On the Irrationality of the Odd Zeta Values

We prove the irrationality of the odd zeta values ( zeta(2n+1),,ninmathbb{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.