The Lie Bracket and the Arithmetic Derivative

Jun Fan and Sergey Utev Department of Mathematics University of Leicester Leicester LE1 7RH United Kingdom

                     jf249@leicester.ac.uk
                     su35@leicester.ac.uk

Abstract

We apply the Lie bracket approach to characterize the semi-derivations on the positive integers. The approach is motivated by the Stroock-Lie bracket identity commonly used in Malliavin calculus.

1 Introduction

The Malliavin derivative D, the Skorohod integral δ, and the associated Malliavin calculus are powerful tools for the analysis of stochastic processes. The Malliavin calculus, named after P. Malliavin, is also called the stochastic calculus of variations [8, p. VII, p. 1]. The definition of the Malliavin derivative and the Skorohod integral can be found, for example, in [2, p. 25], [3, pp. 20, 27], and [10, pp. 25, 40], respectively. The Stroock-Lie bracket type identity [D, δ] = Dδ − δD = I is a common notion in the Malliavin calculus [6, p. 355], which is also referred to as the fundamental theorem of calculus [3, Thm. 3.18, p. 37]. The Malliavin derivative and the Skorohod integral (the adjoint operator) act in the space of random processes, which are treated as functions of a Gaussian process.

On the one hand, many random structures satisfy the functional Gaussian approximation [9]. On the other hand, the Lie bracket is a powerful tool in the study of differential equations, in particular, quantum stochastic calculus [11]. This motivated us to apply the Lie bracket in a totally different environment. More exactly, we treat D and δ as operators acting on integer sequences or dynamical systems on the natural numbers ℕ. Following Barbeau [1], Ufnarovski and Åhlander [14], and Kovič [7], we study a partial number derivative operator D_{A}. The operator is introduced and characterized as a solution to the modified Stroock-Lie bracket type identity. The Barbeau arithmetic derivative D is then characterized as the dynamics on the positive integers, which satisfies the Stroock-Lie bracket identity [D, ℓ] = I, and which holds for all linear functions ℓ = ℓ_{p} = pn, where p is a prime number. Moreover, the Stroock-Lie bracket characterization is illustrated by examples on several commutative rings without zero divisors. Finally, arithmetic type differential equations driven by D_{A} are briefly analyzed.

2 Lie bracket analysis of linear functions of positive integers

Definition 1. For two functions F, U : ℤ_{+} → ℤ_{+}, we define the Lie bracket or the commutator [F, U] by

[F, U] = F ∘ U − U ∘ F, where F ∘ U(n) = F (U(n)).

Hence [F, U](n) = F (U(n)) − U(F (n)).

Let L = {ℓ_{x} : ℓ_{x}(n) = nx, n ∈ ℤ_{+}} denote the set of linear functions on ℤ_{+}. Notice that L is a commutative semiring with unity ℓ_{1} = I and zero ℓ_{0} = 0 with respect to the multiplication and addition operations. Then ℓ_{x}∘ℓ_{y} = ℓ_{xy}, ℓ_{x}+ℓ_{y} = ℓ_{x+y}, ℓ_{x}∘ℓ_{1} = ℓ_{1}∘ℓ_{x} = ℓ_{x}, and ℓ_{x} + ℓ_{0} = ℓ_{0} + ℓ_{x} = ℓ_{x}. By construction, the semiring L is isomorphic to the semiring (ℤ_{+}, +, ·) of non-negative integers.

Definition 2. Consider D : ℤ_{+} → ℤ_{+} and define the Lie bracket linearity set of D by

W_{D} = {x : there exists y = y_{x} such that [D, ℓ_{x}] = ℓ_{y}}.  (1)

Lemma 3. W_{D} is a multiplicative semigroup in (ℤ_{+}, ·) that includes 1.

Proof. Let x, z ∈ W_{D}. By the Lie bracket definition and algebraic manipulations,

[D, ℓ_{xz}](n) = D ∘ ℓ_{xz}(n) − ℓ_{xz} ∘ D(n) = [D, ℓ_{x}](zn) + ℓ_{x} ∘ [D, ℓ_{z}](n).

Now we apply to the last line, first, the definition of W_{D} and, then, the semiring properties of linear functions. This leads to [D, ℓ_{xz}](n) = ℓ_{y_{x}} ∘ ℓ_{z}(n) + ℓ_{x} ∘ ℓ_{y_{z}}(n) = ℓ_{y_{x}z+xy_{z}}(n).

Lemma 4. The following statements are equivalent:

(i) D satisfies the Leibnitz rule D(mn) = mD(n) + nD(m); (ii) D(1) = 0 and W_{D} = ℤ_{+}.

Moreover, [D, ℓ_{m}] = ℓ_{D(m)}.

Proof. (i) ⟹ (ii) follows by direct calculations.

(ii) ⟹ (i): Fix m. Then, for any n ∈ ℤ_{+}, we have

y_{m}n = ℓ_{y_{m}}(n) = [D, ℓ_{m}](n) = D ∘ ℓ_{m}(n) − ℓ_{m} ∘ D(n) = D(mn) − mD(n).

Finally, take n = 1. Then y_{m} = D(m) − mD(1) = D(m). Therefore, [D, ℓ_{m}] = ℓ_{D(m)} and D(mn) = mD(n) + nD(m), proving the lemma.

Remark 5. Let x be a linear function l_{x}, and let the function composition ∘ be replaced by the usual multiplication. Then the main Lie bracket characteristic becomes the Pincherle derivative f′ = f ·x−x·f = [f, l_{x}], as introduced in [12]. Tempesta [13] applied the Pincherle derivative and the associated Lie bracket approach in quantum calculus.

We now apply Lemmas 3 and 4 to characterize the arithmetic type derivative D_{A} as the dynamics on the positive integers, which satisfies the Stroock-Lie bracket type identity [D_{A}, ℓ_{x}] = y_{x}I.

Let P denote the set of all primes. We say that sets A, B are orthogonal, if for any x ∈ A and y ∈ B, we have gcd(x, y) = 1, i.e., the sets A and B do not have any common divisors. Notice that disjoint subsets in P are orthogonal.

Lemma 6. Consider the nonempty subset of primes A ⊂ P. Let D_{A} : ℤ_{+} → ℤ_{+} be such that D_{A}(1) = 0. The following properties are equivalent:

(i) [D_{A}, ℓ_{p}] = I, for p ∈ A, and [D_{A}, ℓ_{p}] = 0, for p ∈ Ā = P − A;

(ii) D_{A} = ∑_{p∈A} D_{p}, where D_{p}(n) = jp^{j−1}m, for n = p^{j}m with m ⊥ p. Moreover, D_{A} satisfies the Leibnitz rule and has a representation

D_{A}(n) = n ∑_{p∈A} \frac{n_{p}}{p} , where n = ∏_{p∈P} p^{n_{p}}.  (2)

Proof. (ii) ⟹ (i) follows by direct calculations.

(i) ⟹ (ii): Consider the linearity set W_{D_{A}}. Notice that {0, 1} ⊆ W_{D_{A}} and the smallest multiplicative semigroup, containing all primes, is ℤ_{+} − {0, 1}. By Lemma 3, it then follows that W_{D_{A}} = ℤ_{+}. Thus, by Lemma 4, the function D satisfies the Leibnitz rule. Moreover, D_{A}(p) = 1, for p ∈ A, and D_{A}(p) = 0, for p ∉ A. Now we apply the argument from Ufnarovski and Åhlander [14]. Consider the log transform of D_{A} defined by L_{A}(n) = D_{A}/n. By the Leibnitz rule, we see that L_{A} is a homomorphism of the multiplicative semigroup to the additive semigroup on ℤ_{+}, which shows that

L_{A}(n) = ∑_{p∈P} \frac{n_{p}}{p} D_{A}(p) = ∑_{p∈A} \frac{n_{p}}{p} , where n = ∏_{p∈P} p^{n_{p}}.

This implies representation (2). Take A = {p}. We then derive D_{p}(p^{j}m) = jp^{j−1}m, for m ⊥ p. This proves the representation D_{A} = ∑_{p∈A} D_{p}.

Corollary 7. Let D : ℕ → ℕ such that D(1) = 0. Assume that for each linear function ℓ = ℓ_{p} = pn, where p is a fixed prime, the following Stroock-Lie bracket identity

[D, ℓ] = I, i.e., Dℓ(n) = n + ℓD(n),  (3)

holds for all n ∈ ℕ. Then D is an arithmetic derivative, i.e.,

D (n) = n′ = n ∑_{i=1}^{k} \frac{n_{i}}{p_{i}} , where n = ∏_{i=1}^{k} p_{i}^{n_{i}}.

Moreover, [D, ℓ_{m}] = D(m)I. In particular, we have the following characterization of the Stroock-Lie bracket identity

[D, ℓ_{m}] = I if and only if m is a prime.  (4)

Proof. Let us prove the last statement. The Stroock-Lie bracket equation states that

Dℓ_{m}(n) = n + ℓ_{m}D(n) = n + mD(n) = n + mn′.

The left-hand side of the former equation is computed by

Dℓ_{m}(n) = (mn)′ = D(mn) = m′n + mn′.

Hence, by equating it to the right-hand side of the same equation, we derive m′n + mn′ = n + mn′. Clearly the last equation holds if and only if m′ = 1. Therefore, m is a prime, as proved in [14].

3 Lie bracket properties

3.1 Lie bracket properties for the arithmetic derivative

From Corollary 7, we derive

Corollary 8. Let m_{1} + ⋯ + m_{k} = m. Then [D, ℓ_{m_{1}} + ⋯ + ℓ_{m_{k}}] = I holds if and only if m is a prime.

Remark 9. (i) In particular, [D, ℓ_{p} + ℓ_{2}] = I if and only if p and p + 2 are twin primes.

(ii) According to the Goldbach weak conjecture, every prime number greater than 5 can be expressed as the sum of three primes. Then, for each such triple of primes p_{1}, p_{2}, p_{3} with p_{1} + p_{2} + p_{3} being a prime, we have [D, ℓ_{p_{1}} + ℓ_{p_{2}} + ℓ_{p_{3}}] = I.

Barbeau [1] proved that if the natural number n is not a prime or unity, then n′ ⩾ 2\sqrt{n}. The equality holds if and only if n = p^{2}, where p is a prime. In particular, the equation m′ = 2 does not have solutions in positive integers. Therefore, we obtain the following lemmas.

Lemma 10. For any primes p_{1} and p_{2},

[D, ℓ_{p_{1}} + ℓ_{p_{2}}] ≠ [D, ℓ_{p_{1}}] + [D, ℓ_{p_{2}}] .

Proof. By definition, the right-hand side of the former equation equals to 2n, n ∈ ℕ. The left-hand side of the equation is [D, ℓ_{p_{1}} + ℓ_{p_{2}}] (n) = D ((p_{1} + p_{2}) n) − (p_{1} + p_{2}) D(n) = (p_{1} + p_{2})′ n. It remains to notice that the equation (p_{1} + p_{2})′ = 2 does not have solutions.

Lemma 11. For any primes p_{1} and p_{2},

[D + ℓ_{p_{1}}, ℓ_{p_{2}}] = I.

Proof. The left-hand side of the former equation is [D + ℓ_{p_{1}}, ℓ_{p_{2}}] = [D, ℓ_{p_{2}}] + [ℓ_{p_{1}}, ℓ_{p_{2}}] = I + [ℓ_{p_{1}}, ℓ_{p_{2}}]. Clearly [ℓ_{p_{1}}, ℓ_{p_{2}}] = 0, which completes the proof.

Next lemma shows that there are infinitely many non-linear dynamics or integer sequences satisfying the Stroock-Lie bracket lemma for the arithmetic derivative.

Lemma 12. Let p_{1}, p_{2} and q be three different prime numbers. And let σ(q^{q}) = p_{1}q^{q}, σ(n) = p_{2}n, for n ≠ q^{q}. Then [D, σ] = I and σ is not linear.

Proof. For n ≠ q^{q},

Dσ(n) = (p_{2}n)′ = n + p_{2}(n′) = n + σ(n).

On the other hand, by definition we have D(q^{q}) = q^{q} if q is a prime. Hence, for n = q^{q}, Dσ(q^{q}) = (p_{1}q^{q})′ = q^{q} + p_{1}(q^{q})′ = q^{q} + σ(q^{q}), proving that [D, σ] = I. Clearly σ is not linear, since σ(2q^{q}) = p_{2}(2q^{q}) ≠ 2q^{q} = 2σ(q^{q}).

3.2 Lie bracket properties for other derivatives

Following Ufnarovski and Åhlander [14], we define the generalized arithmetic derivative by

D (x) = x ∑_{i=1}^{k} \frac{x_{i}D (p_{i})}{p_{i}} , where x = ∏_{i=1}^{k} p_{i}^{x_{i}}.

Note that from this definition D (p) = 1 is no longer true for prime p, in general. Then the restriction on the linear function ℓ_{m} in (4) can be weakened to m such that D(m) = 1.

3.2.1 Partial arithmetic derivative D_{p} The function D_{p}, defined in Lemma 6, is originated in Kovič [7] and referred to as the partial arithmetic derivative. Notice that D_{p} = D_{{p}} = D is a generalized arithmetic derivative defined by D(p) = 1 and D(q) = 0 for all other primes q. Lemma 13. Let p, q be two different primes. Then

(i) [D_{p}, ℓ_{p}] = I; (ii) [D_{p}, ℓ_{q}] = 0; (iii) [D_{p}, ℓ_{p} + ℓ_{q}] = 0; (iv) [D_{p} + D_{q}, ℓ_{p} + ℓ_{q}] = 0. Proof. The first two properties follow from Lemma 6 (i). By the Leibnitz rule for D_{p} and since p ∤ p + q, we have [D_{p}, ℓ_{p} + ℓ_{q}] = 0. This shows that D_{p} is not linear and proves (iii). Finally, the last property (iv) follows by the linearity of the Lie bracket.

3.2.2 General arithmetic derivative D_{A} Now consider the general arithmetic derivative D_{A} defined in Lemma 6. Notice that for A ≠ ∅, the derivative D_{A} is not linear. Motivated by Haukkanen et al. [5], we derive the following properties on D_{A}. Lemma 14. Let A, A_{i}, i = 1, 2, … be nonempty subsets of primes. Then

(i) [D_{A_{1}}, D_{A_{2}}] = 0 if and only if A_{1} = A_{2}; (ii) Let n = ∏_{p∈∩A_{i}} p^{n_{p}}. Then D_{A_{i}}(n) = D_{A_{j}}(n), for all i, j. (iii) Let n = p^{j} where p is a prime. Assume that all prime divisors of 2, 3, …, j are not in

⋃_{i} A_{i}. Then, for any positive integers k and i_{1}, …, i_{k}, we have

D_{A_{i_{1}}} ⋯ D_{A_{i_{k}}} (n) = D_{p}^{k}(n).

Proof. It follows by direct calculations.

By inspecting the proof of Lemma 13, we extend it to the general arithmetic derivatives. Lemma 15. Let p ∈ A, q ∈ B and A ⊥ B. Then

(i) [D_{A}, ℓ_{p}] = I; (ii) [D_{A}, ℓ_{q}] = 0; (iii) [D_{A}, ℓ_{p} + ℓ_{q}] = 0; (iv) [D_{A} + D_{B}, ℓ_{p} + ℓ_{q}] = 0.

4 Extensions of the Stroock-Lie bracket lemma to other rings

Motivated by Ufnarovski and Åhlander [14] and Haukkanen et al. [4], we discuss the extensions of Lemma 6 to commutative rings with the unique factorization property.

4.1 Extensions to polynomial rings

Consider a polynomial ring K[ℂ], which is a unique factorization domain. By F_{i} denote monic irreducible polynomials, i.e., single-variable polynomials with leading coefficients 1. Any F ∈ K[ℂ] admits the unique factorization F = z ∏_{i=1}^{k} F_{i}^{n_{i}}, where n_{i} ∈ ℕ and z ∈ ℂ. Following Ufnarovski and Åhlander [14], define the derivative of polynomials as

DF = F ∑_{i=1}^{k} \frac{n_{i}}{F_{i}}

and D (z) = 0, for z ∈ ℂ. Since D (zF) = zD (F) = zF ∑_{i=1}^{k} (n_{i}/F_{i}), it follows that D satisfies the Leibnitz rule. Notice that D (G) = 1 if and only if G is a monic irreducible polynomial. Consider a linear functional ℓ = ℓ_{G}(H) = GH, where G, H ∈ K[ℂ]. Then, by direct calculations similar to (4), the Stroock-Lie bracket identity holds [D, ℓ_{G}] = I if and only if D(G) = 1, i.e., G is the monic irreducible polynomial. Moreover, the sum of k monic irreducible polynomials is an irreducible polynomial with the leading coefficient k. Therefore, similar to Corollary 8, we then derive the following result.

Corollary 16. Let F_{i}, i = 1, …, k, be the monic irreducible polynomials in the polynomial ring K[ℂ]. Then [D, ℓ_{F_{1}} + ⋯ + ℓ_{F_{k}}] = kI.

4.2 Extensions to integers and rational numbers

Following Ufnarovski and Åhlander [14], we define

D (x) = x ∑_{i=1}^{k} \frac{x_{i}}{p_{i}} ,

where 0 < x = ∏_{i=1}^{k} p_{i}^{x_{i}} ∈ ℚ, p_{i} are different primes, and x_{i} are integers. Then, for 0 > x ∈ ℚ, we take D(x) = −D(x).

The function D is a map from ℚ to ℚ. As proved in Ufnarovski and Åhlander [14], the map D satisfies the Leibnitz rule. In particular, the Stroock-Lie bracket identity [D, ℓ_{x}] = I holds if and only if D(x) = 1. In this case, x does not need to be a prime, for example one can take x = −5/4 (see [14]).

5 Arithmetic type differential equations

5.1 First order linear arithmetic type differential equations

We begin by considering several cases of the arithmetic type differential equations.

Lemma 17. Let A be a nonempty set of primes and x be a positive integer. Then

(i) D_{A}(x) = 0 if and only if x ⊥ A;

(ii) D_{A}(x) = 1 if and only if x is a prime in A;

(iii) D_{A}(x) = x if and only if x = p^{p}k for p ∈ A and a positive integer k ⊥ A;

(iv) pD_{A}(x) = x, where p ∈ A, if and only if x = pk for a positive integer k ⊥ A.

Proof. (i) Notice that D_{A} = ∑_{p∈A} D_{p}, D_{p}(x) ≥ 0. Haukkanen et al. [5, Theorem 1] proved that D_{p}(x) = 0 if and only if p ∤ x. Thus, D_{A} = 0 if and only if x ⊥ A.

(ii) For the above representation, D_{A}(x) = 1 if and only if there exists p ∈ A such that D_{p}(x) = 1, which is equivalent to x = p.

(iii) We follow arguments in the proofs of [14, Theorem 4 and 5]. Assume that x = p^{j}k, where p ∤ k and p ∈ A. Then D_{A}(x) = p^{j−1}(jk + pD_{A}(k)). And so, if 0 < j < p, then D_{A}(x) = p^{j−1}k̃, where p ∤ k̃, implying that D_{A}(x) ≠ x. On the other hand, assume that x = p^{p}k with k > 1. Then

D_{A}(x) = p^{p}(k + D_{A}(k)) = p^{p}k

if and only if D_{A}(k) = 0, which holds by (i) if and only if k ⊥ A.

(iv) Clearly p|x. Let x = np, n ∈ ℕ. By assumption p ∈ A and, hence, D_{A}(p) = 1. By the Leibnitz rule, we obtain

pD_{A}(x) = pD_{A}(np) = p(pD_{A}(n) + n) ⩾ pn = x,

where the equality holds if and only if D_{A}(n) = 0. By (i), then n ⊥ A, proving (iv).

5.2 Lie bracket arithmetic type differential equations

Fix a nonempty subset A of primes and consider the following equation

x = [D_{A}, ℓ_{x}] (a) = D_{A}(x)a,

where a, x are positive integers.

In the next lemma, we characterize those pairs (a, x) that satisfy the equation.

Lemma 18. The arithmetic differential equation aD_{A}(x) = x has a solution in natural numbers a, x if and only if one of the following statements is satisfied:

(i) a = p and x = kp;

(ii) a = 1 and x = kp^{p},

where p is a prime in A and k ⊥ A.

Proof. To identify the pair (a, x), we first determine a, and then solve the equation aD_{A}(x) = x. First, assume a = 1. The equation becomes D_{A}(x) = x. According to property (iii) of Lemma 17, then x = p^{p}k, where p is a prime in A and k ⊥ A.

Next, assume a = p with p ∈ A. Then the equation becomes pD_{A}(x) = x. By property (iv) of Lemma 17, we then have x = pk, where p is a prime in A and k ⊥ A.

Now assume that a ≠ 1. We show that a = p, for some p ∈ A. Firstly, consider a subcase a ⊥ A. Then the equation becomes aD_{A}(x) = x. Hence,

x = aD_{A}(x) = aD_{A}[aD_{A}(x)] = a^{2}D_{A}^{2}(x) .

For any j, we then derive x = a^{j}D_{A}^{j}(x), which is not possible. Hence, a = pc, for some p ∈ A and c ≥ 1. The equation becomes pcD_{A}(x) = x and it remains to show that c = 1. We proceed by absurd and assume that c > 1. Let D_{A}(x) = b. Then

pcD_{A}(x) = pcD_{A}(pcb) = pc(cb + pD_{A}(cb)) > pcb = x,

because c > 1 and D_{A}(cb) ≥ 0. Thus, the statement x = pcD_{A}(x) is not possible. Hence, c = 1 and the proof is complete.

6 Acknowledgments

We are grateful to the Editor-in-Chief and the referee for their valuable comments and suggestions.

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2000 Mathematics Subject Classification: Primary 11A25; Secondary 11A41, 11R27.

Keywords: arithmetic derivative, unique factorization, prime number, Leibnitz rule, integer sequence.

Received July 26 2019; revised version received February 19 2020. Published in Journal of Integer Sequences, February 19 2020.

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