Ramanujan-type formulae for 1/π: A second wind?∗

In 1914 S. Ramanujan recorded a list of 17 series for 1/π. We survey the methods of proofs of Ramanujan’s formulae and indicate recently discovered generalisations, some of which are not yet proven. Let us start with two significant events of the 20th century, in the opposite historical order. At first glance, the stories might be thought of a different nature. In 1978, R. Apéry showed the irrationality of ζ(3) (see [4] and [13]). His rational approximations to the number in question (known nowadays as the Apéry constant) have the form vn/un ∈ Q for n = 0, 1, 2, . . . , where the denominators {un} = {un}n=0,1,... and numerators {vn} = {vn}n=0,1,... satisfy the same polynomial recurrence (n + 1)un+1 − (2n + 1)(17n + 17n + 5)un + nun−1 = 0 (1) with the initial data u0 = 1, u1 = 5, v0 = 0, v1 = 6. (2) Then lim n→∞ vn un = ζ(3) (3) ∗A talk at Number Theory Launch Seminar (Max Planck Institute for Mathematics in Bonn, April 25, 2006).