The method of creative microscoping (Analytic Number Theory and Related Topics)

Abstract

Ramanujan's formulae for l/π and their generalisations remain an amazing topic, with many mathematical challenges. Recently it was observed that the formulae possess spectacular 'supercongruence' counterparts. For example, truncating the sum in Ramanujan's formula Σ∞k=0(4k2k)(2kk)2/28k32k(8k + 1)= 2√3 /π to the first p terms correspond to the congruenceΣp-1k=0(4k2k)(2kk)2/28k32k (8k+ 1)三p(-3/p)(modp3) valid for any prime p > 3. Some supercongruences were shown to be true through a tricky use of classical hypergeometric identities or the Wilf-Zeilberger method of creative telescoping. The particular example displayed above (and many other entries) were resistant to such techniques. In joint work with Victor Guo we develop a new method of 'creative microscoping' that provides conseptual reasons for and simultaneous proofs of both the underlying Ramanujan's formula and its finite supercongruence counterparts. The main ingredient is an asymptotic analysis of suitable q-deformations of Ramanujan's formulas at all roots of unity. Here we outline the method on the concrete example of supercongruence above.