Fundamentos da aritmética formal

Abstract

In this work I proposed the union of Peano's axiomatics to the axiomatics of vector spaces, through the concept of ordered basis, culminating in the de nition of arithmetic space. This union allowed a universal systematization of the most common procedures in the study of number theory through generating functions, developing a comprehensive cohesive language. I de ned the notion of arithmetic, successor function and generator, iterative and endomorphic arithmetic operations, and monoids of operations and their homomorphisms, as the self-similar notion of meta-arithmetic. I developed the concept of algebra of arithmetic operations, completely translating the theory of arithmetic operations into the theory of linear transformations. I showed how the de ned algebras can be understood in several well-established ways of Algebra, and the relationship of these structures with convolutional algebras. I studied their homomorphisms when they are Banach algebras and, in particular, the problem of arithmetic inversion in these algebras. I proved the decomposition of the group of its invertible elements into elementary factors, a theorem considerably more useful than the Fundamental Theorem of Algebra. I investigated some relations combinatorially, in particular the construction of prime multiplications by natural ones, the Law of Natural Factorizations and some primitive formulas. I created the notion of symmetrized algebra of operations and the vague theory of symmetrized correspondents. I described how the algebra of circular additive operations gives rise, in the most natural way, to the concept of the Discrete Fourier Transform, a fundamental notion of the discipline of signal processing. I obtain representations of arithmetically remarkable functions, such as the Mertens function, in an abstract way, without resorting to the Zeta function, through harmonic analysis applied to groups of invertible multiplicative operations. Finally, I show a heuristic argument for obtaining an asymptote closely linked to the Riemann hypothesis, using complex residues of the classical theory.