Notes on and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 24, 2018, No. 3, 68–76 DOI: 10.7546/nntdm.2018.24.3.68-76

The arithmetic and Leibniz-additive functions

Pentti Haukkanen1, Jorma K. Merikoski1 and Timo Tossavainen2

1 Faculty of Natural Sciences FI-33014 University of Tampere, Finland e-mails: [email protected], [email protected] 2 Department of Arts, Communication and Education Lulea University of Technology, SE-97187 Lulea, Sweden e-mail: [email protected]

Received: 16 May 2018 Accepted: 19 August 2018

Abstract: An 푓 is Leibniz-additive if there is a completely multiplicative function ℎ푓 such that

푓(푚푛) = 푓(푚)ℎ푓 (푛) + 푓(푛)ℎ푓 (푚) for all positive 푚 and 푛. A motivation for the present study is the fact that Leibniz- additive functions are generalizations of the arithmetic derivative 퐷; namely, 퐷 is Leibniz- additive with ℎ퐷(푛) = 푛. We study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function 푓 is totally determined by the values of 푓 and

ℎ푓 at primes. We also find connections of Leibniz-additive functions to the usual product, compo- sition and Dirichlet convolution of arithmetic functions. The arithmetic partial derivative is also considered. Keywords: Arithmetic derivative, Arithmetic partial derivative, Arithmetic function, Completely additive function, Completely multiplicative function, Leibniz rule, Dirichlet convolution. 2010 Mathematics Subject Classification: 11A25, 11A41.

68 1 Introduction

Let 푛 be a positive . Its arithmetic derivative 퐷(푛) = 푛′ is defined as follows: (i) 푝′ = 1 for all primes 푝, (ii) (푚푛)′ = 푚푛′ + 푚′푛 for all positive integers 푚 and 푛. Given ∏︁ 푛 = 푞휈푞(푛), 푞∈P where P is the set of primes, the formula for computing the arithmetic derivative of 푛 is (see, e.g., [2, 13]) ∑︁ 휈푝(푛) 푛′ = 푛 . 푝 푝∈P A brief summary on the history of arithmetic derivative and its generalizations to other number sets can be found, e.g., in [2, 13, 5]. Similarly, one can define the arithmetic partial derivative (see, e.g., [7, 5]) via

휈 (푛) 퐷 (푛) = 푛′ = 푝 푛, 푝 푝 푝 and the arithmetic logarithmic derivative [13] as

퐷(푛) ld(푛) = . 푛 An arithmetic function 푓 is said to be additive if 푓(푚푛) = 푓(푚)+푓(푛), whenever gcd(푚, 푛) = 1, and multiplicative if 푓(1) = 1 and 푓(푚푛) = 푓(푚)푓(푛), whenever gcd(푚, 푛) = 1. Additive and multiplicative functions are totally determined by their values at prime powers. Further, an arithmetic function 푓 is said to be completely additive if 푓(푚푛) = 푓(푚) + 푓(푛) for all posi- tive integers 푚 and 푛, and completely multiplicative if 푓(1) = 1 and 푓(푚푛) = 푓(푚)푓(푛) for all positive integers 푚 and 푛. These functions are widely studied in the literature, see, e.g., [1, 6, 8, 9, 10, 11, 12]. We say that an arithmetic function 푓 is Leibniz-additive (or, L-additive, in short) if there is a

completely multiplicative function ℎ푓 such that

푓(푚푛) = 푓(푚)ℎ푓 (푛) + 푓(푛)ℎ푓 (푚) (1)

for all positive integers 푚 and 푛. Then 푓(1) = 0, since ℎ푓 (1) = 1. The property (1) may be considered a generalized Leibniz rule. This terminology arises from the observation that the

arithmetic derivative 퐷 is L-additive with ℎ퐷(푛) = 푛; it satisfies the usual Leibniz rule

퐷(푚푛) = 퐷(푚)푛 + 퐷(푛)푚

for all positive integers 푚 and 푛, and the function ℎ퐷(푛) = 푛 is completely multiplicative. In

addition, the arithmetic partial derivative with respect to the prime 푝 is L-additive with ℎ퐷푝 (푛) =

69 푛. Further, all completely additive functions 푓 are L-additive with ℎ푓 (푛) = 1. For example, the logarithmic derivative of 푛 is completely additive, since

ld(푚푛) = ld(푚) + ld(푛).

The term “L-additive function” seems to be new in the literature, yet Chawla [3] has defined the concept of a completely distributive arithmetic function meaning the same as we do with an L-additive function. However, this is a somewhat misleading term, since a distributive arithmetic function usually refers to the property that

푓(푢 * 푣) = (푓푢) * (푓푣), (2) i.e., the function 푓 distributes over the Dirichlet convolution. This is satisfied by completely multiplicative arithmetic functions, not by completely distributive functions as Chawla defined them. An arithmetic function 푓 is said to generalized additive [4] if there is a multiplicative func- tion ℎ푓 such that (1) holds for all positive integers 푚 and 푛 with gcd(푚, 푛) = 1. Chawla [3] refers to these functions as distributive arithmetic functions. Some properties of these functions are presented in [3, 4]. It is clear that L-additive functions is a subclass of the class of general- ized additive functions, and, in this sense, we could refer to L-additive functions as generalized complete additive functions. We, however, prefer the term “L-additive function”, since our main purpose is to attain new aspects on the arithmetic derivative and the arithmetic partial derivative. For the same reason, in this paper, we do not consider wider classes of arithmetic functions such as generalized additive functions, or multiplicative analogues of the present concepts. In this paper, we consider L-additive functions especially from the viewpoint that they are generalizations of the arithmetic derivative and the arithmetic partial derivative. In Section 2, we study basic properties of L-additive functions and thereby provide some new insight into the arithmetic derivative and the arithmetic partial derivative. In Section 3, we study L-additivity, the arithmetic derivative and the arithmetic partial derivative in terms of the Dirichlet convolution.

2 Basic properties

Theorem 2.1. Let 푓 be an arithmetic function. If 푓 is L-additive and ℎ푓 is nonzero-valued, then

푓/ℎ푓 is completely additive. Conversely, if there is a completely multiplicative nonzero-valued function ℎ such that 푓/ℎ is completely additive, then 푓 is L-additive and ℎ푓 = ℎ.

Proof. If 푓 satisfies (1) and ℎ푓 is never zero, then

푓(푚푛) 푓(푚)ℎ (푛) + 푓(푛)ℎ (푚) 푓(푚) 푓(푛) = 푓 푓 = + , ℎ푓 (푚푛) ℎ푓 (푚)ℎ푓 (푛) ℎ푓 (푚) ℎ푓 (푛)

verifying the first part. The second part follows by substituting ℎ푓 = ℎ in the above and exchang- ing the sides of the second equation.

70 Theorem 2.2. Let 푓 be an arithmetic function. If 푓 is L-additive and ℎ푓 is nonzero-valued, then

푓 = 푔푓 ℎ푓 ,

where 푔푓 is completely additive. Conversely, if 푓 is of the form

푓 = 푔ℎ, (3) where 푔 is completely additive and ℎ is completely multiplicative, then 푓 is L-additive with ℎ푓 = ℎ.

Proof. If 푓 is L-additive such that ℎ푓 (푛) ̸= 0 for all positive integers 푛, then denoting 푔푓 = 푓/ℎ푓

in Theorem 2.1 we obtain 푓 = 푔푓 ℎ푓 , where 푔푓 is completely additive. Conversely, assume that 푓 is of the form (3). Then

푓(푚푛) = ℎ(푚)ℎ(푛)[푔(푚) + 푔(푛)] = (ℎ푔)(푚)ℎ(푛) + (ℎ푔)(푛)ℎ(푚) = 푓(푚)ℎ(푛) + 푓(푛)ℎ(푚).

Example 2.1. If an arithmetic function 푓 satisfies the Leibniz rule

푓(푚푛) = 푓(푚)푛 + 푓(푛)푚 (4) for all positive integers 푚, 푛, then 푓(푛) = 푔푓 (푛)푛, where 푔푓 is completely additive. Also the

converse holds. In particular, the arithmetic derivative 퐷 and the arithmetic partial derivative 퐷푝 satisfy (4).

Theorem 2.2 shows that each L-additive function 푓 such that ℎ푓 is always nonzero can be

represented as a pair of a completely additive function 푔푓 and a completely multiplicative function

ℎ푓 . In this case, we write 푓 = (푔푓 , ℎ푓 ). However, if ℎ푓 (푝) = 0 for some prime 푝 and 푓(푝) is

nonzero, then there is not any function 푔푓 such that 푓 = 푔푓 ℎ푓 and, consequently, a representation of this kind is not possible. On the other hand, the representation of an L-additive function as such a pair is not always

unique either. For instance, if 푓 and 푔푓 are identically zero, then ℎ푓 may be any completely multiplicative function. The next theorem will however show that this representation is unique

whenever 푓 and ℎ푓 are nonzero for all primes. Observe that the latter condition is equivalent to

assuming that ℎ푓 is nonzero for all positive integers.

Theorem 2.3. If 푓 is L-additive such that 푓(푝) ̸= 0 and ℎ푓 (푝) ̸= 0 for all primes 푝, then the

representation 푓 = (푔푓 , ℎ푓 ) is unique. ˜ Proof. Let 푓 = (푔푓 , ℎ푓 ) = (˜푔푓 , ℎ푓 ). Then for all primes 푝, ˜ 푓(푝) = 푔푓 (푝)ℎ푓 (푝) =푔 ˜푓 (푝)ℎ푓 (푝).

On the other hand, by the definition of L-additivity,

2 ˜ 푓(푝 ) = 2푓(푝)ℎ푓 (푝) = 2푓(푝)ℎ푓 (푝), ˜ which implies that ℎ푓 (푝) = ℎ푓 (푝) and, consequently, 푔푓 (푝) =푔 ˜푓 (푝).

71 Example 2.2. The arithmetic derivative can be represented as 퐷 = (푔퐷, ℎ퐷), where 푔퐷(푝) =

1/푝 = ld(푝) and ℎ퐷(푝) = 푝 for all primes 푝. In other words, the arithmetic derivative has the representation as the pair of the logarithmic derivative and the identity function. Similarly, for the arithmetic partial derivative with respect to the prime 푝, we have 퐷푝 = (푔퐷푝 , ℎ퐷푝 ), where

푔퐷푝 (푞) = 0 if 푞 ̸= 푝, 푔퐷푝 (푝) = 1/푝, and ℎ퐷푝 (푞) = 푞 for all primes 푞 (that is, ℎ퐷푝 is the identity function).

Completely additive and completely multiplicative functions are totally determined by their values at primes as follows. Let

푛1 푛2 푛푠 푛 = 푞1푞2 ··· 푞푟 = 푝1 푝2 ··· 푝푠 , (5) where 푞1, 푞2, . . . , 푞푟 are primes and 푝1, 푝2, . . . , 푝푠 are distinct primes. If 푓 is completely additive, then 푓(1) = 0 and 푟 푠 ∑︁ ∑︁ 푓(푛) = 푓(푞푖) = 푛푖푓(푝푖), (6) 푖=1 푖=1 and if 푓 is completely multiplicative, then 푓(1) = 1 and

푟 푠 ∏︁ ∏︁ 푛푖 푓(푛) = 푓(푞푖) = 푓(푝푖) . (7) 푖=1 푖=1

If 푓 is only L-additive, then we must also know the values of ℎ푓 at primes. A generalization of (6) (and an analogue of (7)) is given below.

Theorem 2.4. Let 푛 be as in (5). If 푓 is L-additive, then

푟 ∑︁ 푓(푛) = ℎ푓 (푞1) ··· ℎ푓 (푞푖−1)푓(푞푖)ℎ푓 (푞푖+1) ··· ℎ푓 (푞푟). 푖=1

If ℎ푓 (푝1), . . . , ℎ푓 (푝푠) ̸= 0, then

푟 푠 ∑︁ 푓(푞푖) ∑︁ 푛푖푓(푝푖) 푓(푛) = ℎ (푛) = ℎ (푛) . 푓 ℎ (푞 ) 푓 ℎ (푝 ) 푖=1 푓 푖 푖=1 푓 푖 Proof. In order to verify the first claim, it suffices to notice that

푓(푛) = 푓(푞1)ℎ푓 (푞2 ··· 푞푟) + 푓(푞2 ··· 푞푟)ℎ푓 (푞1)

= 푓(푞1)ℎ푓 (푞2) ··· ℎ푓 (푞푟) + 푓(푞2 ··· 푞푟)ℎ푓 (푞1) [︀ ]︀ = 푓(푞1)ℎ푓 (푞2) ··· ℎ푓 (푞푟) + ℎ푓 (푞1) 푓(푞2)ℎ푓 (푞3 ··· 푞푟) + 푓(푞3 ··· 푞푟)ℎ푓 (푞2) [︀ ]︀ = 푓(푞1)ℎ푓 (푞2) ··· ℎ푓 (푞푟) + ℎ푓 (푞1) 푓(푞2)ℎ푓 (푞3) ··· ℎ푓 (푞푟) + 푓(푞3 ··· 푞푟)ℎ푓 (푞2) = ··· 푟 ∑︁ = ℎ푓 (푞1) ··· ℎ푓 (푞푖−1)푓(푞푖)ℎ푓 (푞푖+1) ··· ℎ푓 (푞푟). 푖=1

The rest of the theorem follows from the equation ℎ푓 (푛) = ℎ푓 (푞1) ··· ℎ푓 (푞푟), which holds,

since ℎ푓 is completely multiplicative.

72 Example 2.3. Since 퐷(푝) = 1 and ℎ퐷(푝) = 푝 for all primes 푝, Theorem 2.4 gives the well-known formula 푟 푟 푠 ∑︁ ∑︁ 1 ∑︁ 푛푖 퐷(푛) = 푞 ··· 푞 푞 ··· 푞 = 푛 = 푛 . 1 푖−1 푖+1 푟 푞 푝 푖=1 푖=1 푖 푖=1 푖 It also implies that 푘 푘−1 푓(푝 ) = 푘ℎ푓 (푝) 푓(푝)

for all primes 푝 and nonnegative integers 푘. (For 푘 = 0, ℎ푓 (푝) must be nonzero.) In particular, for the arithmetic derivative, this reads

퐷(푝푘) = 푘푝푘−1.

For the arithmetic partial derivative 퐷푝, Theorem 2.4 reduces to its definition. Theorem 2.5. If 푢 is L-additive and 푣 is completely multiplicative, then their product function 푢푣

is L-additive with ℎ푢푣 = ℎ푢푣. Proof. For all positive integers 푚 and 푛, (︀ )︀ (푢푣)(푚푛) = 푢(푚푛)푣(푚푛) = 푢(푚)ℎ푢(푛) + 푢(푛)ℎ푢(푚) 푣(푚)푣(푛)

= (푢푣)(푚)(ℎ푢푣)(푛) + (푢푣)(푛)(ℎ푢푣)(푚).

Thus 푢푣 is L-additive with completely multiplicative part equaling to ℎ푢푣. Example 2.4. The function 푓(푛) = 퐷(푛)푛푘, where 푘 is a nonnegative integer, is L-additive with 푘+1 ℎ푓 (푛) = 푛 . The same property holds for the arithmetic partial derivative 퐷푝, since ℎ퐷 = ℎ퐷푝 . Theorem 2.6. If 푣 is L-additive and 푢 is completely multiplicative with positive integer values, then their composite function 푣 ∘ 푢 is L-additive with ℎ푣∘푢 = ℎ푣 ∘ 푢 . Proof. For all positive integers 푚 and 푛,

(푣 ∘ 푢)(푚푛) = 푣(푢(푚푛)) = 푣(푢(푚)푢(푛)) = 푣(푢(푚))ℎ푣(푢(푛)) + 푣(푢(푛))ℎ푣(푢(푚))

= (푣 ∘ 푢)(푚)(ℎ푣 ∘ 푢)(푛) + (푣 ∘ 푢)(푛)(ℎ푣 ∘ 푢)(푚).

It remains to show that ℎ푣 ∘ 푢 is completely multiplicative. We have

(ℎ푣 ∘ 푢)(푚푛) = ℎ푣(푢(푚푛)) = ℎ푣(푢(푚)푢(푛)) = ℎ푣(푢(푚))ℎ푣(푢(푛)) = (ℎ푣 ∘ 푢)(푚)(ℎ푣 ∘ 푢)(푛).

This completes the proof. Example 2.5. Let 푢 be as in Theorem 2.6. Since 퐷 is L-additive, then, by this theorem, 퐷 ∘ 푢 is

L-additive with ℎ퐷∘푢 = 푢; that is, 퐷(푢(푚푛)) = 푢(푛)퐷(푢(푚)) + 푢(푚)퐷(푢(푛))

for all positive integers 푚, 푛. In particular,

퐷((푚푛)푘) = 푛푘퐷(푚푘) + 푚푘퐷(푛푘),

where 푘 is a nonnegative integer. These equations follow also from the fact that 퐷 satisfies the

Leibniz rule. The above formulas also hold for the arithmetic partial derivative 퐷푝.

73 3 L-additive functions in terms of the Dirichlet convolution

Above we have seen that many fundamental properties of the arithmetic derivative, e.g., the for- mula for computing the arithmetic derivative of a given positive integer, are rooted to the fact that 퐷 is L-additive. We complete this article by changing our point of view slightly and demonstrate that L-additive functions can also be studied in terms of the Dirichlet convolutions. Let 푢 and 푣 be arithmetic functions. Their Dirichlet convolution is ∑︁ (푢 * 푣)(푛) = 푢(푑)푣(푛/푑). 푑|푛

We let 푓(푢 * 푣) denote the product function of 푓 and 푢 * 푣, i.e.,

(푓(푢 * 푣))(푛) = 푓(푛)(푢 * 푣)(푛).

Theorem 3.1. An arithmetic function 푓 is completely additive if and only if

푓(푢 * 푣) = (푓푢) * 푣 + 푢 * (푓푣)

for all arithmetic functions 푢 and 푣.

Proof. See [11, Proposition 2].

The next theorems show that L-additive functions can also be characterized in an analogous way.

Theorem 3.2. Let 푓 be an arithmetic function. If 푓 is L-additive and ℎ푓 is nonzero-valued, then

푓(푢 * 푣) = (푓푢) * (ℎ푓 푣) + (ℎ푓 푢) * (푓푣) (8) for all arithmetic functions 푢 and 푣. Conversely, if there is a completely multiplicative nonzero- valued function ℎ such that

푓(푢 * 푣) = (푓푢) * (ℎ푣) + (ℎ푢) * (푓푣) (9) for all arithmetic functions 푢 and 푣, then 푓 is L-additive and ℎ푓 = ℎ.

Proof. Under the assumptions of the first part, Theorems 2.1 and 3.1 imply

(푓/ℎ푓 )(푢 * 푣) = (푓푢/ℎ푓 ) * 푣 + 푢 * (푓푣/ℎ푓 ).

Multiplying by ℎ푓 , the left-hand side becomes 푓(푢 * 푣). Since

ℎ푓 ((푓푢/ℎ푓 ) * 푣) = (푓푢) * (ℎ푓 푣), ℎ푓 (푢 * (푓푣/ℎ푓 )) = (ℎ푓 푢) * (푓푣)

by (2), the claim follows. To prove the second part, multiply (9) by 1/ℎ and perform a simple modification of the above.

74 Corollary 3.1. If 푢 and 푣 are arithmetic functions, then

퐷(푢 * 푣) = (퐷푢) * (푁푣) + (푁푢) * (퐷푣),

where 푁(푛) = 푛.

Proof. It suffices to notice that ℎ퐷 = 푁.

On the other hand, taking 푢 = 푣, equation (8) becomes

푓(푢 * 푢) = 2[(푓푢) * (ℎ푓 푢)].

For 푢 = 퐸, where 퐸(푛) = 1 for all positive integers 푛, this reads

푓휏 = 2(푓 * ℎ푓 ), where 휏 is the divisor-number-function. Especially, we have

퐷(푢 * 푢) = 2[(퐷푢) * (푁푢)]

and, with 푢 = 퐸, 퐷휏 = 2(퐷 * 푁). Some remarks can be made also in the opposite direction. Assume now that

푓휏 = 2(푓 * ℎ)

for some completely multiplicative function ℎ that is nonzero for all positive integers. Then

(푓/ℎ)휏 = 2((푓/ℎ) * 퐸).

Thus, again according to [11, Proposition 2], we see that 푓/ℎ is completely additive, which

shows that 푓 is L-additive with ℎ푓 = ℎ. In particular, if

푓휏 = 2(푓 * 푁), then 푓 is L-additive with ℎ푓 = 푁. For example, 퐷 satisfies this condition. All of the above results for the arithmetic derivative 퐷 hold also for the arithmetic partial derivative 퐷푝, since ℎ퐷 = ℎ퐷푝 . Further properties of L-additive functions in terms of the Dirichlet convolution can be derived from the results in [8, 11]. It would be possible to obtain some properties of L-additive functions in terms of the unitary convolution as well from the results in [8].

References

[1] Apostol, T. M. (1976) Introduction to Analytic Number Theory, Springer-Verlag, New York.

[2] Barbeau, E. J. (1961) Remarks on an arithmetic derivative, Canad. Math. Bull. 4(2), 117–122.

75 [3] Chawla, L. M. (1973) A note on distributive arithmetical functions, J. Nat. Sci. Math. 13(1), 11–17.

[4] Haukkanen, P. (1992) A note on generalized multiplicative and generalized additive arith- metic functions, Math. Stud. 61, 113–116.

[5] Haukkanen, P., Merikoski, J. K. & Tossavainen, T. (2016) On arithmetic partial differential equations, J. Integer Seq. 19, Article 16.8.6.

[6] Kiuchi, I. & Minamide, M. (2014) On the Dirichlet convolution of completely additive functions. J. Integer Seq. 17, Article 14.8.7.

[7] Kovic,ˇ J. (2012) The arithmetic derivative and antiderivative, J. Integer Seq. 15, Arti- cle 12.3.8.

[8] Laohakosol, V. & Tangsupphathawat, P. (2016) Characterizations of additive functions. Lith. Math. J. 56(4), 518–528.

[9] McCarthy, P. J. (1986) Introduction to Arithmetical Functions, Springer-Verlag, New York.

[10] Sándor, J. & Crstici, B. (2004) Handbook of Number Theory II, Kluwer Academic, Dor- drecht.

[11] Schwab, E. D. (1995) Dirichlet product and completely additive arithmetical functions, Nieuw Arch. Wisk. 13(2), 187–193.

[12] Shapiro, H. N. (1983) Introduction to the Theory of Numbers, Wiley InterScience, New York.

[13] Ufnarovski, V. & Åhlander, B. (2003) How to differentiate a number, J. Integer Seq. 6, Article 03.3.4.

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