GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 4, 1998, 315-320

MULTIPLICATIVE B-PRODUCT AND ITS PROPERTIES

D. BHATTACHARJEE

Abstract. The author considers a special kind of B-product which is called the multiplicative B-product. The most important property of multiplicative B-product is that it preserves multiplicativity of functions.

In the previous work [1], the author showed 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 the arithmetical functions f and g their B-product f ∗_{B} g is as follows:

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

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 this paper the author considers a special kind of B-product which is called the multiplicative B-product. It is defined that a B-product is multiplicative if for every pair (m, n) of relatively prime natural numbers

B_{mn} = {(r_{1}r_{2}, s_{1}s_{2}) : (r_{1}, s_{1}) ∈ B_{m}, (r_{2}, s_{2}) ∈ B_{n}}.

This definition can also be formulated as follows. For every k and n denote by k^{(n)} the g.c.d. of k and n. Then the B-product is multiplicative iff 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}.  (1)

We say that an arithmetical function f is multiplicative iff f(mn) = f(m)f(n) for every pair (m, n) of relatively prime natural numbers.

315

1991 Mathematics Subject Classification. 11A25.

Key words and phrases. Arithmetical function, convolution product, multiplicative function, B-product, multiplicative B-product.

1072-947X/98/0700-0315$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

A multiplicative B-product is determined uniquely by the sets B_{p^{n}}, where p is a prime and n ≥ 1. These sets can be given arbitrarily. Moreover, for every multiplicative B-product we have B_{1} = {(1, 1)} provided that B_{n} ≠ ∅ for some n. As for multiplicative function, it is determined uniquely by its values on arguments of the form p^{n}, where p is a prime and n ≥ 1. These values can be given arbitrarily. Moreover, for every multiplicative function f we have f(1) = 1 provided that f(n) ≠ 0 for some n.

In this paper we discuss some properties of this multiplicative B-product. The most important property is that this product preserves multiplicativity of functions. More precisely, the following theorem holds.

Theorem 1.1. The B-product of multiplicative functions is a multiplicative function iff the B-product is multiplicative.

Proof. ⇐ Suppose that the B-product is multiplicative. Let f, g be multiplicative functions and let (m, n) be a pair of relatively prime natural numbers. Then

(f ∗_{B} g)(mn)= ∑_{(r,s)∈B_{mn}} f(r)g(s)=  ∑_{(r^{(m)},s^{(m)})∈B_{m}} ∑_{(r^{(n)},s^{(n)})∈B_{n}} f(r^{(m)}, r^{(n)})×

×g(s^{(m)}, s^{(n)})=  ∑_{(r^{(m)},s^{(m)})∈B_{m}} f(r^{(m)})g(s^{(m)}) ∑_{(r^{(n)},s^{(n)})∈B_{n}} f(r^{(n)})g(s^{(n)})=
= (f ∗_{B} g)(m) · (f ∗_{B} g)(n).

⇒ Let (m, n) be a pair of relatively prime natural numbers and let r|mn, s|mn. We shall prove by induction on rs that (1) holds:

Let rs = 1, i.e., r = s = 1. For every d we have

(e_{1} ∗_{B} e_{1})(d) = ∑_{(r,y)∈B_{d}} e_{1}(r) e_{1}(y) = \begin{cases}1 if (1, 1) ∈ B_{d}, \\ 0 otherwise. \end{cases}

By assumption, the function e_{1} ∗_{B} e_{1} is multiplicative. Therefore (e_{1} ∗_{B} e_{1})(mn) = (e_{1} ∗_{B} e_{1})(m) · (e_{1} ∗_{B} e_{1})(n). Consequently, (e_{1} ∗_{B} e_{1})(mn) = 1 iff (e_{1} ∗_{B} e_{1})(m) = (e_{1} ∗_{B} e_{1})(n) = 1. Thus (1, 1) ∈ B_{mn} iff (1, 1) ∈ B_{m} and (1, 1) ∈ B_{n}.

Now let r_{0}|mn, s_{0}|mn, r_{0}s_{0} > 1 and assume that for every r|mn, s|mn with rs < r_{0}s_{0} (1) holds. Put

f(d) = \begin{cases}1 & if d|r_{0}, \\ 0 & otherwise, \end{cases} g(d) = \begin{cases}1 & if d|s_{0}, \\ 0 & otherwise. \end{cases}

The functions f and g are multiplicative. Therefore, by assumption, the function f ∗_{B} g is multiplicative. In particular, we have

(f ∗_{B} g)(mn) = (f ∗_{B} g)(m) · (f ∗_{B} g)(n).  (2)

MULTIPLICATIVE b-PRODUCT AND ITS PROPERTIES 317

For every k we have

(f ∗_{B} g)(k)= ∑_{(r,y)!∈B_{k}}f(r)g(y)= ∑_{\begin{matrix}(r,y)∈B_{k} \\ r|r_{0}^{(k)},y|s_{0}^{(k)} \end{matrix}}1= ∑_{(r_{0}^{(k)},s_{0}^{(k)})∈B_{k}}1+ ∑_{\begin{matrix}(r,y)∈B_{k}r|r_{0}^{(k)},s|s_{0}^{(k)} \\ rs<r_{0}^{(k)}s_{0}^{(k)} \end{matrix}}1.

Substituting this into (2), for k = mn, m, and n, and using the multiplicativity of the B-product, we get

∑_{(r_{0},s_{0})∈B_{mn}} 1 + ∑_{\begin{matrix}(r_{0},s)∈B_{mn} \\ r|r_{0},s|s_{0} \\ rs<r_{0}s_{0} \end{matrix}} 1 = ∑_{\begin{matrix}(r_{0}^{(m)},s_{0}^{(m)})∈B_{m} \\ (r_{0}^{(n)},s_{0}^{(n)})∈B_{n} \end{matrix}} 1 + ∑_{\begin{matrix}(r,s)∈B_{mn} \\ r|r_{0},s|s_{0} \\ rs<r_{0}s_{0} \end{matrix}} 1.

Thus it follows that (r_{0}, s_{0}) ∈ B_{mn} iff (r_{0}^{(m)}, s_{0}^{(m)}) ∈ B_{m} and (r_{0}^{(n)}, s_{0}^{(n)}) ∈ B_{n}.

Now we give conditions for the commutativity and associativity of the multiplicative B-product analogous to those given in [1] for an arbitrary B-product. We omit the proofs since they are straightforward.

Theorem 1.2. If a B-product is multiplicative, then it is commutative iff the following condition holds:

For every nonnegative integer k, m, n and every prime number p we have (p^{k}, p^{m}) ∈ B_{p^{n}} iff (p^{m}, p^{k}) ∈ B_{p^{n}}.

Theorem 1.3. If a B-product is multiplicative, then it is associative iff the following condition is satisfied.

For every nonnegative integer j, k, m, n and a prime number p we have

∑_{\begin{matrix}b \\ (p^{b},p^{j})∈B_{p^{n}} ,(p^{k},p^{m})∈B_{p^{b}} \end{matrix}} 1 = ∑_{\begin{matrix}c \\ (p^{k},p^{c})∈B_{p^{n}} ,(p^{m},p^{j})∈B_{p^{c}} \end{matrix}} 1.

Theorem 1.4. If e is the unit in the system R_{B} = ⟨ℂ^{ℕ}, +, ∗_{B}⟩, where ℂ^{ℕ} stands for the set of all arithmetical functions, i.e., the set of all complex valued functions defined on the set of natural numbers and ∗_{B} is a multiplicative B-product, then the function e is multiplicative.

Proof. Let m and n be relatively prime natural numbers. We shall prove that e(mn) = e(m) e(n) proceeding by induction on mn. For mn = 1, this equality follows from Corollary 2.5 of [1]. Let mn > 1. In view of Theorem 2.4 of [1], for every r|mn and k|mn we have

e_{k^{(m)}}(m) = ∑_{\begin{matrix}r^{(m)} \\ (r^{(m)},k^{(m)})∈B_{m} \end{matrix}} e(r^{(m)}), e_{k^{(n)}}(n) = ∑_{\begin{matrix}r^{(n)} \\ (r^{(n)},k^{(n)})∈B_{n} \end{matrix}} e(r^{(n)}).

Therefore from the multiplicativity of the B-product we get

e_{k}(mn) = e_{k^{(m)}}(m) · e_{k^{(n)}}(n) = ∑_{\begin{matrix}r \\ (r,k)∈B_{mn} \end{matrix}} e(r^{(m)}) · e(r^{(n)}).  (3)

In view of Corollary 2 of Theorem 2.5 of [1] there exists k such that (mn, k) ∈ B_{mn}. For this k, in view of (3) and the inductive assumption, we have e_{k}(mn) = ∑_{\begin{matrix}r \\ (r,k)∈B_{mn},r<mn \end{matrix}} e(r) + e(m) · e(n). On the other hand, from

Theorem 2.4 of [1] it follows that

e_{k}(mn) = ∑_{\begin{matrix}r \\ (r,k)∈B_{mn} \end{matrix}} e(r) = ∑_{\begin{matrix}r \\ (r,k)∈B_{mn},r<mn \end{matrix}} e(r) + e(mn).

The two latter relations lead to e(mn) = e(m) e(n).

Theorem 1.5. If a B-product is multiplicative, then a multiplicative function e is the unit of the system R_{B} iff, for every nonnegative integer m and t and a prime number p, it satisfies the condition

∑_{\begin{matrix}s \\ (p^{t},p^{s})∈B_{p^{m}} \end{matrix}} e(p^{s}) = ∑_{\begin{matrix}s \\ (p^{s},p^{t})∈B_{p^{m}} \end{matrix}} e(p^{s}) = e_{t}(m).

Corollary 1.6. If the B-product is multiplicative, then the function e_{1} is the unit of the system R_{B} iff (p^{k}, 1) ∈ B_{p^{n}} ⇔ k = n ⇔ (1, p^{k}) ∈ B_{p^{n}} for every nonnegative integer k and n and a prime number p.

Proof. ⇒ The result follows immediately from Corollary 2.6 of [1]. We have

∑_{\begin{matrix}s \\ (p^{k},p^{s})∈B_{p^{n}} \end{matrix}} e_{1}(p^{s}) = \begin{cases}1 if (p^{k}, 1) ∈ B_{p^{n}} \\ 0 otherwise \end{cases} = \begin{cases}1 if k = n \\ 0 otherwise \end{cases} = e_{p^{k}}(p^{n}).

Similarly, one can prove that ∑_{\begin{matrix}s \\ (p^{s},p^{k})∈B_{p^{n}} \end{matrix}} e_{1}(p^{s}) = e_{p^{k}}(p^{n}). Therefore

from Theorem 1.5 it follows that e_{1} is the unit of the system R_{B}.

By extending Corollary 2.5 of [1] we shall give below some necessary conditions for a function e to be the unit of the system R_{B}.

Proposition 1.7. If e is the unit of the system R_{B}, then

(i) e(1) = 1 and B_{1} = {(1, 1)}.

(ii) For every prime number p we have e(p) = 1 and B_{p} = {(p, p)} or e(p)=−1 and B_{p} ={(1, 1), (1, p), (p, 1)} or

e(p)=0 and B_{p} ={(1, p), (p, 1)} or {(1, p), (p, 1), (p, p)}.

(iii) For every prime number p, e(p^{2}) = 0, 1, or −1.

Proof. To prove (i) see Corollary 2.5 of [1]. As for (ii), denote for fixed n,

x_{rs} = x_{rs}^{(n)} = \begin{cases}1 if (r, s) ∈ B_{n}, \\ 0 otherwise. \end{cases}

For n = p Theorem 1.5 gives

x_{11} + x_{1p}e(p) = x_{11} + x_{p1}e(p) = 0, x_{p1} + x_{pp}e(p) = x_{1p} + x_{pp}e(p) = 1.

From these equalities it easily follows that:

If x_{11} = 1, then x_{1p} = x_{p1} = 1, e(p) = −1,

If x_{11} = 0 and x_{1p} = 0, then x_{p1} = 0, x_{pp} = 1, e(p) = 1,

If x_{11} = 0 and x_{1p} = 1, then x_{p1} = 1, e(p) = 0 and x_{pp} = 0 or 1.

To prove (iii) a reasoning similar to the above leads for n = p^{2} to the equalities

x_{11} + x_{1p} e(p) + x_{1p^{2}} e(p^{2}) = x_{11} + x_{p1} e(p) + x_{p^{2}1}e(p^{2}) = 0,

x_{p1} + x_{pp} e(p) + x_{pp^{2}} e(p^{2}) = x_{1p} + x_{pp} e(p) + x_{p^{2}p} e(p^{2}) = 0,  (4)

x_{p^{2}1} + x_{p^{2}p} e(p) + x_{p^{2}p^{2}} e(p^{2}) = x_{1p^{2}} + x_{pp^{2}} e(p) + x_{p^{2}p^{2}} e(p^{2}) = 1.

Suppose that e(p^{2}) ≠ 0, 1, −1. Then from the last equality it follows that x_{p^{2}1} = 1 or x_{p^{2}p} = 1. Therefore from the first two equalities of (4) we deduce respectively that e(p^{2}) = −x_{11} − x_{p1}e(p) or e(p^{2}) = −x_{1p} − x_{pp}e(p).

Since in view of (ii) we have e(p) = 0, 1, or −1, we conclude that in both cases e(p^{2}) = −2, e(p) = 1 and respectively x_{11} = x_{p1} = 1 or x_{1p} = x_{pp} = 1.

In the first case from the first equality of (4) we deduce that x_{1p^{2}} = 1 and from the second one that x_{pp} = x_{pp^{2}} = 1. Hence the last part of the third equality of (4) takes the form 1 + 1 − 2x_{p^{2}p^{2}} = 1. This is a contradiction. In the second case we obtain a contradiction in a similar way.

Theorem 1.8. If e is the unit in the system R_{B}, where ∗_{B} is a multiplicative B-product, then |e(n)| < n for every n.

Proof. In view of Theorem 1.4 the function e is multiplicative. Therefore it is sufficient to prove that |e(p^{n})| < p^{n} for every prime number p and every natural number n.

From Theorem 1.5 we have

∑_{\begin{matrix}r \\ (p^{r},p^{k})∈B_{p^{n}} \end{matrix}} e(p^{r}) = e_{p^{k}}(p^{n}) = ∑_{\begin{matrix}r \\ (p^{k},p^{r})∈B_{p^{n}} \end{matrix}} e(p^{r}).  (5)

From (10), for k = n, it follows that (p^{t}, p^{n}) ∈ B_{p^{n}} for some t. Take the least value of t. If t = n, then from (5), for k = n, it follows that e(p^{n}) = 1.

If t < n, then from (5), for k = t (in the second equality) we get

0 = ∑_{\begin{matrix}r \\ (p^{t},p^{r})∈B_{p^{n}} \end{matrix}} e(p^{r}) = ∑_{\begin{matrix}r<n \\ (p^{t},p^{r})∈B_{p^{n}} \end{matrix}} e(p^{r}) + e(p^{n}).

Therefore, in view of the inductive assumption, we get

|e(p^{n})| ≤ ∑_{r=0}^{n−1} |e(p^{r})| ≤ ∑_{r=0}^{n−1} p^{r} < p^{n}.

Remark. In fact, the above proof jointly with the Proposition 1.7 gives a stronger inequality |e(p^{n})| ≤ 2^{n−1} for n ≥ 1, and therefore |e(n)| ≤ 2^{Ω(n)−ω(n)} for every n, where ω(n) is the number of different prime factors of n, and Ω(n) is the total number of prime factors of n.

Acknowledgements

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 the Institute of Mathematics, Warsaw University, Warsaw, Poland. The author wishes to express his sincere gratitude to Prof. Browkin for his invaluable guidance and suggestions 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. of Pure and Appl. Sci. 16 E(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. American J. Math. 53(1931), 843–854.

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

(Received 19.10.1995; revised 11.06.1997)

Author’s address:

Department of Mathematics

Pachhunga University College

North-Eastern Hill University

Aizawl 796001, Mizoram

India