The Convolution Ring of Arithmetic Functions and Symmetric Polynomials

TL;DR

This paper introduces new operators LOG and EXP for arithmetic functions, establishing a structure that connects convolution and addition, and explores identities and matrices related to polynomials and their derivatives.

Contribution

It develops a novel theory of logarithms and exponentials in arithmetic functions, linking convolution and addition, and introduces new identities and matrix representations in the isobaric ring.

Findings

LOG and EXP operators create a new algebraic structure for arithmetic functions.

Hyperbolic functions are constructed within this framework, satisfying classical identities.

The paper characterizes locally and globally representable arithmetic functions.

Abstract

Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas polynomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to produce a theory of logarithms and exponentials within arithmetic functions giving another proof of the fact that the group of multiplicative functions under convolution product is isomorphic to the group of additive functions under addition. The hyperbolic trigonometric functions are constructed from the EXP operator, again, in the usual way. The usual hyperbolic trigonometric identities hold. We exhibit new structure and identities in the isobaric ring. Given a monic polynomial, its infinite companion matrix can be embedded in the group of weighted isobaric polynomials. The derivative of the monic polynomial and its companion matrix give us the different matrix and the infinite different matrix. The determinant of the different matrix is the discriminate of the monic polynomial up to sign. In fact, the LOG operating on the infinite companion matrix is the infinite different matrix. We prove that an arithmetic function is locally representable if an only if it is a multiplicative function. An arithmetic function is both locally and globally representable if it is trivially globally represented.