composite number, then: n (︁n)︁}︁ {︂ (︂n)︂}︂ max f(d)g = max g(p)f (4) * d2Dn d p p n (︁n)︁}︁ {︂ (︂n)︂}︂ min f(d)g = min f(p)g (5) * d2Dn d p p where p runs over all prime divisors of n: 3 Main results Let f; g 2 M and n > 1 be a composite number. Let Y 훼q def n = q ; where 훼q = ordqn (6) qjn be the canonical prime factorization of n; i.e. q runs over all prime divisors of n: Then if p is a prime divisor of n we have: 0 1 0 1 (︂n)︂ B 훼p−1 Y 훼q C (︀ 훼p−1)︀ BY 훼q C f(p)g = f(p)g Bp q C = f(p)g p g B q C = p @ A @ A qjn qjn q6=p q6=p 0 1 0 1 훼p−1 훼p−1 g (p ) 훼 BY 훼 C g (p ) Y 훼 = f(p) g (p p ) g B q q C = f(p) g @ q q A = g (p훼p ) @ A g (p훼p ) qjn qjn q6=p g (p훼p−1) = f(p) g(n) g (p훼p ) From the above and from(2),(3), (4)and (5) we get the first two main results of the paper: 20 Theorem 3. Let n > 1 be a composite number. If f; g 2 M and the pair (f; g) has the property S (satisfying (ii)), then ( ) n (︁n)︁}︁ g (︀p−1+ordpn)︀ max f(d)g = g(n) max f(p) (7) * ordpn d2Dn d p g (p ) ( ) n (︁n)︁}︁ f (︀p−1+ordpn)︀ min f(d)g = f(n) min g(p) (8) * ordpn d2Dn d p f (p ) where p runs over all prime divisors of n: Theorem 4. Let n > 1 be a composite number. If f; g 2 M and the pair (f; g) has the property S (satisfying (i)), then ( ) n (︁n)︁}︁ f (︀p−1+ordpn)︀ max f(d)g = f(n) max g(p) (9) * ordpn d2Dn d p f (p ) ( ) n (︁n)︁}︁ g (︀p−1+ordpn)︀ min f(d)g = g(n) min f(p) (10) * ordpn d2Dn d p g (p ) where p runs over all prime divisors of n: Let n > 1 be a squarefree composite number. In this case, each proper divisor of n is * ** a unitary proper divisor of n: Therefore, Dn coincides with Dn : Then as a corollary from Theorem 3 and Theorem 4 we obtain: Theorem 5. Let f; g 2 M and the pair (f; g) has the property S (satisfying (ii)). Then for every composite squarefree number n > 1 it is fulfilled: n (︁n)︁}︁ {︂f(p)}︂ max f(d)g = g(n) max * d2Dn d p g(p) n (︁n)︁}︁ {︂ g(p) }︂ min f(d)g = f(n) min * d2Dn d p f(p) where p runs over all prime divisors of n: Theorem 6. Let f; g 2 M and the pair (f; g) has the property S (satisfying (i)). Then for every composite squarefree number n > 1 it is fulfilled: n (︁n)︁}︁ {︂ g(p) }︂ max f(d)g = f(n) max * d2Dn d p f(p) n (︁n)︁}︁ {︂f(p)}︂ min f(d)g = g(n) min * d2Dn d p g(p) where p runs over all prime divisors of n: As a corollary from Theorem 5 and Theorem 6 we obtain: 21 Theorem 7. Let f; g 2 M and the pair (f; g) has the property S (satisfying (ii)). Then for every composite squarefree number n > 1 it is fulfilled: {︂f(d)}︂ {︂f(p)}︂ max = max * d2Dn g(d) p g(p) {︂f(d)}︂ f(n) {︂ g(p) }︂ min = min * d2Dn g(d) g(n) p f(p) where p runs over all prime divisors of n: Theorem 8.