A correspondence between the isobaric ring and multiplicative arithmetic functions
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. Introduction.In this paper we give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials.The Dirichlet ring of arithmetic functions A * is well known to be a unique factorization domain (see Cashwell and Everett [4]).Its ring theoretic properties have been investigated in, e.g., Rearick [29,30], Shapiro [33], Carroll and Gioia [3], MacHenry [22], MacHenry and Tudose [25].The multiplicative arithmetic functions are units in this ring, and their properties can be described locally, that is, at each prime number p, (see, e.g., McCarthy [27], Sivaramakrishnan [35] and Vaidyanathswamy,[36]).It is this local behavior which we take advantage of to construct a representation in terms of a certain class of symmetric polynomials called weighted isobaric polynomials [25].It is advantageous to use the isobaric basis as a basis for the 2010 AMS Mathematics subject classification.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".