On the Kesava Menon norm of semimultiplicative functions

The Kesava Menon norm of an arithmetical function f is defined by N(f)(n)=(f∗λf)(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(f)(n) = (f*\lambda f)(n^2)$$\end{document}, where ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} denotes the Dirichlet convolution and λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} denotes Liouville’s function. The mth power Kesava Menon norm of f is defined inductively by N0(f)=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^0(f) = f$$\end{document}, Nm(f)=N(Nm-1(f))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^m(f)=N\big (N^{m-1}(f)\big )$$\end{document}, m=1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=1,2,\ldots $$\end{document} In this paper we prove that the mth power Kesava Menon norm of a semimultiplicative function is semimultiplicative and that the mth power Kesava Menon norm distributes over the Dirichlet convolution of semimultiplicative functions. In addition we show that the mth power Kesava Menon norm of a rational arithmetical function of degree (r, s) is a rational arithmetical function of the same degree.


Introduction
Let f be an arithmetical function (that is, a real-or complex-valued function on the set of positive integers). In 1963, Kesava Menon [4,Section 3] defined the norm of f as the arithmetical function N (f ) given by where * is the Dirichlet convolution (see (1)) and f is the conjugate of f . The conjugate is defined as where λ is Liouville's function (see (3)). The norm N (f ) is referred to as the Kesava Menon norm in the literature [6,Section 5]. Redmond and Sivaramakrishnan [9,Section 4]  In this paper we investigate the conjugate, the Kesava Menon norm and the mth power Kesava Menon norm of semimultiplicative functions. Semimultiplicative functions form a superclass of the class of the usual multiplicative functions. Quasimultiplicative functions lie between multiplicative and semimultiplicative functions. Rational arithmetical functions form the subgroup of the group of multiplicative functions under the Dirichlet convolution generated by completely multiplicative functions. An arithmetical function f is said to be a rational arithmetical function of degree (r, s) if it is the Dirichlet convolution of r completely multiplicative functions and the inverse of s completely multiplicative functions. For details of these various types of multiplicativity, see Sect. 2.
This paper is organized as follows. In Sect. 2 we review the known properties of arithmetical functions needed in this paper. In Sect. 3 we present new results. In Sect. 3 we first note that the conjugate of a semimultiplicative function is semimultiplicative and that the conjugate distributes over the Dirichlet convolution of any two arithmetical functions. We continue by applying these results to show that the mth power Kesava Menon norm of a semimultiplicative function is semimultiplicative, that is, the mth power Kesava Menon norm preserves semimultiplicativity. As special cases we obtain the same properties for quasimultiplicative and multiplicative functions. Therefore our result generalizes the result of Sivaramakrishnan [11,Section 2], namely that the usual Kesava Menon norm of a multiplicative function is multiplicative. In Sect. 3 we also prove that the mth power Kesava Menon norm distributes over the Dirichlet convolution of semimultiplicative functions, extending the result of Laohakosol and Pabhapote [6,Section 5], who proved that the mth power Kesava Menon norm distributes over the Dirichlet convolution of rational arithmetical functions. We apply our distributivity property to show that the mth power Kesava Menon norm preserves the Dirichlet inverse of a quasimultiplicative function.
In Sect. 4 of this paper we utilize the properties presented in Sect. 3 to prove that the mth power Kesava Menon norm of a rational arithmetical function of degree (r, s) is a rational arithmetical function of the same degree. Laohakosol and Pabhapote [6, Section 5] proved the same result in a different way. We also present the analogous results for the conjugate of semimultiplicative functions and rational arithmetical functions of degree (r, s).

Preliminaries on arithmetical functions
The Dirichlet convolution of arithmetical functions f and g is defined as (1) We may also interpret that an arithmetical function f is defined on the set of positive real numbers so that f (x) = 0 if x is not a positive integer. This makes it possible to present the Dirichlet convolution in the form This expression is useful in some calculations presented in this paper. The The Dirichlet convolution of multiplicative functions is multiplicative. The same applies to quasimultiplicative and semimultiplicative functions. To be more precise [8,Section 5], if f and g are semimultiplicative, then f * g is semimultiplicative with

P. Haukkanen AEM
A multiplicative function f is said to be completely multiplicative if f (mn) = f (m)f (n) for all positive integers m, n. Liouville's function λ is an example of a completely multiplicative function. It is defined as where Ω(n) represents the total number of prime factors of n, each counted according to multiplicity [1,Section 2.12].
A multiplicative function f is said to be a rational arithmetical function of degree (r, s) if for some completely multiplicative functions g 1 , . . . , g r , h 1 , . . . , h s (see [6] and [13, Section III]). A rational arithmetical function of degree (2, 0) is referred to as a specially multiplicative function [9]. If f = g 1 * g 2 is a specially multiplicative function, we denote f A = g 1 g 2 . The function f A is termed as the associated completely multiplicative function. For example, the divisor functions σ a and Ramanujan's τ -function are specially multiplicative functions. Euler's totient function φ is a rational arithmetical function of degree (1,1).

The mth power Kesava Menon norm of semimultiplicative functions
In this section we first note in Theorems 3.1 and 3.2 that the conjugate of a semimultiplicative function is semimultiplicative and that the conjugate distributes over the Dirichlet convolution of any two arithmetical functions. We then apply these theorems to prove Theorems 3.3, 3.4 and 3.5, which state that the mth power Kesava Menon norm of a semimultiplicative function is semimultiplicative and that the mth power Kesava Menon norm distributes over the Dirichlet convolution of semimultiplicative functions.

Theorem 3.2.
For all arithmetical functions f and g, Theorems 3.1 and 3.2 follow directly from the definitions of conjugate and semimultiplicative function and from complete multiplicativity of λ.
In order to prove that the mth power Kesava Menon norm preserves semimultiplicativity, we first present this result in the case m = 1, since this case is needed in various stages of the proof of the general case.
Proof. By the definitions of the Kesava Menon norm and the Dirichlet convolution, Applying the definition of a semimultiplicative function and Theorem 3.1, we obtain By the definitions of the conjugate and the Kesava Menon norm, we see that . Since the Kesava Menon norm of a multiplicative function is multiplicative, N (f M ) is multiplicative. We thus obtain the result.
Proof. We proceed by induction on m. For m = 0 the theorem holds, since N 0 (f ) = f . The case m = 1 is presented in Theorem 3.3. Suppose that the theorem is true for m = k. Thus N k (f ) is semimultiplicative, and then applying Theorem 3.3 we see that N (N k (f )) is semimultiplicative, that is, the function N k+1 (f ) is semimultiplicative.
Further, from Theorem 3.3, we have By the induction hypothesis, . 76 P. Haukkanen AEM

By Theorem 3.3 and the induction hypothesis,
This completes the proof.
Proof. If f is quasimultiplicative, then a f = 1, and thus This completes the proof.

Corollary 3.2. If f is multiplicative, then
Corollary 3.2 follows directly from Corollary 3.1, since each multiplicative function f is quasimultiplicative with f (1) = 1.
Proof. Suppose first that f and g are multiplicative. Then, by the definition of the Kesava Menon norm, for all prime powers p e , By Theorem 3.2, we obtain Since f and g are multiplicative, N (f * g) and N (f ) * N (g) are also multiplicative. A multiplicative function is totally determined by its values at prime powers. Therefore

Now, applying induction on m gives
Consider now the general case that f and g are semimultiplicative. Then, by Theorem 3.4 and Eq. (2), N m (f * g) and N m (f ) * N m (g) are semimultiplicative. In addition, using Theorem 3.4 we have On the basis of (2), Since λ is completely multiplicative, λ(a f a g ) = λ(a f )λ(a g ). Therefore Using Theorem 3.4 and Eq.
(2), we get On the basis of the first part of this proof on multiplicative functions, By Theorem 3.4 and Eq. (2), Finally, combining (4), (5) and (6) gives This completes the proof.
Corollaries 3.3 and 3.4 follow directly from Theorem 3.5, since each quasimultiplicative function is semimultiplicative and each multiplicative function is quasimultiplicative.
Proof. By Theorem 3.5, Applying induction, we obtain This completes the proof.

Corollary 3.5. If f is multiplicative, then
Corollary 3.5 follows directly from Theorem 3.6, since each multiplicative function is quasimultiplicative. Proof. We have

The mth power Kesava Menon norm of rational arithmetical functions
Laohakosol and Pabhapote [6] proved that the mth power Kesava Menon norm of a rational arithmetical function of degree (r, s) is also a rational arithmetical function of degree (r, s). In this paper we present a short proof (applying Corollaries 3.4 and 3.5; see the proof of Theorem 4.1). Redmond and Sivaramakrishnan [9] proved this result for rational arithmetical functions of degree (2, 0), that is, for specially multiplicative functions. We note in Corollary 4.2 a similar result for the conjugate of a rational arithmetical function of degree (r, s).
Proof. For a completely multiplicative function g we have for all prime powers p e , where u(n) = 1 for all positive integers n. Here Therefore N (g)(p e ) = g(p 2e ) = g 2 (p e ).
Since N (g) and g 2 are multiplicative functions, this implies Applying induction on m gives Now, by Corollaries 3.4 and 3.5, we obtain Theorem 4.1.

Corollary 4.1. Suppose that f is a specially multiplicative function. Then
Proof. Let f = g 1 * g 2 , where g 1 and g 2 are completely multiplicative functions. Then f A = g 1 g 2 . On the other hand, by Theorem 4.1, N m (f ) = g 2 m 1 * g 2 m 2 , and thus Further, since f A is completely multiplicative, by Theorem 4.1, we obtain This completes the proof.

Corollary 4.2.
If f is a rational arithmetical function of degree (r, s), then its conjugate is also a rational arithmetical function of degree (r, s) (given as in Theorem 4.2).
Proof. The conjugate of a completely multiplicative function is completely multiplicative. Therefore this corollary follows from Theorem 4.2.
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