On the Irrationality of the π-Normalized Odd Zeta Values

We prove the irrationality of a family of normalized odd zeta values of the form \( \dfrac{\zeta(2n+1)}{\pi^{2n+1}},\,n\in\mathbb{N},\,n\geq 3. \) Our approach is based on constructing explicit integer linear forms in the quantities \( I_n=4(4^n-1)\left[\dfrac{\zeta(2n)\zeta(2n+2)}{\zeta(2n+1)^2}-1\right]-1 \), and applying a refinement of Dirichlet's approximation theorem. The construction of the \( I_n \) is probabilistic in origin. We prove that the sequence of denominators produced by successive rational approximations yields infinitely many nontrivial integer relations of the type \( \Lambda_m^{(q)}=A_m^{(q)} I_n-B_m^{(q)}, \) 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 \( I_n \) is irrational for each \( n\geq 3 \). Consequently, each corresponding normalized odd zeta value is irrational. Our method combines ideas from probability theory, analytic combinatorics and Diophantine approximation, and complements earlier work of Apéry, Beukers, Rivoal, and Zudilin.