A Representation of Multiplicative Arithmetic Functions by Symmetric Polynomials

TL;DR

This paper introduces a novel representation of multiplicative arithmetic functions using symmetric polynomials, specifically Schur polynomials, providing new insights into their structure and classical identities.

Contribution

It presents a local, symmetric polynomial-based framework for multiplicative functions, generalizing classical results and enabling new structural understanding.

Findings

Representation of multiplicative functions via Schur polynomials

Clarification and generalization of the Busche-Ramanujan identity

Structural description of the convolution group of multiplicative functions

Abstract

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 $p$.