One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means

Abstract

Available proofs of the result of the type "at least one of the odd zeta values ζ(5), ζ(7),…, ζ(s) is irrational" make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are, however, counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to the general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result — a proof whose technical ingredients are limited to the prime number theorem and Stirling's approximation formula for the factorial.