Arithmetic Subderivatives and Leibniz-Additive Functions

TL;DR

This paper introduces the concept of arithmetic subderivatives and Leibniz-additive functions, generalizing derivatives in number theory, and explores their properties, bounds, and conditions for Leibniz-additivity.

Contribution

It defines Leibniz-additive functions and extends the notion of arithmetic derivatives, providing foundational properties and bounds for these generalized functions.

Findings

Introduced arithmetic subderivatives with respect to prime sets.

Established conditions for Leibniz-additivity of arithmetic functions.

Derived bounds for Leibniz-additive functions.

Abstract

We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then define that an arithmetic function $f$ is Leibniz-additive if there is a nonzero-valued and completely multiplicative function $h_f$ satisfying $f(mn)=f(m)h_f(n)+f(n)h_f(m)$ for all positive integers $m$ and $n$. We study some basic properties of such functions. For example, we present conditions when an arithmetic function is Leibniz-additive and, generalizing well-known bounds for the arithmetic derivative, establish bounds for a Leibniz-additive function.

Equations

$$n=\prod _{p\in \mathbb{P}}{p}^{{\nu }_p(n)}.$$

$$D_S(n)=n_S^{\prime }=n\sum _{p\in S}\frac{{\nu }_p(n)}{p}.$$

$${\mathrm{ld}}_S(n)=\frac{D_S(n)}{n}=\sum _{p\in S}\frac{{\nu }_p(n)}{p}.$$

$$g(n)=\sum _{p\in \mathbb{P}}{\nu }_p(n)x_p.$$

$$h(n)=\prod _{p\in \mathbb{P}}{y}_p^{{\nu }_p(n)}.$$

$$f(mn)=f(m)h_f(n)+f(n)h_f(m)$$

$$f(p^a)=af(p)h(p{)}^{a-1}$$

$$f(n)=\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(n)\frac{x_{p}}{y_{p}}\Big{)}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(n)}.$$

$$\{p_1,\dots ,p_s\}=\{p\in \mathbb{P}\mid {\nu }_p(n)>0\}$$

$$a_i={\nu }_{p_i}(n),i=1,\dots ,s,$$

$$f(n)=\sum _{i=1}^s{h}_f(p_1{)}^{a_1}\cdots h_f(p_{i-1}{)}^{a_{i-1}}f(p_i^{a_i})h_f(p_{i+1}{)}^{a_{i+1}}\cdots h_f(p_r{)}^{a_r}=$$

$$\sum _{i=1}^s{h}_f(p_1{)}^{a_1}\cdots h_f(p_{i-1}{)}^{a_{i-1}}{a}_{i}f(p_i)h_f(p_i{)}^{a_i-1}{h}_f(p_{i+1}{)}^{a_{i+1}}\cdots h_f(p_r{)}^{a_r}=$$

$$\displaystyle\sum_{p\in\mathbb{P}}\Big{(}\nu_{p}(n)f(p)h_{f}(p)^{\nu_{p}(n)-1}\prod_{\begin{subarray}{c}q\in\mathbb{P}\\ q\neq p\end{subarray}}h_{f}(q)^{\nu_{q}(n)}\Big{)}=\sum_{p\in\mathbb{P}}\Big{(}\nu_{p}(n)\frac{f(p)}{h_{f}(p)}\prod_{q\in\mathbb{P}}h_{f}(q)^{\nu_{q}(n)}\Big{)}=$$

$$\displaystyle\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(n)\frac{x_{p}}{y_{p}}\Big{)}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(n)}.$$

$$h(n)=\prod _{p\in \mathbb{P}}{y}_p^{{\nu }_p(n)}.$$

$$\displaystyle f(mn)=\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(mn)\frac{x_{p}}{y_{p}}\Big{)}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(mn)}=$$

$$\displaystyle\Big{(}\sum_{p\in\mathbb{P}}(\nu_{p}(m)+\nu_{p}(n))\frac{x_{p}}{y_{p}}\Big{)}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(m)+\nu_{p}(n)}=$$

$$\displaystyle\Big{(}\sum_{p\in\mathbb{P}}(\nu_{p}(m)+\nu_{p}(n))\frac{x_{p}}{y_{p}}\Big{)}\Big{(}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(m)}\Big{)}\Big{(}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(n)}\Big{)}=$$

$$\displaystyle\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(m)\frac{x_{p}}{y_{p}}\Big{(}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(m)}\Big{)}\Big{)}\Big{(}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(n)}\Big{)}+$$

$$\displaystyle\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(n)\frac{x_{p}}{y_{p}}\Big{(}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(n)}\Big{)}\Big{)}\Big{(}\prod_{p\in\mathbb{P}}y_{p}^{\nu_{p}(m)}\Big{)}=$$

$$f(m)h(n)+f(n)h(m).$$

$$U_f=\{p\in \mathbb{P}\mid f(p)\ne 0\},V_f=\{p\in \mathbb{P}\mid f(p)=0\}.$$

$$f(p^2)=2f(p)h_f(p)$$

$$h_f(p)=\frac{f(p^2)}{2f(p)}\mathrm{for}p\in U_f.$$

$$f(pq)=f(p)h_f(q)+f(q)h_f(p)=f(q)h_f(p).$$

$$h_f(p)=\frac{f(pq)}{f(q)}\mathrm{for}p\in V_f,$$

$$\displaystyle h_{f}(n)=\Big{(}\prod_{p\in U_{f}}\Big{(}\frac{f(p^{2})}{2f(p)}\Big{)}^{\nu_{p}(n)}\Big{)}\Big{(}\prod_{p\in V_{f}}\Big{(}\frac{f(pq)}{f(q)}\Big{)}^{\nu_{p}(n)}\Big{)},$$

$$\displaystyle f(n)=\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(n)\frac{f(p)}{h(p)}\Big{)}\prod_{p\in\mathbb{P}}h(p)^{\nu_{p}(n)}=\Big{(}\sum_{p\in\mathbb{P}}\nu_{p}(n)\frac{f(p)}{h(p)}\Big{)}h(n).$$

$$g(n)=\sum _{p\in \mathbb{P}}{\nu }_p(n)\frac{f(p)}{h(p)}$$

$$f=gh,$$

$$f(mn)=g(mn)h(mn)=(g(m)+g(n))h(m)h(n)=$$

$$g(m)h(m)h(n)+g(n)h(n)h(m)=f(m)h(n)+f(n)h(m).$$

$$f(p^{a+1})=(a+1)f(p)h_f(p{)}^a,f(p^{b+1})=(b+1)f(p)h_f(p{)}^b,$$

$$f(p^{a+1}{)}^b=(a+1{)}^{b}f(p{)}^b{h}_f(p{)}^{ab},f(p^{b+1}{)}^a=(b+1{)}^{a}f(p{)}^a{h}_f(p{)}^{ba}.$$

$$\frac{f(p^{a+1}{)}^b}{f(p^{b+1}{)}^a}=\frac{(a+1{)}^{b}f(p{)}^b}{(b+1{)}^{a}f(p{)}^a}$$

Full text

1 Introduction

We let $\mathbb{P}$, $\mathbb{Z}_{+}$, $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ stand for the set of primes, positive integers, nonnegative integers, integers, and rational numbers, respectively.

Let $n\in\mathbb{Z}_{+}$. There is a unique sequence $(\nu_{p}(n))_{p\in\mathbb{P}}$ of nonnegative integers (with only finitely many positive terms) such that

[TABLE]

We use this notation throughout.

Let $\emptyset\neq S\subseteq\mathbb{P}$. We define the arithmetic subderivative of $n$ with respect to $S$ as

[TABLE]

In particular, $n^{\prime}_{\mathbb{P}}$ is the arithmetic derivative of $n$, defined by Barbeau [1] and studied further by Ufnarovski and Åhlander [6]. Another well-known special case is $n^{\prime}_{\{p\}}$, the arithmetic partial derivative of $n$ with respect to $p\in\mathbb{P}$, defined by Kovič [5] and studied further by the present authors and Mattila [2, 3].

We define the arithmetic logarithmic subderivative of $n$ with respect to $S$ as

[TABLE]

In particular, ${\rm ld}_{\mathbb{P}}(n)$ is the arithmetic logarithmic derivative of $n$. This notion was originally introduced by Ufnarovski and Åhlander [6].

An arithmetic function $g$ is completely additive (or c-additive, for short) if $g(mn)=g(m)+g(n)$ for all $m,n\in\mathbb{Z}_{+}$. It follows from the definition that $g(1)=0$. An arithmetic function $h$ is completely multiplicative (or c-multiplicative, for short) if $h(1)=1$ and $h(mn)=h(m)h(n)$ for all $m,n\in\mathbb{Z}_{+}$. The following theorems recall that these functions are totally determined by their values at primes. The proofs are simple and omitted.

Theorem 1.1**.**

Let $g$ be an arithmetic function, and let $(x_{p})_{p\in\mathbb{P}}$ be a sequence of real numbers. The following conditions are equivalent:

1. (a)

$g$* is c-additive and $g(p)=x_{p}$ for all $p\in\mathbb{P}$;* 2. (b)

for all $n\in\mathbb{Z}_{+}$,

[TABLE]

Theorem 1.2**.**

Let $h$ be an arithmetic and nonzero-valued function, and let $(y_{p})_{p\in\mathbb{P}}$ be a sequence of nonzero real numbers. The following conditions are equivalent:

1. (a)

$h$* is c-multiplicative and $h(p)=y_{p}$ for all $p\in\mathbb{P}$;* 2. (b)

for all $n\in\mathbb{Z}_{+}$,

[TABLE]

We say that an arithmetic function $f$ is Leibniz-additive (or L-additive, for short) if there is a nonzero-valued and c-multiplicative function $h_{f}$ such that

[TABLE]

for all $m,n\in\mathbb{Z}_{+}$. Then $f(1)=0$, since $h_{f}(1)=1$. The property (2) may be considered a generalized Leibniz rule. Substituting $m=n=p\in\mathbb{P}$ and applying induction, we get

[TABLE]

for all $p\in\mathbb{P}$, $a\in\mathbb{Z}_{+}$.

The arithmetic subderivative $D_{S}$ is L-additive with $h_{D_{S}}=N$, where $N$ is the identity function $N(n)=n$. A c-additive function $g$ is L-additive with $h_{g}=E$, where $E(n)=1$ for all $n\in\mathbb{Z}_{+}$. The arithmetic logarithmic subderivative ${\rm ld}_{S}$ is c-additive and hence L-additive.

This paper is a sequel to [4], where we defined L-additivity without requiring that $h_{f}$ is nonzero-valued. We begin by showing how the values of an L-additive function $f$ are determined in $\mathbb{Z}_{+}$ by the values of $f$ and $h_{f}$ at primes (Section 2) and then study under which conditions an arithmetic function $f$ can be expressed as $f=gh$, where $g$ is c-additive and $h$ is nonzero-valued and c-multiplicative (Section 3). It turns out that the same conditions are necessary for L-additivity (Section 4). Finally, extending Barbeau’s [1] and Westrick’s [7] results, we present some lower and upper bounds for an L-additive function (Section 5). We complete our paper with some remarks (Section 6).

2 Constructing $f(n)$ and $h_{f}(n)$

An L-additive function $f$ is not totally defined by its values at primes. Also, the values of $h_{f}$ at primes must be known.

Theorem 2.1**.**

Let $f$ be an arithmetic function, and let $(x_{p})_{p\in\mathbb{P}}$ and $(y_{p})_{p\in\mathbb{P}}$ be as in Theorems 1.1 and 1.2. The following conditions are equivalent:

1. (a)

$f$* is L-additive and $f(p)=x_{p}$, $h_{f}(p)=y_{p}$ for all $p\in\mathbb{P}$;* 2. (b)

for all $n\in\mathbb{Z}_{+}$,

[TABLE]

Proof.

(a)$\Rightarrow$(b). Since $f(1)=0$, (b) holds for $n=1$. So, let $n>1$. Denoting

[TABLE]

and

[TABLE]

we have

[TABLE]

The first equation can be proved by induction on $r$, the second holds by (3), and the remaining equations are obvious.

(b)$\Rightarrow$(a). We define now

[TABLE]

Let $m,n\in\mathbb{Z}_{+}$. Then

[TABLE]

So, $f$ is L-additive with $h_{f}=h$. It is clear that $f(p)=x_{p}$ and $h_{f}(p)=y_{p}$ for all $p\in\mathbb{P}$. ∎

Next, we construct $h_{f}$ from $f$. Let us denote

[TABLE]

If $f=\theta$, where $\theta(n)=0$ for all $n\in\mathbb{Z}_{+}$, then any $h_{f}$ applies. Hence, we now assume that $f\neq\theta$. Then $U_{f}\neq\emptyset$.

Since

[TABLE]

by (3), we have

[TABLE]

The case $p\in V_{f}$ remains. Let $q\in\mathbb{P}$. Then (2) implies that

[TABLE]

Therefore,

[TABLE]

where $q\in U_{f}$ is arbitrary. Now, by Theorem 1.2,

[TABLE]

where $q\in U_{f}$ is arbitrary. (If $V_{f}=\emptyset$, then the latter factor is the “empty product” one.) We have thus proved the following theorem.

Theorem 2.2**.**

If $f\neq\theta$ is L-additive, then $h_{f}$ is unique and determined by (5).

3 Decomposing $f=gh$

Let $f$ be an arithmetic function and let $h$ be a nonzero-valued and c-multiplicative function. By Theorem 2.1, $f$ is L-additive with $h_{f}=h$ if and only if

[TABLE]

The function

[TABLE]

is c-additive by Theorem 1.1.

We say that an arithmetic function $f$ is gh-decomposable if it has a gh decomposition

[TABLE]

where $g$ is c-additive and $h$ is nonzero-valued and c-multiplicative. We saw above that L-additivity implies $gh$-decomposability. Also, the converse holds.

Theorem 3.1**.**

Let $f$ be an arithmetic function. The following conditions are equivalent:

1. (a)

$f$* is L-additive;* 2. (b)

$f$* is $gh$-decomposable.*

Proof.

(a)$\Rightarrow$(b). We proved this above.

(b)$\Rightarrow$(a). For all $m,n\in\mathbb{Z}_{+}$,

[TABLE]

Consequently, $f$ is L-additive with $h_{f}=h$. ∎

Corollary 3.1**.**

Let $f\neq\theta$ be an arithmetic function. The following conditions are equivalent:

1. (a)

$f$* is L-additive;* 2. (b)

$f$* is uniquely $gh$-decomposable.*

Proof.

In proving (a)$\Rightarrow$(b), $h_{f}$ is unique by Theorem 2.2. Since $h_{f}$ is nonzero-valued, also $g=f/h_{f}$ is unique. ∎

For example, if $f=D_{S}$, then $g={\rm ld}_{S}$ and $h=N$.

By Theorem 2.2, an L-additive function $f\neq\theta$ determines $h_{f}$ uniquely. We consider next the converse problem: Given a nonzero-valued and c-multiplicative function $h$, find an L-additive function $f$ such that $h_{f}=h$.

Theorem 3.2**.**

Let $(x_{p})_{p\in\mathbb{P}}$ be a sequence of real numbers and let $h$ be nonzero-valued and c-multiplicative. There is a unique L-additive function $f$ with $h_{f}=h$ such that $f(p)=x_{p}$ for all $p\in\mathbb{P}$.

Proof.

If at least one $x_{p}\neq 0$, then apply Theorem 2.1 and Corollary 3.1. Otherwise, $f=\theta$. ∎

We can now characterize $D_{S}$ and ${\rm ld}_{S}$.

Corollary 3.2**.**

Let $f$ be an arithmetic function and $\emptyset\neq S\subseteq\mathbb{P}$. The following conditions are equivalent:

1. (a)

$f$* is L-additive, $h_{f}=N$, $f(p)=1$ for $p\in S$, and $f(p)=0$ for $p\in\mathbb{P}\setminus S$;* 2. (b)

$f=D_{S}$.

Corollary 3.3**.**

Let $g$ be an arithmetic function and $\emptyset\neq S\subseteq\mathbb{P}$. The following conditions are equivalent:

1. (a)

$g$* is c-additive, $g(p)=1/p$ for $p\in S$, and $g(p)=0$ for $p\in\mathbb{P}\setminus S$;* 2. (b)

$g={\rm ld}_{S}$.

4 Conditions for L-additivity

Let $f\neq\theta$ be L-additive and $a,b\in\mathbb{N}$.

First, let $p\in\mathbb{P}$. By (3),

[TABLE]

and, further,

[TABLE]

Assume now that $p\in U_{f}$. Then the right-hand sides of the equations in (7) are nonzero and $f(p^{a+1}),f(p^{b+1})\neq 0$. Therefore, by (8),

[TABLE]

or, equivalently,

[TABLE]

Second, assume that $U_{f}$ has at least two elements. If $p,q\in U_{f}$, then (2) and (3) imply that

[TABLE]

Third, assume additionally that $V_{f}\neq\emptyset$. Let $p\in V_{f}$ and $q_{1},q_{2}\in U_{f}$. By (4) and the fact that $h_{f}$ is nonzero-valued,

[TABLE]

In other words, we can “cancel” $p$ in

[TABLE]

Fourth, both the nonzero-valuedness of $h_{f}$ and (5) imply that

[TABLE]

We have thus found necessary conditions for L-additivity.

Theorem 4.1**.**

Let $f\neq\theta$ be L-additive and $a,b\in\mathbb{N}$.

1. (i)

If $p\in U_{f}$, then

[TABLE] 2. (ii)

If $p,q\in U_{f}$, then

[TABLE] 3. (iii)

If $p\in V_{f}$ and $q_{1},q_{2}\in U_{f}$, then

[TABLE] 4. (iv)

If $p\in U_{f}$, then

[TABLE]

The question about the sufficiency of these conditions remains open.

To find sufficient conditions for L-additivity, we study under which conditions we can apply the procedure described in the proof of Theorem 2.2 to a given arithmetic function $f\neq\theta$. The function $h$, defined as $h_{f}$ in (5), must be ($\alpha$) well-defined, ($\beta$) c-multiplicative, and ($\gamma$) nonzero-valued. Condition ($\alpha$) follows from (iii), ($\beta$) is obvious, and ($\gamma$) follows from (iii) and (iv). If the function $g=f/h$ is also c-additive, then $f$ is L-additive by Theorem 3.1. So, we have found sufficient conditions for L-additivity, and they are obviously also necessary.

Theorem 4.2**.**

An arithmetic function $f\neq\theta$ is L-additive if and only if (iii) and (iv) in Theorem 4.1 are satisfied and the function $f/h$ is c-additive, where

[TABLE]

5 Bounds for an L-additive function

Let us express (1) as

[TABLE]

where $q_{1},\dots,q_{r}\in\mathbb{P}$, $q_{1}\leq\dots\leq q_{r}$. We first recall the well-known bounds for $D(n)$ using $n$ and $r$ only.

Theorem 5.1**.**

Let $n$ be as in (9). Then

[TABLE]

Equality is attained in the upper bounds if and only if $n$ is a power of $2$, and in the lower bound if and only if $n$ is a prime or a power of $2$.

Proof.

See [1, pp. 118–119], [6, Theorem 9]. ∎

The first upper bound can be improved using the same information. Westrick [7, Ineq. (6)] presented in her thesis the following bound without proof.

Theorem 5.2**.**

Let $n$ be as in (9). Then

[TABLE]

Equality is attained if and only if $n\in\mathbb{P}$ or $q_{1}=\dots=q_{r-1}=2$.

Proof.

If $r=1$ (i.e., $n\in\mathbb{P}$), then (11) clearly holds with equality. So, assume that $r>1$.

Case 1. $q_{1}=\dots=q_{r-1}=2$. Then

[TABLE]

where “rhs” is short for “the right-hand side”.

Case 2. $q_{1}=\dots=q_{r-2}=2$ (omit this if $r=2$) and $q_{r-1}>2$. Since

[TABLE]

we have

[TABLE]

Case 3. $q_{r-2}>2$. Then $r\geq 3$ and

[TABLE]

The claim with equality conditions is thus verified. Because

[TABLE]

the upper bound (11) indeed improves (10). ∎

We extend the upper bounds (10) and (11) under the assumption

[TABLE]

Let $n$ in (9) have $q_{i_{1}},\dots,q_{i_{s}}\in U_{f}$. We denote

[TABLE]

and

[TABLE]

Theorem 5.3**.**

Let $f\neq\theta$ be nonnegative and L-additive satisfying (12). Then

[TABLE]

where $s$ is as in (13) and $M$ is as in (14). Equality is attained if and only if $n$ is a power of $2$.

Proof.

By (6) and simple manipulation,

[TABLE]

The equality condition is obvious. ∎

Theorem 5.4**.**

Let $f\neq\theta$ be nonnegative and L-additive satisfying (12). Then

[TABLE]

where $s$ is as in (13) and $M$ is as in (14). Equality is attained if and only if $n\in\mathbb{P}$ or $p_{1}=\dots=p_{s-1}=2=h_{f}(2)$.

Proof.

If $s=1$ (i.e., $n\in\mathbb{P}$), then (16) clearly holds with equality. So, assume that $s>1$.

Case 1. $p_{1}=\dots=p_{s-1}=2$. Then

[TABLE]

Case 2. $p_{1}=\dots=p_{s-2}=2$ (omit this if $s=2$) and $p_{s-1}>2$. If $s\geq 3$, then

[TABLE]

The last expression is obviously an upper bound for $f(n)$ also if $s=2$. If

[TABLE]

i.e.,

[TABLE]

then (16) follows. Since

[TABLE]

we actually have a stronger inequality

[TABLE]

Case 3. $p_{s-2}>2$. Then $s\geq 3$ and

[TABLE]

Since

[TABLE]

we also have

[TABLE]

Similarly,

[TABLE]

Because

[TABLE]

it follows from (17) that

[TABLE]

In other words, (16) holds strictly.

The proof is complete. It also includes the equality conditions. ∎

If we do not know $s$ (but know $r$), we can substitute $s=r$ in (15) and (16). We complete this section by extending the lower bound (10).

Theorem 5.5**.**

Let $f$ be nonnegative and L-additive, and let $n$ be as in (9) with

[TABLE]

Then

[TABLE]

where

[TABLE]

Equality is attained if and only if $n$ is a prime or a power of $2$.

Proof.

By (6) and the arithmetic-geometric mean inequality,

[TABLE]

The equality condition is obvious. ∎

6 Concluding remarks

In order to extend the concepts of arithmetic derivative and arithmetic partial derivative, we first defined the concept of arithmetic subderivative. As a further extension, we defined the concept of L-additive function. For simplicity, we stated (contrary to [4]) that $h_{f}$ must be nonzero-valued. If we allow $h_{f}$ to be zero, it turns out that we then just meet extra work without gaining in results.

Which properties of the arithmetic derivative can be extended to arithmetic subderivatives and, further, to L-additive functions? As we saw above, this question can be answered, at least, within certain bounds for the arithmetic derivative.