New Results on Some Multiplicative Functions

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 multiplicative functions with strictly positive values and more precisely, over the pairs (푓, 푔) of such functions, which have a special property, called in the paper property S. For every such pair (푓, 푔) and for every composite number 푛 > 1, the problem of (︀ 푛 )︀ finding the maximum and minimum of the numbers 푓(푑)푔 푑 , when 푑 runs over all proper divisors of 푛, is completely solved. Since some classical multiplicative functions like Euler’s totient function 휙, Dedekind’s function 휓, the sum of all divisors of 푚, i.e. 휎(푚), the number of all divisors of 푚, i.e. 휏(푚), and 2휔(푚) (where 휔(푚) is the number of all prime divisors of 푚) form pairs having property S, we apply our results to these functions and also resolve the questions of finding the maximum (︀ 푛 )︀ (︀ 푛 )︀ (︀ 푛 )︀ 휔(푑) (︀ 푛 )︀ and minimum of the numbers 휙(푑)휎 푑 , 휙(푑)휓 푑 , 휏(푑)휎 푑 , 2 휎 푑 , where 푑 runs over all proper divisors of 푛. 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 푛 ∈ N 퐷푛 denotes the set of all ** proper divisors of 푛, i.e. different than 1 and 푛; 퐷푛 - the set of all proper unitary divisors of 푛 (i.e. different than 1 and 푛)(see [2]); for 푛 > 1 휔(푛) denotes the number of all prime def divisors of 푛 (휔(1) = 0); for 푎, 푏 ∈ N gcd(푎, 푏) denotes the greatest common divisor of 푎 푘 and 푏; for 푝 ∈ P ord푝푛 denotes the largest exponent 푘 for which 푝 is a divisor of 푛.

1 Introduction

The present paper is a continuation of the research from [1]. We remind that an artihmetic function 퐹 is said to be multiplicative if for every 푎, 푏 ∈ N such that gcd(푎, 푏) = 1 it is

18 fulfilled 퐹 (푎푏) = 퐹 (푎)퐹 (푏) Therefore, 퐹 (1) = 1 if 퐹 ̸≡ 0. Some classical examples of multiplicative functions that have an importnat meaning in Number Theory are Euler’s totient function (the function 휙), Dedekind’s function (the function 휓), 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 푛 > 1 these functions admit the following multiplicative representations:

∏︁ (︂ 1)︂ 휙(푛) = 푛 1 − ; 푝 푝|푛

∏︁ (︂ 1)︂ 휓(푛) = 푛 1 + ; 푝 푝|푛 ∏︁ 푝1+ord푝푛 − 1 휎(푛) = ; 푝 − 1 푝|푛 ∏︁ 휏(푛) = (1 + ord푝푛) , 푝|푛 where 푝 runs over all prime divisors of 푛. 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 {︀푓(푑)푔 (︀ 푛 )︀}︀ and min {︀푓(푑)푔 (︀ 푛 )︀}︀ * 푑 * 푑 푑∈D푛 푑∈D푛 when 푛 > 1 is a composite number. Since some pairs of classical multiplicative functions (like (휙, 휎), (휙, 휓), (휏, 휎),(2휔(푚), 휎) ) have property S, we apply our theorems to them to obtain some new results.

2 Preliminary results

Definition. Let 푓, 푔 ∈ M. We say that the ordered pair (푓, 푔) has the property S when one of the following two cases is fulfilled:

(i) ∀푝 ∈ P & ∀푚 ∈ Z+ 푓,푔 def 푘 푚−푘 퐻푝,푚(푘) = 푓(푝 )푔(푝 ) (1) is an increasing function (not necessarily strictly) with respect to 푘 ∈ [0, 푚] ∩ Z+

+ 푓,푔 (ii) ∀푝 ∈ P & ∀푚 ∈ Z the function 퐻푝,푚 from (1) is a decreasing function (not necessarily strictly) with respect to 푘 ∈ [0, 푚] ∩ Z+ In [1] the following result is established (see [1, Theorem 5]):

19 Theorem 1. Let 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (ii)). If 푛 > 1 is a composite number, then: {︁ (︁푛)︁}︁ {︂ (︂푛)︂}︂ max 푓(푑)푔 = max 푓(푝)푔 (2) * 푑∈D푛 푑 푝 푝

{︁ (︁푛)︁}︁ {︂ (︂푛)︂}︂ min 푓(푑)푔 = min 푔(푝)푓 (3) * 푑∈D푛 푑 푝 푝 where 푝 runs over all prime divisors of 푛.

Remark 1. If 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (i)), then the pair (푔, 푓) has the property S (satisfying (ii)). So we are able to apply Theorem 1 by exchanging the places of 푓 and 푔. According to Remark 1 the following result is true (see also [1, Theorem 5]):

Theorem 2. Let 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (i)). If 푛 > 1 is a composite number, then: {︁ (︁푛)︁}︁ {︂ (︂푛)︂}︂ max 푓(푑)푔 = max 푔(푝)푓 (4) * 푑∈D푛 푑 푝 푝

{︁ (︁푛)︁}︁ {︂ (︂푛)︂}︂ min 푓(푑)푔 = min 푓(푝)푔 (5) * 푑∈D푛 푑 푝 푝 where 푝 runs over all prime divisors of 푛.

3 Main results

Let 푓, 푔 ∈ M and 푛 > 1 be a composite number. Let

∏︁ 훼푞 def 푛 = 푞 , where 훼푞 = ord푞푛 (6) 푞|푛 be the canonical prime factorization of 푛, i.e. 푞 runs over all prime divisors of 푛. Then if 푝 is a prime divisor of 푛 we have: ⎛ ⎞ ⎛ ⎞ (︂푛)︂ ⎜ 훼푝−1 ∏︁ 훼푞 ⎟ (︀ 훼푝−1)︀ ⎜∏︁ 훼푞 ⎟ 푓(푝)푔 = 푓(푝)푔 ⎜푝 푞 ⎟ = 푓(푝)푔 푝 푔 ⎜ 푞 ⎟ = 푝 ⎝ ⎠ ⎝ ⎠ 푞|푛 푞|푛 푞̸=푝 푞̸=푝 ⎛ ⎞ ⎛ ⎞ 훼푝−1 훼푝−1 푔 (푝 ) 훼 ⎜∏︁ 훼 ⎟ 푔 (푝 ) ∏︁ 훼 = 푓(푝) 푔 (푝 푝 ) 푔 ⎜ 푞 푞 ⎟ = 푓(푝) 푔 ⎝ 푞 푞 ⎠ = 푔 (푝훼푝 ) ⎝ ⎠ 푔 (푝훼푝 ) 푞|푛 푞|푛 푞̸=푝 푔 (푝훼푝−1) = 푓(푝) 푔(푛) 푔 (푝훼푝 ) From the above and from(2),(3), (4)and (5) we get the first two main results of the paper:

20 Theorem 3. Let 푛 > 1 be a composite number. If 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (ii)), then {︃ }︃ {︁ (︁푛)︁}︁ 푔 (︀푝−1+ord푝푛)︀ max 푓(푑)푔 = 푔(푛) max 푓(푝) (7) * ord푝푛 푑∈D푛 푑 푝 푔 (푝 )

{︃ }︃ {︁ (︁푛)︁}︁ 푓 (︀푝−1+ord푝푛)︀ min 푓(푑)푔 = 푓(푛) min 푔(푝) (8) * ord푝푛 푑∈D푛 푑 푝 푓 (푝 ) where 푝 runs over all prime divisors of 푛.

Theorem 4. Let 푛 > 1 be a composite number. If 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (i)), then {︃ }︃ {︁ (︁푛)︁}︁ 푓 (︀푝−1+ord푝푛)︀ max 푓(푑)푔 = 푓(푛) max 푔(푝) (9) * ord푝푛 푑∈D푛 푑 푝 푓 (푝 )

{︃ }︃ {︁ (︁푛)︁}︁ 푔 (︀푝−1+ord푝푛)︀ min 푓(푑)푔 = 푔(푛) min 푓(푝) (10) * ord푝푛 푑∈D푛 푑 푝 푔 (푝 ) where 푝 runs over all prime divisors of 푛.

Let 푛 > 1 be a squarefree composite number. In this case, each proper divisor of 푛 is * ** a unitary proper divisor of 푛. Therefore, D푛 coincides with D푛 . Then as a corollary from Theorem 3 and Theorem 4 we obtain:

Theorem 5. Let 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (ii)). Then for every composite squarefree number 푛 > 1 it is fulfilled:

{︁ (︁푛)︁}︁ {︂푓(푝)}︂ max 푓(푑)푔 = 푔(푛) max * 푑∈D푛 푑 푝 푔(푝)

{︁ (︁푛)︁}︁ {︂ 푔(푝) }︂ min 푓(푑)푔 = 푓(푛) min * 푑∈D푛 푑 푝 푓(푝) where 푝 runs over all prime divisors of 푛.

Theorem 6. Let 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (i)). Then for every composite squarefree number 푛 > 1 it is fulfilled:

{︁ (︁푛)︁}︁ {︂ 푔(푝) }︂ max 푓(푑)푔 = 푓(푛) max * 푑∈D푛 푑 푝 푓(푝)

{︁ (︁푛)︁}︁ {︂푓(푝)}︂ min 푓(푑)푔 = 푔(푛) min * 푑∈D푛 푑 푝 푔(푝) where 푝 runs over all prime divisors of 푛.

As a corollary from Theorem 5 and Theorem 6 we obtain:

21 Theorem 7. Let 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (ii)). Then for every composite squarefree number 푛 > 1 it is fulfilled:

{︂푓(푑)}︂ {︂푓(푝)}︂ max = max * 푑∈D푛 푔(푑) 푝 푔(푝)

{︂푓(푑)}︂ 푓(푛) {︂ 푔(푝) }︂ min = min * 푑∈D푛 푔(푑) 푔(푛) 푝 푓(푝) where 푝 runs over all prime divisors of 푛.

Theorem 8. Let 푓, 푔 ∈ M and the pair (푓, 푔) has the property S (satisfying (i)). Then for every composite squarefree number 푛 > 1 it is fulfilled:

{︂푓(푑)}︂ 푓(푛) {︂ 푔(푝) }︂ max = max * 푑∈D푛 푔(푑) 푔(푛) 푝 푓(푝)

{︂푓(푑)}︂ {︂푓(푝)}︂ min = min * 푑∈D푛 푔(푑) 푝 푔(푝) where 푝 runs over all prime divisors of 푛.

Other corollaries from Theorem 3 and Theorem 4, are obtained below for the case when 푔(푚) = 푚 ∀푚 ∈ N :

Theorem 9. Let 푓 ∈ M and the pair (푓, 푔), with 푔(푚) = 푚 ∀푚 ∈ N, has the property S (satisfying (ii)). Then for every composite number 푛 > 1 it is fulfilled:

{︂푓(푑)}︂ {︂푓(푝)}︂ max = max * 푑∈D푛 푑 푝 푝

⎧ (︁ 푛 )︁⎫ {︂푓(푑)}︂ ⎨푓 푝 ⎬ min = min 푛 푑∈ * 푝 D푛 푑 ⎩ 푝 ⎭ where 푝 runs over all prime divisors of 푛.

Theorem 10. Let 푓 ∈ M and the pair (푓, 푔), with 푔(푚) = 푚 ∀푚 ∈ N, has the property S (satisfying (i)). Then for every composite number 푛 > 1 it is fulfilled:

⎧ (︁ 푛 )︁⎫ {︂푓(푑)}︂ ⎨푓 푝 ⎬ max = max 푛 푑∈ * 푝 D푛 푑 ⎩ 푝 ⎭

{︂푓(푑)}︂ {︂푓(푝)}︂ min = min * 푑∈D푛 푑 푝 푝 where 푝 runs over all prime divisors of 푛.

22 We shall apply (7) in the particular case: 푓 = 휙, 푔 = 휎. In this case one may verify that the pair (휙, 휎) has the property S (satisfying (ii)) and obtain: {︃ }︃ {︁ (︁푛)︁}︁ 휎 (︀푝−1+ord푝푛)︀ max 휙(푑)휎 = 휎(푛) max 휙(푝) , (11) * ord푝푛 푑∈D푛 푑 푝 휎 (푝 )

where 푝 runs over all prime divisors of 푛. Since 푝훼푝 − 1 푝훼푝+1 − 1 휙(푝) = 푝 − 1; 휎 (︀푝−1+훼푝 )︀ = ; 휎 (푝훼푝 ) = , 푝 − 1 푝 − 1 def where 훼푝 = ord푝푛, (11) yields:

{︁ (︁푛)︁}︁ {︂ 푝훼푝 − 1 }︂ max 휙(푑)휎 = 휎(푛) max (푝 − 1) , (12) * 훼푝+1 푑∈D푛 푑 푝 푝 − 1

where 푝 runs over all prime divisors of 푛. Using the identity

푝훼푝 − 1 1 (︂ (푝 − 1)2 )︂ (푝 − 1) = 1 − 1 + 푝훼푝+1 − 1 푝 푝훼푝+1 − 1

we may rewrite (12) in the form:

{︁ (︁푛)︁}︁ (︂ {︂1 (︂ (푝 − 1)2 )︂}︂)︂ max 휙(푑)휎 = 휎(푛) 1 − min 1 + , * 1+ord푝푛 푑∈D푛 푑 푝 푝 푝 − 1

where 푝 runs over all prime divisors of 푛. Hence it is proved that:

{︁ (︁푛)︁}︁ (︂ {︂1 (︂ 휙(푝) )︂}︂)︂ max 휙(푑)휎 = 휎(푛) 1 − min 1 + , (13) * ord푝푛 푑∈D푛 푑 푝 푝 휎(푝 )

where 푝 runs over all prime divisors of 푛. Also it is easy to see that the identity:

푝훼푝 − 1 (︂ 1)︂ (︂ 1 )︂ (푝 − 1) = 1 − 1 − 푝훼푝+1 − 1 푝 휎(푝ord푝푛)

holds. From the above identity and (12) it follows that:

{︁ (︁푛)︁}︁ {︂(︂ 1)︂ (︂ 1 )︂}︂ max 휙(푑)휎 = 휎(푛) max 1 − 1 − (14) * ord푝푛 푑∈D푛 푑 푝 푝 휎(푝 ) where 푝 runs over all prime divisors of 푛. Since (13) and (14) are true, we may consider as proved the following third main result of the present paper:

23 Theorem 11. Let 푛 > 1 be a composite number. Then the following two representations are fulfilled: {︁ (︁푛)︁}︁ (︂ {︂1 (︂ 휙(푝) )︂}︂)︂ max 휙(푑)휎 = 휎(푛) 1 − min 1 + (*) * ord푝푛 푑∈D푛 푑 푝 푝 휎(푝 )

{︁ (︁푛)︁}︁ {︂(︂ 1)︂ (︂ 1 )︂}︂ max 휙(푑)휎 = 휎(푛) max 1 − 1 − , (**) * ord푝푛 푑∈D푛 푑 푝 푝 휎(푝 ) where 푝 runs over all prime divisors of 푛. Corollary 1. Let 푛 > 1 be a composite number.If 푑* is an arbitrary proper unitary divisor of 푛, then the inequalities 휙(푑*) {︂1 (︂ 휙(푝) )︂}︂ ≤ 1 − min 1 + 휎(푑*) 푝 푝 휎(푝ord푝푛) 휙(푑*) {︂(︂ 1)︂ (︂ 1 )︂}︂ ≤ max 1 − 1 − , 휎(푑*) 푝 푝 휎(푝ord푝푛) where 푝 runs over all prime divisors of 푛, hold. Proof. The assertion follows from (*) and (**) because of the relation:

(︀ 푛 )︀ 휎 * 1 푑 = , 휎(푛) 휎(푑*) which is true since 푑* is unitary divisor of 푛. Corollary 2. Under the conditions of Corollary 1 the following two inequalities hold: 휙(푑) {︂1 (︂ 휙(푝) )︂}︂ max ≤ 1 − min 1 + ** ord푝푛 푑∈D푛 휎(푑) 푝 푝 휎(푝 ) 휙(푑) {︂(︂ 1)︂ (︂ 1 )︂}︂ max ≤ max 1 − 1 − , ** ord푝푛 푑∈D푛 휎(푑) 푝 푝 휎(푝 ) where 푝 runs over all prime divisors of 푛.

3.1 Further applications of the results Let 푛 > 1 be a composite number and let us choose again: 푓 = 휙, 푔 = 휎. Since for a prime 푝 we have: 푝 + 1 2 푝 + 1 1 휎(푝) = 푝 + 1; = 1 + ; = 1 + ; 푝 − 1 푝 − 1 푝 푝 휙(푝훼푝−1) {︂ 1 for 훼 = 1, = 푝−1 푝 , 훼푝 1 휙(푝 ) 푝 for 훼푝 > 1 def where 훼푝 = ord푝푛, it is easy to observe that

휙(푝−1+ord푝푛) {︂ 1 + 2 for ord 푛 = 1, 휎(푝) = 푝−1 푝 ord푝푛 1 휙(푝 ) 1 + 푝 for ord푝푛 > 1

24 Let 푝′ be the largest number among all prime divisors 푝 of 푛, satisfying the condition ′′ ord푝푛 = 1 and 푝 be the largest number among all prime divisors 푝 of 푛, satisfying the condition ord푝푛 > 1. Then putting (︂ 2 1 )︂ 훿 def= min 1 + , 1 + (15) 1 푝′ − 1 푝′′ it is trivial to see that {︂ 휙(푝−1+ord푝푛)}︂ 훿1 = min 휎(푝) (16) 푝 휙(푝ord푝푛) where 푝 runs over all prime divisors of 푛. Since the pair (휙, 휎) has the property S (satisfying (ii)), (8) and (16) yield {︁ (︁푛)︁}︁ min 휙(푑)휎 = 훿1휙(푛) (17) * 푑∈D푛 푑

Here we remind that max {︀휙(푑)휎 (︀ 푛 )︀}︀ was found with the help of Theorem 11 (see (*) * 푑 푑∈D푛 and (**)). Let 휓 be the Dedekind’s function. Since 휓 ∈ M and for a prime 푝 we have 휓(푝) = 휎(푝) = 푝 + 1, using the fact that the pair (휙, 휓) has the property S (satisfying (ii)), we obtain that for a composite 푛 > 1 the representation {︁ (︁푛)︁}︁ min 휙(푑)휓 = 훿1휙(푛) (18) * 푑∈D푛 푑 holds. 푛 Let 푑 be a proper unitary divisor of 푛. Then 푑 is also a proper unitary divisor of 푛 and we have (︁푛)︁ 휎 (︀ 푛 )︀ 휙(푑)휎 = 푑 (19) 푑 (︀ 푛 )︀ 휙 푑 푛 Therefore, when 푑 runs over all proper unitary divisors of 푛, so does 푑 and (17), (18) and (19) yield for every proper unitary divisor 푑 of 푛 the inequality:

(︂휎 (푑) 휓 (푑))︂ min , ≥ 훿 휙(푑) 휙(푑) 1 Using the facts that for a prime 푝 : 푝 − 1 2 푝 − 1 1 휙(푝) = 푝 − 1; = 1 + ; = 1 − ; 푝 + 1 푝 + 1 푝 푝

휓(푝ord푝푛−1) {︂ 1 for ord 푛 = 1, = 푝+1 푝 , ord푝푛 1 휓(푝 ) 푝 for ord푝푛 > 1 we find 휓(푝−1+ord푝푛) {︂ 1 − 2 for ord 푛 = 1, 휙(푝) = 푝+1 푝 ord푝푛 1 휓(푝 ) 1 − 푝 for ord푝푛 > 1

25 The above equality yields:

{︂ 휓(푝−1+ord푝푛)}︂ 훿2 = max 휙(푝) , 푝 휓(푝ord푝푛)

def (︁ 2 1 )︁ where 푝 runs over all prime divisors of 푛 and 훿2 = max 1 − 푝′+1 , 1 − 푝′′ . The last equality and (7) (for 푓 = 휙, 푔 = 휓) yield: {︁ (︁푛)︁}︁ max 휙(푑)휓 = 훿2휓(푛) * 푑∈D푛 푑 The last equality and (18) prove the following fourth main result in the paper: Theorem 12. Let 푛 > 1 be a composite number. Then it is fulfilled: {︁ (︁푛)︁}︁ max 휙(푑)휓 = 훿2휓(푛) * 푑∈D푛 푑 {︁ (︁푛)︁}︁ min 휙(푑)휓 = 훿1휙(푛) * 푑∈D푛 푑 Corollary 3. For every proper unitary divisor 푑 of 푛 it is fulfilled (︁푛)︁ 훿 휙(푛) ≤ 휙(푑)휓 ≤ 훿 휓(푛) 1 푑 2 Moreover, there exist values of 푑 for which the left and right inequalities become equalities. Corollary 4. Let 푛 > 1 be a composite number and 푑 be an arbitrary proper unitary divisor of 푛. Then the inequalities: 휓(푑) ≥ 훿 휙(푑) 1 휙(푑) ≤ 훿 휓(푑) 2 hold. Remark 2. We must note that if 푛 > 1 is a composite number such that for every prime 2 2 divisor 푝 of 푛 the condition ord푝푛 = 1 is satisfied, then 훿1 = 1 + 푝′−1 and 훿2 = 1 − 1+푝′ . Remark 3. If 푛 > 1 is a composite number such that for every prime divisor 푝 of 푛 the 1 1 condition ord푝푛 > 1 is satisfied, then 훿1 = 1 + 푝′′ and 훿2 = 1 − 푝′′ . From the above remarks it follows: Corollary 5. Let 푛 > 1 be a composite square-free number. Then for every proper divisor 푑 of 푛 it is fulfilled (︂ 2 )︂ (︁푛)︁ (︂ 2 )︂ 1 + 휙(푛) ≤ 휙(푑)휓 ≤ 1 − 휓(푛) 푝 − 1 푑 푝 + 1 where 푝 is the largest prime divisor of 푛. Moreover, there exist values of 푑 for which the left and right inequalities become equalities.

26 Corollary 6. Let 푛 > 1 be a composite number such that if 푞 is prime divisor of 푛, 푞2 is also a divisor of 푛. Then for every proper divisor 푑 of 푛 it is fulfilled

(︂ 1)︂ (︁푛)︁ (︂ 1)︂ 1 + 휙(푛) ≤ 휙(푑)휓 ≤ 1 − 휓(푛) 푝 푑 푝

where 푝 is the largest prime divisor of 푛. Moreover, there exist values of 푑 for which the left and right inequalities become equalities.

One may conjecture: “For every composite 푛 > 1 max 휙(푑)휎 (︀ 푛 )︀ is reached for 푑 = 푝*, where 푝* is the * 푑 푑∈D푛 maximal prime divisor of 푛.” Indeed this is true for infinitely many 푛, but unfortunately the conjecture is untrue for infinitely many values of 푛 too. Below we construct one possible counter-example. Let 푝 > 3 be prime and 푞 is a prime number for which 푝 < 푞 ≤ 2푝 − 3. Such prime 푞 always exists according to Bertrand’s postulate (see e.g. [3]): “For 푝 > 3, there always exists a prime number 푞 between 푝 and 2푝−2 (i.e. 푞 ∈ (푝, 2푝−2)).” In this form Bertrand’s postulate follows immediately from Nagura’s result (see [4]): “For every 푝 ≥ 25, there 6 always exists a prime 푞 ∈ (푝, 5 푝).” It is clear that when 푝 runs over all primes greater than 3, we obtain infinitely many pairs (푝, 푞) like the described above. For every such pair (푝, 푞) 2 * we put 푛 = 푝 푞. Then all elements of the set 퐷푛 ordered by increasing magnitude are: 푝, 푞, 푝2, 푝푞. Let 푞 = 푝 + 푥. Then it is easy to see that the inequality

(︂푛)︂ (︂푛)︂ 휙(푝)휎 > 휙(푞)휎 (20) 푝 푞

is equivalent to 푝2 − 푝 푥 < . 푝 + 2 푝2−푝 And since we have, 푥 ≤ 푝 − 3, and 푝 − 3 < 푝+2 , then (20) holds. Also a direct check shows that the inequalities:

(︂푛)︂ (︂ 푛 )︂ (︂푛)︂ (︂ 푛 )︂ 휙(푝)휎 > 휙(푝2)휎 ; 휙(푝)휎 > 휙(푝푞)휎 푝 푝2 푝 푝푞

hold too. Therefore, max 휙(푑)휎 (︀ 푛 )︀ is reached for 푑 = 푝, i.e. for the smallest prime divisor * 푑 푑∈D푛 of 푛, while in the considered case 푝* = 푞. It is easy to observe that min 휙(푑)휎 (︀ 푛 )︀ in each one of the above infinitely many counter * 푑 푑∈D푛 푛 examples is reached for 푑 = 푝 and based on that the following conjecture can be formulated: “For every composite 푛 > 1 if max 휙(푑)휎 (︀ 푛 )︀ is reached for 푑 = 푑,^ then min 휙(푑)휎 (︀ 푛 )︀ * 푑 * 푑 푑∈D푛 푑∈D푛 is reached at 푑 = 푛 .” 푑^ Unfortunately, this conjecture is also untrue. One may verify that a counter-example is every 푛 given by: 푛 = 푝2(2푝 − 1), where the numbers 푝 > 3 and 2푝 − 1 are simultaneously prime.

27 Let us consider the classical arithmetic function 휏 given by: ∑︁ 휏(푚) = 1 푑|푚

where 푑 runs over all divisors of 푚. Then 휏(푚) coincides with the number of all divisors of 푚. Moreover, 휏 ∈ M. It can be checked directly that the pair (휏, 휎) has the property S (satisfying (ii)). Therefore, we may substitute 푓 = 휏, 푔 = 휎 and applying (7) and (8) obtain: {︃ }︃ {︁ (︁푛)︁}︁ 휎 (︀푝−1+ord푝푛)︀ max 휏(푑)휎 = 휎(푛) max 휏(푝) (21) * ord푝푛 푑∈D푛 푑 푝 휎 (푝 ) {︃ }︃ {︁ (︁푛)︁}︁ 휏 (︀푝−1+ord푝푛)︀ min 휏(푑)휎 = 휏(푛) min 휎(푝) (22) * ord푝푛 푑∈D푛 푑 푝 휏 (푝 ) where 푝 runs over all prime divisors of 푛. Since 휏(푝) = 2 and

(︀ −1+ord 푛)︀ 휎 푝 푝 푝ord푝푛 − 1 1 (︂ 푝 − 1 )︂ 1 (︂ 1 )︂ = = 1 − = 1 − (23) 휎 (푝ord푝푛) 푝1+ord푝푛 − 1 푝 푝1+ord푝푛 − 1 푝 휎 (푝ord푝푛)

we may rewrite (21) in the form

{︁ (︁푛)︁}︁ {︂1 (︂ 1 )︂}︂ max 휏(푑)휎 = 2휎(푛) max 1 − (24) * ord푝푛 푑∈D푛 푑 푝 푝 휎 (푝 ) where 푝 runs over all prime divisors of 푛. Since 휎(푝) = 푝 + 1 and

(︀ −1+ord푝푛)︀ (︀ ord푝푛)︀ 휏 푝 = ord푝푛; 휏 푝 = 1 + ord푝푛

we may rewrite (22) in the form

{︁ (︁푛)︁}︁ {︂ (︂ 1 )︂}︂ min 휏(푑)휎 = 휏(푛) min (푝 + 1) 1 − (25) * 푑∈D푛 푑 푝 1 + ord푝푛 where 푝 runs over all prime divisors of 푛. Then (24) and (25) yield the fifth main result of the paper:

Theorem 13. Let 푛 > 1 be a composite number. Then the relations:

{︁ (︁푛)︁}︁ {︂1 (︂ 1 )︂}︂ max 휏(푑)휎 = 2휎(푛) max 1 − * ord푝푛 푑∈D푛 푑 푝 푝 휎 (푝 )

{︁ (︁푛)︁}︁ {︂ (︂ 1 )︂}︂ min 휏(푑)휎 = 휏(푛) min (푝 + 1) 1 − , * 푑∈D푛 푑 푝 1 + ord푝푛 where 푝 runs over all prime divisors of 푛, hold.

28 훼1 훼2 Corollary 7. Let 푛 = 푝1 푝2 , where 푝1, 푝2 ∈ P, 푝1 ̸= 푝2 and 훼1, 훼2 ∈ N. Then: {︁ (︁푛)︁}︁ (︂ 1 (︂ 1 )︂ 1 (︂ 1 )︂)︂ max 휏(푑)휎 = 2휎(푛) max 1 − , 1 − ; * 훼1 훼2 푑∈D푛 푑 푝1 휎 (푝1 ) 푝2 휎 (푝2 ) {︁ (︁푛)︁}︁ (︂ (︂ 1 )︂ (︂ 1 )︂)︂ min 휏(푑)휎 = 휏(푛) min (푝1 + 1) 1 − , (푝2 + 1) 1 − * 푑∈D푛 푑 훼1 + 1 훼2 + 1

It is well known that 2휔(푚) is multiplicative function. Since 2휔(푚) ∈ M, it is easy to verify that the pair (2휔(푚), 휎) has the property S (satisfying (ii)). Let 푛 > 1 be a composite number. Then applying (7) with 푓 = 2휔(푚) and 푔 = 휎 we obtain: {︃ }︃ {︁ (︁푛)︁}︁ 휎 (︀푝−1+ord푝푛)︀ max 2휔(푑)휎 = 휎(푛) max 2휔(푝) , * ord푝푛 푑∈D푛 푑 푝 휎 (푝 ) where 푝 runs over all prime divisors of 푛. Hence: {︃ }︃ {︁ (︁푛)︁}︁ 휎 (︀푝−1+ord푝푛)︀ max 2휔(푑)휎 = 2휎(푛) max (26) * ord푝푛 푑∈D푛 푑 푝 휎 (푝 )

(where 푝 runs over all prime divisors of 푛), since 휔(푝) = 1. Since 휏(푝) = 2 (21) yields {︃ }︃ {︁ (︁푛)︁}︁ 휎 (︀푝−1+ord푝푛)︀ max 휏(푑)휎 = 2휎(푛) max , * ord푝푛 푑∈D푛 푑 푝 휎 (푝 )

where 푝 runs over all prime divisors of 푛. Comparing the last equality with (26) we conclude that: {︁ (︁푛)︁}︁ {︁ (︁푛)︁}︁ max 휏(푑)휎 = max 2휔(푑)휎 (27) * * 푑∈D푛 푑 푑∈D푛 푑 Also, from (23) and (26) we obtain {︁ (︁푛)︁}︁ {︂1 (︂ 1 )︂}︂ max 2휔(푑)휎 = 2휎(푛) max 1 − , (28) * ord푝푛 푑∈D푛 푑 푝 푝 휎 (푝 ) where 푝 runs over all prime divisors of 푛. Putting in (8) 푓 = 2휔(푚) and 푔 = 휎 we obtain

{︃ −1+ord푝푛 }︃ {︁ (︁푛)︁}︁ 2휔(푝 ) min 2휔(푑)휎 = 2휔(푛) min (푝 + 1) , * 휔 푝ord푝푛 푑∈D푛 푑 푝 2 ( ) where 푝 runs over all prime divisors of 푛. Hence:

{︁ (︁푛)︁}︁ {︂ 2휈푛(푝) }︂ min 2휔(푑)휎 = 2휔(푛) min (푝 + 1) (29) * 푑∈D푛 푑 푝 2 (where 푝 runs over all prime divisors of 푛 ), where {︂ def 0 for ord푝푛 = 1, 휈푛(푝) = 1 for ord푝푛 > 1

29 Let * 푝 = {푝 : 푝 ∈ P, 푝|푛 & ord푝푛 > 1} ** 푝 = {푝 : 푝 ∈ P, 푝|푛 & ord푝푛 = 1} Then (27), (28) and (29) prove the sixth main result of this paper:

Theorem 14. Let 푛 > 1 be a composite number. Then the following relations hold:

{︁ (︁푛)︁}︁ {︂1 (︂ 1 )︂}︂ max 2휔(푑)휎 = 2휎(푛) max 1 − * ord푝푛 푑∈D푛 푑 푝 푝 휎 (푝 )

(where 푝 runs over all prime divisors of 푛);

{︁ (︁푛)︁}︁ (︂ 1 + 푝** )︂ min 2휔(푑)휎 = 2휔(푛) min 1 + 푝* , ; * 푑∈D푛 푑 2 {︁ (︁푛)︁}︁ {︁ (︁푛)︁}︁ max 휏(푑)휎 = max 2휔(푑)휎 . * * 푑∈D푛 푑 푑∈D푛 푑 We conclude this paper with a demonstration of the effectiveness of the established results by providing a simple example. 999 9999 * Let 푛 = 2 3 . Then the set 퐷푛 has exactly 9999998 in number elements. That means that the number of all proper divisors of 푛 is 9999998. Now it is clear that to find min {︀휏(푑)휎 (︀ 푛 )︀}︀ is not a simple problem. But after using the second formula of the * 푑 푑∈D푛 Corollary 7, we find immediately that

{︁ (︁푛)︁}︁ (︂ (︂ (︂ 1 )︂ (︂ 1 )︂)︂)︂ min 휏(푑)휎 = 107 min 3 1 − , 4 1 − = 29970000 * 푑∈D푛 푑 1000 10000

References

[1] Vassilev-Missana, V. Some Results on Multiplicative Functions. Notes on Number The- ory and Discrete Mathematics, Vol. 16, 2010, No. 4, 29-40

[2] Weisstein, Eric W. “Unitary Divisor.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/UnitaryDivisor.html

[3] Sondow, Jonathan and Weisstein, Eric W. “Bertrand’s Postulate.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/BertrandsPostulate.html

[4] Nagura, J. “On the interval containing at least one prime number”. Proc. Japan Acad. Vol. 28, 1952, No.4, 177-181