On the Irrationality of the Odd Zeta Values

We prove the irrationality of the odd zeta values \( \zeta(2n+1),\,n\in\mathbb{N} \). Our approach is based on constructing explicit integer linear forms in \( \zeta(2n+1) \), and applying a refinement of Dirichlet's approximation theorem. We prove that the sequence of denominators produced by successive rational approximations yields infinitely many nontrivial integer relations of the type \( \Lambda_m^{(q)}=A_m^{(q)}\zeta(2n+1)-B_m^{(q)} \), with \( |\Lambda_m^{(q)}| \) (\( q \) being a parameter) decaying towards zero as \( m \) approaches infinity. This permits us to invoke a general irrationality criterion and thereby deduce that each \( \zeta(2n+1) \) is irrational. Our method combines ideas from probability theory and Diophantine approximation, and complements earlier work of Apéry, Beukers, Rivoal, and Zudilin.