Abstract

Bruce C. Berndt(University of Illinois), Ramanujan's Series for 1/pi


  • Abstract:
  • In his famous paper, Modular Equations and Approximations to Pi, Ramanujan recorded 17 hypergeometric-like series representations for 1/pi. These were not completely proved until 1987 when Jonathan and Peter Borwein found proofs. In the past 20 years, several authors have found new hypergeometric-like series for 1/pi. In particular, in the past two years, Heng Huat Chan, Nayandeep Deka Baruah, and the speaker have returned to Ramanujan's paper and used his ideas, which are based on Eisenstein series, more so than previous authors to establish proofs of most of Ramanujan's formulas and to discover many new such formulas as well. A historical survey of attempts to prove Ramanujan's formulas will be given, emphasizing the contributions of S. Chowla, R. William Gosper, Jr., Jonathan and Peter Borwein, David and Gregory Chudnovsky, Heng Huat Chan, Nayandeep Baruah, and others. Ramanujan's ideas arising from Eisenstein series will be explained.