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How Ramanujan may have discovered the mock theta functions
1The Pennsylvania State University, University Park, PA 16802, USA.
Summary
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.
- 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.
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