An elementary proof of Ramanujan's identity for odd zeta values

Abstract

The Riemann zeta function ζ(s) is one of the most important special functions of Mathematics. While the critical strip 0 < R (s) < 1 is undoubtedly the most important region in the complex plane on account of the unsolved problem regarding location of non-trivial zeros of ζ(s), namely, the Riemann Hypothesis, the right-half plane R (s) > 1 also has its own share of interesting unsolved problems to contribute to. It is quite well known that many number theoretic properties of odd zeta values are nowadays still unsolved mysteries, such as the rationality, transcendence and existence of closed-forms. Only in 1978 did Apéry [2] famously proved that ζ(3) is irrational. This was later reproved in a variety of ways by several authors, in particular Beukers [10] who devised a simple approach involving certain integrals over [0, 1]. In the early 2000s, an important work of Rivoal [21], and Ball and Rivoal [4] determined that infinitely many values of ζ at odd integers are irrational, and the work of Zudilin [27] proved that at least one among ζ(5), ζ(7), ζ(9) and ζ(11) is irrational. A very recent result due to Rivoal and Zudilin [22] states that at least two of the numbers ζ(5), ζ(7), . . . , ζ(69) are irrational. Moreover, for any pair of positive integers a and b, Haynes and Zudilin [17, Theorem 1] have shown that either there are infinitely many m ∈ N for which ζ(am+ b) is irrational, or the sequence {qm} ∞ m=1 of common denominators of the rational elements of the set {ζ(a+ b), . . . , ζ(am+ b)} grows super-exponentially, that is, q 1/m m → ∞ as m → ∞. Despite these advances, to this day no value of ζ(2n+ 1) with n > 2 is known to be irrational. A folklore conjecture states that the numbers π, ζ(3), ζ(5), ζ(7), . . . are algebraically independent over the rationals. This conjecture is predicted by the Grothendieck’s period conjecture for mixed Tate motives. But both conjectures are far out of reach and we do not even know the transcendence of a single odd zeta value. One should mention that Brown [11] has in the past few years outlined a simple geometric approach to understand the structures involved in Beukers’s proof of irrationality of ζ(3) and how this may generalize to other odd zeta values. Ramanujan made many beautiful and elegant discoveries in his short life of 32 years. One of the most remarkable formulas suggested by Ramanujan that has attracted the attention of several mathematicians over the years is the following intriguing identity involving the odd values of the Riemann zeta function [6, 1.2]: