Sobre a equação funcional da função zeta de Riemann.

Abstract

In his epoch-making memoir of 1859 Riemann given two proofs of the functional equation of the zeta function. In 1932 Siegel published an account of the work relating to the zeta function and analytic number theory found in Riemann’s private papers, where he shows that we may call of third proof of functional equation deduced starting of the so-called the Riemann-Siegel integral formula. The bridge between the second and the third proofs of the functional equation is hinted by Kusmin’s proof, in 1934, of the Riemann-Siegel integral formula. As consequence of the three proofs given we deduced, of each them, a specific kind of the functional equation, viz., respectively, the symmetric functional equation, the approximated functional equation and the parametric functional equation. The three are “totally equivalents”each other. As application of the symmetric equation acquired by the third proofs given along the methods used we showed the Hardy’s theorem that ζ (1/2 + ti) has infinitely many roots for t ∈ R comparing it with the way used in Landau to deduction of the same. Finally, we present three equivalences to the Riemann hypothesis.