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}},,ninmathbb{N},,ngeq 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}-1right]-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 ( ngeq 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.