Logarithmic Operators and Characterizations of Completely Multiplicative Functions

“…While the isobaric ring is a not so classical version of the well known ring of symmetric functions, the elements in the ring of arithmetic functions have long been objects of study, but not usually from a structural point of view (but see [4], [28], [29][32], and recently, [16], [15], [17], [7], [8], [9], [10], [11]). It is possible that the relation between the two structures is implicitly well understood, but it is rather surprising that the relationship, to our knowledge, has not been made explicit in the literature.…”

Section: The Ring Of Arithmetic Functionsmentioning

confidence: 99%

A Representation of Multiplicative Arithmetic Functions by Symmetric Polynomials

MacHenry,

Wong

2007

Preprint

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We give a representation of the classical theory of multiplicative arithmetic functions (MF)in the ring of symmetric polynomials. The basis of the ring of symmetric polynomials that we use is the isobaric basis, a basis especially sensitive to the combinatorics of partitions of the integers. The representing elements are recursive sequences of Schur polynomials evaluated at subrings of the complex numbers. The multiplicative arithmetic functions are units in the Dirichlet ring of arithmetic functions, and their properties can be described locally, that is, at each prime number p. Our representation is, hence, a local representation. One such representing sequence is the sequence of generalized Fibonacci polynomials. In general the sequences consist of Schur-hook polynomials. This representation enables us to clarify and generalize classical results, e.g., the Busche-Ramanujan identity, as well as to give a richer structural description of the convolution group of multiplicative functions. It is a consequence of the representation that the MF's can be defined in a natural way on the negative powers of the prime p.

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Distributive property of completely multiplicative functions

Pabhapote

Laohakosol

2010

Lith Math J

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Logarithmic Operators and Characterizations of Completely Multiplicative Functions

Cited by 9 publications

References 5 publications

Independence measures of arithmetic functions

Independence measures of arithmetic functions

A Representation of Multiplicative Arithmetic Functions by Symmetric Polynomials

Distributive property of completely multiplicative functions

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