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How Ramanujan may have discovered the mock theta functions.

George E Andrews1

  • 1The Pennsylvania State University, University Park, PA 16802, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|December 10, 2019
PubMed
Summary
This summary is machine-generated.

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This paper explores the origins of mock theta functions, a concept introduced by Srinivasa Ramanujan. It investigates the mathematical context and potential inspirations behind his groundbreaking work.

Area of Science:

  • Number Theory
  • Mathematical Physics
  • History of Mathematics

Background:

  • Mock theta functions were unexpectedly introduced by Srinivasa Ramanujan in his final correspondence with G.H. Hardy.
  • Their sudden appearance in Ramanujan's work lacks a clear, documented explanation, prompting historical and mathematical inquiry.

Purpose of the Study:

  • To investigate the potential reasons and mathematical context that led Srinivasa Ramanujan to conceive of mock theta functions.
  • To provide a plausible historical and theoretical basis for Ramanujan's introduction of these novel mathematical objects.

Main Methods:

  • Historical analysis of Ramanujan's mathematical work and correspondence.
  • Exploration of contemporary mathematical concepts and theories available to Ramanujan.
Keywords:
Ramanujanmock theta functionstheta functions

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  • Contextualization within the mathematical landscape of the early 20th century.

    • Main Results:

    • The study posits that Ramanujan's work on related series and transformations likely influenced his conceptualization of mock theta functions.
    • Potential connections to modular forms and other advanced number theoretic concepts are explored as contributing factors.

    Conclusions:

    • Ramanujan's introduction of mock theta functions was likely a result of deep insights into the properties of certain infinite series and their transformations.
    • This exploration offers a reasoned perspective on a pivotal moment in the development of mock theta function theory.