GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 4, 1999, 299-306

ON SOME RINGS OF ARITHMETICAL FUNCTIONS

D. BHATTACHARJEE

Abstract. In this paper we consider several constructions which from a given B-product ∗_{B} lead to another one _{∗}_{B}. We shall be interested in finding what algebraic properties of the ring R_{B} = ⟨C^{ℕ}, +, ∗_{B} ⟩ are shared also by the ring R_{B} = ⟨C^{ℕ}, +, ∗_{B} ⟩. In particular, for some constructions the rings R_{B} and R_{B} will be isomorphic and therefore have the same algebraic properties.

§ 1. Introduction

In [1] the author shows a new kind of convolution product called the B-product defined as follows. For every natural number n let B_{n} be the set of some pairs (r, s) of divisors of n.

For arithmetical functions f and g we define their B-product f∗_{B} as

(f ∗_{B} g)(n) = ∑_{(r,s)∈B_{n}} f(r)g(s) for n = 1, 2, 3, ….  (1)

This B-product generalizes simultaneously the A product of W. Narkiewicz [2] and the l.c.m. product and has a nonempty intersection with the ψ-product of D. H. Lehmer [3]. The τ-product of H. Scheid [4] is also a particular case of the B-product.

In [5] the author consideres a special kind of the B-product called the “multiplicative B-product”. A B-product is multiplicative iff the following condition holds:

For every pair (m, n) of relatively prime natural numbers we have

(r, s) ∈ B_{mn} iff (r^{(m)}, s^{(m)}) ∈ B_{m} and (r^{(n)}, s^{(n)}) ∈ B_{n},  (2)

where k^{(n)} denotes the g.c.d. of k and n.

299

1991 Mathematics Subject Classification. 11A25.

Key words and phrases. Convolution product, multiplicative B-product, twisted product, strong associativity, unitary B-product, Narkiewicz product.

1072-947X/99/0700-0299$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1999 Plenum Publishing Corporation

In this paper we consider several constructions which, given a B-product ∗_{B}, lead to another one ˜∗_{B}. We shall be interested in finding what algebraic properties of the ring R_{B} = ⟨C^{ℕ}, +, ∗_{B}⟩ are shared by the ring R_{B} = ⟨C^{ℕ}, +, ∗_{B}⟩, where C^{ℕ} denotes the set of all arithmetical functions. In particular, for some constructions the rings R_{B} and R_{B} will be isomorphic and therefore have the same algebraic properties.

§ 2. Twisted Products

Let R_{B} be a commutative and associative ring with a unit e. For a fixed invertible element h ∈ R_{B}, let f^{(h)} = f ∗_{B} h, where f ∈ R_{B}.

Evidently, f ↦ f^{(h)} is the one-to-one mapping of R_{B} onto itself preserving the set of invertible elements. It is also an isomorphism of the additive group of R_{B}.

We define the twisted product ∗h_{B} as follows:

f ∗h_{B} g = f ∗_{B} g ∗_{B} h for f, g ∈ R_{B}.

In other words,

f^{(h)} ∗_{B} g^{(h)} = (f ∗h_{B} g)(h).

This means that the rings R_{B}^{h} = ⟨C^{ℕ}, +, ∗_{B}⟩ are isomorphic. The isomorphism R_{B} → R_{B}^{h} is given by the twisting f ↦ f^{(h^{−1})}, where h^{−1} is the inverse of h in R_{B}.

Therefore the ring R_{B}^{h} is also commutative and associative and e^{(h^{−1})} = e ∗_{B} h^{−1} = h^{−1} is its unit element.

Let us remark that if the product ∗_{B} is multiplicative and if the function h is multiplicative, then the product ∗h_{B} is multiplicative. In fact, if functions f and g are multiplicative, then the function f ∗h_{B} g = f ∗_{B} g ∗_{B} h is multiplicative, since ∗_{B} preserves the multiplicativity.

In general, the twisted multiplication ∗h_{B} is not a B-product. We shall give below some conditions on ∗_{B} and on h for ∗h_{B} to be a B-product.

Theorem 2.1. The twisted product ∗h_{B} is a B-product iff for every r, s, and n

∑_{\begin{matrix}d_{1}, d_{2} \\ (r,s)∈B_{d_{1}} \\ (d_{1},d_{2})∈B_{n} \end{matrix}} h(d_{2}) = 0 or 1.  (3)

Moreover if we denote by B_{n}^{h} the set corresponding to the B-product ∗h_{B}, then (r, s) ∈ B_{n}^{h} iff sum (3) is equal to 1.

Proof. ⟹ We have

(e_{r} ∗_{B} e_{s})(n) = \begin{cases}1 & if (r, s) ∈ B_{n}, \\ 0 & otherwise. \end{cases}

Therefore, since by the assumption ∗h_{B} is a B-product, we have

(e_{r} ∗h_{B} e_{s})(n) = \begin{cases}1 & if (r, s) ∈ B_{n}^{n}, \\ 0 & otherwise. \end{cases}

On the other hand,

(e_{r} ∗h_{B} e_{s})(n) = (e_{r} ∗_{B} e_{s} ∗_{B} h)(n) = ∑_{\begin{matrix}d_{1}, d_{2} \\ (d_{1},d_{2})∈B_{n} \end{matrix}}(e_{r} ∗_{B} e_{S})(d_{1})h(d_{2}) =

= ∑_{\begin{matrix}d_{1}, d_{2} \\ (d_{1},d_{2})∈B_{n} \\ (r,s)∈B_{d_{1}} \end{matrix}} h(d_{2}).  (4)

Hence the result follows.

⟸ Define B_{n}^{h} as the set of pairs (r, s) such that sum (3) is equal to 1. In view of (4), for any functions f and g, we have

(f ∗h_{B} g)(n) = [( ∑_{r=1}^{n} f(r)e^{r}) ∗h_{B} ∗h_{B}( ∑_{s=1}^{n} g(s)e_{s})](n) =

= ∑_{r,s=1}^{n} f(r)g(s)(e_{r} ∗h_{B} e_{s})(n) = ∑_{(r,s)∈B_{n}^{h}}f(r)g(s) = (f ∗_{B^{h}} g)(n).

If we start from the Dirichlet convolution ∗, then condition (3) of Theorem 2.1 takes the form

∑_{\begin{matrix}d_{1}, d_{2} \\ r_{s}=d_{1} \\ d_{1}d_{2}=n \end{matrix}} h(d_{2}) = h(\frac{n}{r_{s}} ) = 0 or 1

for every r, s, and n.

Thus h(m) = 0 or 1 for every m. Moreover, since h is invertible, we conclude that h(1) ≠ 0, i.e., h(1) = 1.

Corollary 2.2. Let h(1) = 1 and h(n) = 0 or 1 for every n. Then the multiplication ∗^{h} defined by

f ∗^{h} g = f ∗ g ∗ h,

where ∗ is the Dirichlet convolution, is a B-product and

B_{n}^{h} = \{(r, s) : rs|n and h(\frac{n}{rs}) = 1\}.

If, moreover, the function h is multiplicative, then the twisted product ∗^{h} preserves the multiplicativity.

Rings R = ⟨C^{ℕ}, +, ∗⟩ and R^{h} = ⟨C^{ℕ}, +, ∗^{h}⟩ are isomorphic. The isomorphism is given by f ↦ f ∗ h^{−1}, where h^{−1} is the Dirichlet inverse of h. The function h^{−1} is the unit of the ring R^{h}.

Consequently R^{h} is a local ring without zero divisors.

A function f is invertible in R iff it is invertible in R^{h} iff f(1) ≠ 0, and the inverse f^{(−1)} of f in R^{h} is given by

f^{(−1)} = f^{−1} ∗ h^{−2}.

§ 3. Strong Associativity

We introduce the important notion of a strong associativity. We say that a B-product is strongly associative iff for fixed d_{1}, d_{2}, d_{3}, n, the fulfilment of

(r, d_{1}) ∈ B_{n} and (d_{2}, d_{3}) ∈ B_{r}  (5)

for some r implies that w = \frac{d_{1}r}{d_{2}} (which is evidently a natural number) satisfies the condition

(d_{2}, w) ∈ B_{n} and (d_{3}, d_{1}) ∈ B_{w}  (6)

and, conversely, the fulfilment of (6) for w implies that r = \frac{d_{2}w}{d_{1}} (which is a natural number) satisfies (5).

From this definition and Theorem 2.1 of [1] it follows that every strongly associative B-product is associative. The converse does not hold in generaL. Nevertheless the following theorem is true.

Theorem 3.1. An associative τ-product is strongly associative iff

d_{2}τ(d_{3}, d_{1}) = τ(d_{2}, d_{3})d_{1} for all d_{1}, d_{2}, d_{3} satisfying

τ(τ(d_{2}, d_{3}), d_{1}) ≠ 0. (7)

Proof. ⟸ Let r = τ(d_{2}, d_{3}) and n = τ(r, d_{1}) = τ(τ(d_{2}, d_{3}), d_{1}) ≠ 0. Then (5) holds and by the strong associativity we have (6), i.e., w = τ(d_{3}, d_{1}), where w = \frac{d_{1}r}{d_{2}} . Therefore we get (7).

⟹ Suppose that (5) holds for some d_{1}, d_{2}, d_{3}, n. Then r = τ(d_{2}, d_{3}) and n = τ(r, d_{1}) = τ(τ(d_{2}, d_{3}), d_{1}). Since n ≠ 0, we have (7) by the assumption. Hence

w = \frac{d_{1}r}{d_{2}} = τ(d_{3}, d_{1}), i.e., (d_{3}, d_{1}) ∈ B_{w}.

Therefore

τ(d_{2}, w) = τ(d_{2}, τ(d_{3}, d_{1})) = τ(τ(d_{2}, d_{3}), d_{1}) = n,

i.e., (d_{2}, w) ∈ B_{n}. Thus (6) holds.

Similarly, we can prove that (6) implies (5).

§ 4. Unitary B-products

For a given B-product ∗_{B} we define the corresponding unitary B-product denoted by ∘_{B} as follows. Let

B_{n}^{0} = \{(r, s) : (r, s) ∈ B_{n}, (r, s) = 1, (rs, \frac{n}{rs}) = 1\}.

Then (f ∘_{B} g)(n) = ∑_{(r,s)∈B_{n}^{0}} f(r)g(s).

We shall investigate relations between the corresponding properties of B-products ∗_{B} and ∘_{B}.

Theorem 4.1.

(i) If ∗_{B} is commutative, then ∘_{B} is commutative.

(ii) If ∗_{B} is strongly associative, then ∘_{B} is associative.

Proof. (i) is clear. To prove (ii) we have to show that for every d_{1}, d_{2}, d_{3}, and n

∑′ ; \begin{matrix}r \\ (r,d_{1})∈B_{n}^{0} \\ (d_{2},d_{3})∈B_{r}^{0} \end{matrix} = ^{∑′} ; \begin{matrix}w \\ (d_{2},w)∈B_{n}^{0} \\ (d_{3},d_{1})∈B_{w}^{0} \end{matrix} .  (8)

Suppose that r satisfies (r, d_{1}) ∈ B_{n}^{0}, (d_{1}, d_{2}) ∈ B_{r}^{0}, i.e., (r, d_{1}) ∈ B_{n}, (d_{2}, d_{3}) ∈ B_{r} and, moreover,

(r, d_{1}) = 1, (rd_{1}, \frac{n}{rd_{1}} ) = 1, (d_{2}, d_{3}) = 1, (d_{2}d_{3}, \frac{r}{d_{2}d_{3}} ) = 1. (∗)

By the strong associativity of ∗_{B} we find for w = \frac{rd_{1}}{d_{2}} that (d_{2}, w) ∈ B_{n}, (d_{3}, d_{1}) ∈ B_{w}.

To prove that w satisfies

(d_{2}, w) ∈ B_{n}^{0}, (d_{3}, d_{1}) ∈ B_{w}^{0},

it is sufficient to show that

\begin{matrix}(d_{2}, w) = 1, (d_{2}w, \frac{n}{d_{2}w} ) = 1, \\ (d_{3}, d_{1}) = 1, (d_{3}d_{1}, \frac{w}{d_{3}d_{1}} ) = 1 \end{matrix}  (∗∗)

in view of (6). The formulas (∗∗) follow from (∗). We have proved that to every summand of L, H, S of (8) there corresponds a summand in R, H, S. Similarly, one can give the inverse correspondence.

Theorem 4.2. If e_{1} is the unit in the ring R_{B}, then e_{1} is also the unit for the corresponding unitary product ∘_{B}.

Proof. In view of Corollary 2.6 of [1] e_{1} is the unit in the ring R_{B} iff (1, n), (n, 1) ∈ B_{n} for n ≥ 1 and (k, 1) ∉ B_{n}, (1, k) ∉ B_{n} for k ≠ n, n > 1.

These conditions clearly imply that (1, n), (n, 1) ∈ B_{n}^{0} and (k, 1) ∉ B_{n}^{0} and (1, k) ∉ B_{n}^{0} for k ≠ n, n > 1.

Therefore, using once more Corollary 2.6 of [1], we deduce that e_{1} is the unit with respect to ∘_{B}.

Theorem 4.3. If the B-product ∗_{B} is multiplicative, then the corresponding unitary product ∗_{B} is multiplicative.

Proof. Let m and n be coprime natural numbers and suppose that

(r, s) ∈ B_{mn}^{0}.  (A)

Then (r, s) ∈ B_{mn}^{0} and hence by the multiplicativity of ∗_{B} we get (r^{(m)}, s^{(m)}) ∈ B_{m} and, similarly, (r^{(n)}, s^{(n)}) ∈ B_{n}. Moreover, from (r, s)=1 and (rs, \frac{mn}{rs})=1 we conclude that

(r^{(m)}, s^{(m)}) = 1, (r^{(n)}, s^{(n)}) = 1 and futher

(r^{(m)}s^{(m)}, \frac{m}{r^{(m)}s^{(m)}} ) = 1, (r^{(n)}s^{(n)}, \frac{n}{r^{(n)}s^{(n)}} ) = 1.

Thus

(r^{(m)}, s^{(m)}) ∈ B_{m}^{0} and (r^{(n)}, s^{(n)}) ∈ B_{n}^{0}.  (B)

Similarly, one can prove that (B) implies (A).

§ 5. The Narkiewicz Product

For a given B-product ∗_{B} we define a new B-product Δ_{B} as follows: Let

B_{n}^{Δ} = {(r, s) : (r, s) ∈ B_{n}, rs = n}.

Then

(fΔ_{B}g)(n) = ∑_{(r,s)∈B_{n}^{Δ}} f(r)f(s) = ∑_{(r, \frac{n}{r})∈B_{n}} f(r)g(\frac{n}{r}).

Evidently, Δ_{B} is the Narkiewicz product. It will be called the Narkiewicz product corresponding to the B-product ∗_{B}. We shall investigate some important properties of the product Δ_{B}.

Theorem 5.1. If the product ∗_{B} is commutative, then Δ_{B} is also commutative.

Proof is clear.

Theorem 5.2. If the B-product ∗_{B} is strongly associative, then the corresponding Narkiewicz product Δ_{B} is associative.

Proof. For fixed d_{1}, d_{2}, d_{3}, n we have

∑′ ; \begin{matrix}r \\ (r,d_{1})∈B_{n}^{Δ} \\ (d_{2},d_{3})∈B_{r}^{Δ} \end{matrix} 1 = ^{∑′} ; \begin{matrix}r \\ (r,d_{1})∈B_{n}, rd_{1}=n \\ (d_{2},d_{3})∈B_{r}, d_{2}d_{3}=r \end{matrix} 1.

If n ≠ d_{1}d_{2}d_{3}, then this sum is equal to 0. We assume that n = d_{1}d_{2}d_{3}. Then the sum is equal to

∑_{\begin{matrix}( \frac{n}{d_{1}} ,d_{1})∈B_{n} \\ (d_{2},d_{3})∈B_{\frac{n}{d_{1}}} \end{matrix}} 1

for r = \frac{n}{d_{1}} .

By the strong associativity of ∗_{B} for w = \frac{rd_{1}}{D_{2}} the above sum is equal to

∑_{\begin{matrix}w \\ (d_{2},w)∈B_{n} \\ (d_{3},d_{1})∈B_{w} \end{matrix}} 1 = ∑_{\begin{matrix}(d_{2}, \frac{n}{d_{2}} )∈B_{n} \\ (d_{3},d_{1})∈B_{\frac{n}{d_{2}}} \end{matrix}} 1 = ∑_{\begin{matrix}(d_{2}\overline{w})∈B_{n}^{Δ} \\ (d_{3},d_{1})∈BΔ ; \overline{w} \end{matrix}} 1.

Therefore the product Δ_{B} is associative.

Acknowledgement

This paper forms a part of the author’s Ph. D. thesis titled ”Rings of arithmetical functions” written under the supervision of Prof. Dr. Hab. Jerzy Browkin from Institute of Mathematics, Warsaw University, Warsaw, Poland. The author wishes to express his sincere gratitude to Prof. Browkin for his invaluable guidance and suggestion throughout the period of his doctoral work. The author is also thankful to the Polish Government for supporting him with a study grant during his stay in Poland and the Government of India for sponsoring him.

References

1. D. Bhattacharjee, B-product and its properties. Bull. Pure Appl. Sci. Sect. E. Math. 16(1997), No. 2 (to appear).

2. W. Narkiewicz, On a class of arithmetical convolutions. Colloq. Math. 10(1963), 81–94.

3. D. H. Lehmer, A new calculus of numerical functions. Amer. J. Math. 53(1931), 843–854.

4. H. Scheid, Einige Ringe zahlentheoretischer Funktionen. J. reine angew. Math. 237(1969), 1–11.

5. D. Bhattacharjee, Multiplicative B-product and its properties. Georgian Math. J. 5(1998), No. 4, 415–420.

(Received 08.10.1997)

Author’s address: Department of Mathematics Pachhunga University College North-Eastern Hill University Aizawl 796001, Mizoram India