A correspondence between the isobaric ring and multiplicative arithmetic functions
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Bibliographic record
Abstract
We give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials written on the isobaric basis. The representing elements are recursive sequences of Schur-hook polynomials evaluated on subrings of the complex numbers. 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. 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 MFs can be defined in a natural way on the negative powers of prime p which, in turn, leads to a natural extension of Schur-hook polynomials to negatively indexed Schur-hook polynomials.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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