On the Kesava Menon norm of semimultiplicative functions

Aequat. Math. c The Author(s) 2019 https://doi.org/10.1007/s00010-019-00660-x Aequationes Mathematicae On the Kesava Menon norm of semimultiplicative functions Pentti Haukkanen Abstract. The Kesava Menon norm of an arithmetical function fis defined by N(f)(n)= (f∗λf)(n2), where ∗denotes the Dirichlet convolution and λdenotes Liouville’s function. The mthpowerKesavaMenonnormoffis defined inductively by N0(f)=f,Nm(f)= NNm−1(f),m=1,2,... In this paper we prove that the mthpowerKesavaMenonnorm 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. Mathematics Subject Classification. 11A25. Keywords. Semimultiplicative function, Kesava Menon norm, Dirichlet convolution, Rational arithmetical function. 1. Introduction Let fbe an arithmetical function (that is, a realor complex-valued function on the set of positive integers). In 1963, Kesava Menon [4, Section 3] defined the norm of fas the arithmetical function N(f) given by N(f)(n)=(f∗f)(n2), where ∗is the Dirichlet convolution (see (1)) and fis the conjugate of f.The conjugate is defined as f=λf, where λis 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] defined the mth power Kesava Menon norm of f inductively by P. Haukkanen AEM N0(f)=f Nm(f)=NNm−1(f)for m=1,2,... 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 fis said to be a rational arithmetical function of degree (r, s) if it is the Dirichlet convolution of rcompletely multiplicative functions and the inverse of scompletely multiplicative functions. For details of these various types of multiplicativity, see Sect. 2. This paper is organized as follows. In Sect. 2we review the known properties of arithmetical functions needed in this paper. In Sect. 3we present new results. In Sect. 3we 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. 4of this paper we utilize the properties presented in Sect. 3to 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). On the Kesava Menon norm of semimultiplicative functions 2. Preliminaries on arithmetical functions The Dirichlet convolution of arithmetical functions fand gis defined as (f∗g)(n)= d|n f(d)g(n/d).(1) We may also interpret that an arithmetical function fis defined on the set of positive real numbers so that f(x)=0ifxis not a positive integer. This makes it possible to present the Dirichlet convolution in the form (f∗g)(n)= ∞  k=1 f(k)g(n/k). This expression is useful in some calculations presented in this paper. The function δ, defined as δ(1) = 1 and δ(n) = 0 otherwise, serves as the identity under the Dirichlet convolution. The Dirichlet inverse of fexists if and only if f(1) = 0, and it is denoted by f−1. An arithmetical function fis said to be multiplicative if f(1) = 1 and f(mn)=f(m)f(n) for all coprime positive integers m, n. An arithmetical function fis said to be semimultiplicative [8] if there exists a nonzero constant cf, a positive integer afand a multiplicative function fMsuch that f(n)=cffM(n/af). Semimultiplicative functions can also be characterized as the arithmetical functions f(not identically zero) satisfying the functional equation f(m)f(n)=f((m, n))f([m, n]) for all positive integers mand n, where (m, n) and [m, n] are the gcd and lcm of mand n. Semimultiplicative functions are the same as Selberg multiplicative functions (see [2, Section 2.1] and [10]). We do not present the details here. Quasimultiplicative functions (see [2, Section 2.1] and [12, Section XI.2]) are the arithmetical functions fsuch that f(1) =0andf(1)f(mn)=f(m)f(n) for all coprime positive integers m, n. Lahiri [5] refers to these functions as hypo-multiplicative functions. Quasimultiplicative functions are, in fact, the semimultiplicative functions fwith af= 1 (i.e. with f(1) = 0), cf=f(1) and fM(n)=f(n)/f(1). Note that fis multiplicative if and only if fis semimultiplicative with af=cf=1andfM=f. The Dirichlet convolution of multiplicative functions is multiplicative. The same applies to quasimultiplicative and semimultiplicative functions. To be more precise [8, Section 5], if fand gare semimultiplicative, then f∗gis semimultiplicative with cf∗g=cfcg,a f∗g=afag,(f∗g)M=fM∗gM.(2) P. Haukkanen AEM A multiplicative function fis said to be completely multiplicative if f(mn)=f(m)f(n) for all positive integers m, n. Liouville’s function λis an example of a completely multiplicative function. It is defined as λ(n)=(−1)Ω(n),(3) where Ω(n) represents the total number of prime factors of n, each counted according to multiplicity [1, Section 2.12]. A multiplicative function fis said to be a rational arithmetical function of degree (r, s)if f=g1∗···∗gr∗(h1∗···∗hs)−1 for some completely multiplicative functions g1,...,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=g1∗g2is a specially multiplicative function, we denote fA=g1g2. The function fAis termed as the associated completely multiplicative function. For example, the divisor functions σaand Ramanujan’s τ-function are specially multiplicative functions. Euler’s totient function φis a rational arithmetical function of degree (1,1). For general accounts on arithmetical functions, we refer to [1,7,12]. 3. 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 3.1. If fis semimultiplicative, then fis semimultiplicative with cf= λ(af)cf,af=afand fM= (fM)=λfM. Theorem 3.2. For all arithmetical functions fand g, f∗g=f∗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. On the Kesava Menon norm of semimultiplicative functions Theorem 3.3. If fis semimultiplicative, then N(f)is semimultiplicative with cN(f)=cfcf=λ(af)cf2 aN(f)=af N(f)M=N(fM). Proof. By the definitions of the Kesava Menon norm and the Dirichlet convolution, N(f)(n)=(f∗f)(n2)= ∞  k=1 f(k)f(n2/k). Applying the definition of a semimultiplicative function and Theorem 3.1,we obtain N(f)(n)= ∞  k=1 cffM(k/af)λ(af)cf(λfM)n2/(kaf) =λ(af)(cf)2 ∞  k=1 fM(k)(λfM)n2/(af)2/k =λ(af)(cf)2(fM∗(λfM))((n/af)2). By the definitions of the conjugate and the Kesava Menon norm, we see that N(f)(n)=λ(af)(cf)2N(fM)(n/af). By Theorem 3.1,λ(af)(cf)2=cfcf(= 0). Since the Kesava Menon norm of a multiplicative function is multiplicative, N(fM) is multiplicative. We thus obtain the result.  Theorem 3.4. If fis semimultiplicative, then Nm(f)(where m≥0)issemimultiplicative with cNm(f)=cf(cf)2m−1=λ(af)2m−1cf2m aNm(f)=af Nm(f)M=Nm(fM). Proof. We proceed by induction on m.Form= 0 the theorem holds, since N0(f)=f. The case m= 1 is presented in Theorem 3.3. Suppose that the theorem is true for m=k.ThusNk(f) is semimultiplicative, and then applying Theorem 3.3 we see that N(Nk(f)) is semimultiplicative, that is, the function Nk+1(f) is semimultiplicative. Further, from Theorem 3.3,wehave cNk+1(f)=cN(Nk(f)) =λ(aNk(f))cNk(f)2. By the induction hypothesis, cNk+1(f)=λ(af)λ(af)2k−1cf2k2=λ(af)2k+1−1cf2k+1 . P. Haukkanen AEM By Theorem 3.3 and the induction hypothesis, aNk+1(f)=aNNk(f)=aNk(f)=af and Nk+1(f)M=NNk(f)M=NNk(f)M =NNk(fM)=Nk+1(fM). This completes the proof.  Corollary 3.1. If fis quasimultiplicative, then Nm(f)(where m≥0)isquasimultiplicative with Nm(f)(1) = f(1)2m. Proof. If fis quasimultiplicative, then af= 1, and thus aNm(f)=af=1. This shows that Nm(f) is quasimultiplicative. Since λ(af)=λ(1) = 1, Nm(f)(1) = cNm(f)=λ(af)2m−1cf2m =cf2m =f(1)2m. This completes the proof.  Corollary 3.2. If fis multiplicative, then Nm(f)(where m≥0) is multiplicative. Corollary 3.2 follows directly from Corollary 3.1, since each multiplicative function fis quasimultiplicative with f(1) = 1. Theorem 3.5. If fand gare semimultiplicative, then Nm(f∗g)=Nm(f)∗Nm(g),m≥0. Proof. Suppose first that fand gare multiplicative. Then, by the definition of the Kesava Menon norm, for all prime powers pe, N(f∗g)(pe)=(f∗g)∗(f∗g)(p2e). By Theorem 3.2, we obtain N(f∗g)(pe)=(f∗f)∗(g∗g)(p2e)= 2e  i=0 (f∗f)(pi)(g∗g)(p2e−i). But (f∗f)(pi)=0ifiis odd; hence we have N(f∗g)(pe)= e  i=0 (f∗f)(p2i)(g∗g)(p2(e−i)) = e  i=0 N(f)(pi)N(g)(pe−i)=N(f)∗N(g)(pe). On the Kesava Menon norm of semimultiplicative functions Since fand gare multiplicative, N(f∗g)andN(f)∗N(g) are also multiplicative. A multiplicative function is totally determined by its values at prime powers. Therefore N(f∗g)=N(f)∗N(g). Now, applying induction on mgives Nm(f∗g)=Nm(f)∗Nm(g). Consider now the general case that fand gare semimultiplicative. Then, by Theorem 3.4 and Eq. (2), Nm(f∗g)andNm(f)∗Nm(g) are semimultiplicative. In addition, using Theorem 3.4 we have cNm(f∗g)=λ(af∗g)2m−1(cf∗g)2m. On the basis of (2), cNm(f∗g)=λ(afag)2m−1(cfcg)2m. Since λis completely multiplicative, λ(afag)=λ(af)λ(ag). Therefore cNm(f∗g)=λ(af)2m−1(cf)2mλ(ag)2m−1(cg)2m. Using Theorem 3.4 and Eq. (2)weget cNm(f∗g)=cNm(f)cNm(g)=cNm(f)∗Nm(g).(4) Further, applying Theorem 3.4 and Eq. (2)weget aNm(f∗g)=af∗g=afag =aNm(f)aNm(g) =aNm(f)∗Nm(g).(5) Next, applying Theorem 3.4 and Eq. (2), we get Nm(f∗g)M=Nm(f∗g)M=Nm(fM∗gM). On the basis of the first part of this proof on multiplicative functions, Nm(f∗g)M=Nm(fM)∗Nm(gM). By Theorem 3.4 and Eq. (2), Nm(f∗g)M=Nm(f)M∗Nm(g)M=Nm(f)∗Nm(g)M.(6) Finally, combining (4), (5) and (6) gives Nm(f∗g)=Nm(f)∗Nm(g). This completes the proof.  Corollary 3.3. If fand gare quasimultiplicative, then Nm(f∗g)=Nm(f)∗Nm(g),m≥0. P. Haukkanen AEM Corollary 3.4. If fand gare multiplicative, then Nm(f∗g)=Nm(f)∗Nm(g),m≥0. Corollaries 3.3 and 3.4 follow directly from Theorem 3.5, since each quasimultiplicative function is semimultiplicative and each multiplicative function is quasimultiplicative. Theorem 3.6. If fis quasimultiplicative, then Nm(f−1)=Nm(f)−1,m≥0. Proof. By Theorem 3.5, Nm(f)∗Nm(f−1)=Nm(f∗f−1)=Nm(δ). Now, N(δ)=δ∗δ=δ∗δ=δ. Applying induction, we obtain Nm(δ)=δ. This completes the proof.  Corollary 3.5. If fis multiplicative, then Nm(f−1)=Nm(f)−1. Corollary 3.5 follows directly from Theorem 3.6, since each multiplicative function is quasimultiplicative. Theorem 3.7. For all arithmetical functions with f(1) =0, (f−1)=(f)−1. Proof. We have f∗(f−1)=λf ∗λf−1=λ(f∗f−1)=λδ =δ.  4. The mth power Kesava Menon norm of rational arithmetical functions Laohakosol and Pabhapote [6]provedthatthemth 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). On the Kesava Menon norm of semimultiplicative functions Theorem 4.1. Suppose that fis a rational arithmetical function of degree (r, s) given as f=g1∗···∗gr∗(h1∗···∗hs)−1, where g1,...,g r,h 1,...,h sare completely multiplicative functions. Then Nm(f)is a rational arithmetical function of degree (r, s)such that Nm(f)=(g1)2m∗···∗(gr)2m∗(h1)2m∗···∗(hs)2m−1,m≥0. Proof. For a completely multiplicative function gwe have N(g)(pe)=g∗(λg)(p2e)=g(u∗λ)(p2e) for all prime powers pe, where u(n) = 1 for all positive integers n. Here (u∗λ)(p2e)= 2e  i=0 (−1)2e−i= 2e  i=0 (−1)i=1. Therefore N(g)(pe)=g(p2e)=g2(pe). Since N(g)andg2are multiplicative functions, this implies N(g)=g2. Applying induction on mgives Nm(g)=g2m. Now, by Corollaries 3.4 and 3.5, we obtain Theorem 4.1. Corollary 4.1. Suppose that fis a specially multiplicative function. Then Nm(f)is specially multiplicative with Nm(f)A=fA2m . Further, Nm(f)A=Nm(fA). Proof. Let f=g1∗g2, where g1and g2are completely multiplicative functions. Then fA=g1g2. On the other hand, by Theorem 4.1,Nm(f)=g2m 1∗g2m 2, and thus Nm(f)A=g2m 1g2m 2=(g1g2)2m=(fA)2m. Further, since fAis completely multiplicative, by Theorem 4.1, we obtain Nm(fA)=(fA)2m. This completes the proof. 