Linear Algebraic Number Theory, Part I: Foundations

TL;DR

This paper introduces linear algebraic number theory (LANT), a novel framework that reformulates number theory problems as regression models using matrix algebra in log space, offering new insights and potential approaches to longstanding questions.

Contribution

The paper develops the foundational theory of LANT, connecting it to classical number theory and proposing a new perspective that could impact understanding of prime numbers and the Riemann hypothesis.

Findings

LANT reformulates number theory problems as regression models.

Prime numbers are represented as atomic particles in the framework.

Connections to classical number theory and implications for the Riemann hypothesis are discussed.

Abstract

We introduce a new framework called linear algebraic number theory (LANT) that reformulates the number-theoretic problem as a regression model and solves it using matrix algebra. This framework restricts all computations to log space, therefore replaces multiplication with addition and allows to capture variation in the natural numbers from variation in the prime numbers. This automatically puts prime numbers to their designated place of atomic particles of natural numbers and enables fruitful new formulations of number-theoretic functions. We outline the theory, derive some basic results, make connections to standard number theory and give an outlook regarding the Riemann hypothesis, number theory's long-standing enigma.