Linear Algebraic Number Theory, Part I: Foundations

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.