La Zeta de Riemann est irrationnelle aux impairs positifs

TL;DR

The paper proves the irrationality of Riemann zeta function values at positive odd integers using a new functional equation derived from Hurwitz's Zeta Function, establishing linear independence and irrationality for these values.

Contribution

It introduces a novel functional equation for Riemann Zeta Function via Hurwitz's Zeta Function, leading to a proof of irrationality at positive odd integers.

Findings

Proves irrationality of ζ(j) for j=3,4,5,6,7,8,9,...

Derives a new functional equation for Riemann Zeta Function

Establishes linear independence of certain vector elements

Abstract

We found, by Hurwitz's Zeta Function, a new functional equation for Riemann Zeta Function. Considering this equation for $s=2$ and $s=1$, we determine a relation between the values of Riemann zeta Function on positive integers. The Matrix has two dimensions, and the second member is the vector (1, $-\zeta(2)$). The elements of this vector are linearly independent on the Rationals; and from this independence, we determined that $\zeta(j)$ is irrational for each j=3,4,5,6,7,8,9,....