On a subgroup of the group of multiplicative arithmetic functions

Carroll, Timothy B.; Gioia, A. A.

doi:10.1017/s144678870002070x

Cited by 20 publications

(22 citation statements)

References 7 publications

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“…Such functions were first studied by Vaidyanathaswamy [23] in 1931, and later by several authors; see, for example, [4,6,7,9,10,13,16,18,20]. Two important classes of rational functions are Ꮿ(1,1) whose elements are known as totients, and Ꮿ(2,0) whose elements are the so-called specially multiplicative functions.…”

Section: Introductionmentioning

confidence: 99%

“…For f ∈ Ꮽ, f (1) ∈ R + , the Rearick logarithm of f (see [11,14,15]), denoted by Log f ∈ Ꮽ, is defined via (Log f )(1) = log f (1), 4) where df (n) = f (n)logn denotes the log derivation of f . The Hsu's generalized Möbius function (see [2]) µ r , r ∈ R, is defined as µ r (n) = p|n r ν p (n) (−1) νp(n) , (1.5) where ν p (n) is the highest power of the prime p dividing n. It is known (see [8,12]) that for f ∈ ᏹ, 6) and the converse holds under additional hypotheses.…”

Section: Introductionmentioning

confidence: 99%

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Properties of rational arithmetic functions

Laohakosol

Pabhapote

2005

International Journal of Mathematics and Mathematical Sciences

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Rational arithmetic functions are arithmetic functions of the formg1∗⋯∗gr∗h1−1∗⋯∗hs−1, wheregi,hjare completely multiplicative functions and∗denotes the Dirichlet convolution. Four aspects of these functions are studied. First, some characterizations of such functions are established; second, possible Busche-Ramanujan-type identities are investigated; third, binomial-type identities are derived; and finally, properties of the Kesava Menon norm of such functions are proved.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Properties of rational arithmetic functions

Laohakosol

Pabhapote

2005

International Journal of Mathematics and Mathematical Sciences

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“…We use this to generalize the results of [10] to an arbitrary ring and to answer a question posed in [22]. In Section 8, we prove some results about the order of a rational arithmetic function [6]. In Section 9, we address the question: given a multiplicative R-arithmetic function f , which powers of f are rational?…”

Section: Introductionmentioning

confidence: 97%

“…The group of rational R-arithmetic functions, denoted Rat(R), is the subgroup of Mul(R) generated by the completely multiplicative functions [6]. It is a subring of Mul(R) by Theorem 1.1(3a).…”

Section: Introductionmentioning

confidence: 99%

Ring structures on groups of arithmetic functions

Elliott

2008

Journal of Number Theory

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Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powers f α for α ∈ Bin(R) of a multiplicative arithmetic function f . For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.

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“…This is a more natural definition than the standard definition of powers due to Rearick [15] -which assumes that A = R or A = C -in terms of his logarithmic and exponential operators. Moreover, further knowledge of this Bin U (A)-algebra structure can yield useful information, say, about the powers of a given multiplicative function that are rational, in the sense of Carroll-Gioia [6]. This topic will be taken up elsewhere.…”

Section: Introductionmentioning

confidence: 99%

Binomial rings, integer-valued polynomials, and λ-rings

Elliott

2006

Journal of Pure and Applied Algebra

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A commutative ring A is said to be binomial if A is torsion-free (as a Z-module) and the element a(a − 1)(a − 2) · · · (a − n + 1)/n! of A ⊗ Z Q lies in A for every a ∈ A and every positive integer n. Binomial rings were first defined circa 1969 by Philip Hall in connection with his groundbreaking work in the theory of nilpotent groups. They have since had further applications to integer-valued polynomials, Witt vectors, and λ-rings. For any set X , the ring of integer-valued polynomials in Q[X ] is the free binomial ring on the set X . Thus the binomial property provides a universal property for rings of integer-valued polynomials. We give several characterizations of binomial rings and their homomorphic images.For example, we prove that a binomial ring is equivalently a λ-ring A whose Adams operations are all the identity on A. This allows us to construct a right adjoint Bin U for the inclusion from binomial rings to rings which has several applications in commutative algebra and number theory. For example, there is a natural Bin U (A)-algebra structure on the universal λ-ring Λ(A), and likewise on the abelian group of multiplicative A-arithmetic functions. Similarly, there is a natural Bin U (A)-module structure on the abelian group 1 + a for any ideal a in A with respect to which A is complete.

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On a subgroup of the group of multiplicative arithmetic functions

Cited by 20 publications

References 7 publications

Properties of rational arithmetic functions

Properties of rational arithmetic functions

Ring structures on groups of arithmetic functions

Binomial rings, integer-valued polynomials, and λ-rings

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