Ramanujan's tau function

These are the slides as well as the handout of my presentation for the ''Elliptic Functions'' class held by Victor H. Moll at Tulane University in Spring 2007.

Abstract

We will be concerned with Ramanujan's \(\tau\) function defined by $$ \Delta \triangleq q \theta (q)^{24} \triangleq q \prod_{n \geqslant 1} (1 - q^n)^{24} = \sum_{n \geqslant 1} \tau (n) q^n . $$ The Ramanujan numbers \(\tau(n)\) appear as sequence A000594 in Sloane's On-Line Encyclopedia of Integer Sequences. Their first few values are $$ 1, - 24, 252, - 1472, 4830, - 6048, - 16744, 84480, - 113643. $$ We'll have a special interest in congruences for \(\tau(n)\). One of the historical motivations for such congruences has been the hope to establish Lehmer's conjecture, namely that \(\tau(n) \neq 0\) for all \(n\), based on congruence considerations.

On our way, we will have to make use of some machinery involving modular forms. The basic theory we need will be briefly sketched.

Contents

1. A First Class of Simple Congruences
2. A Glance at the Theory of Modular Forms
   1. Basics About Modular Forms
   2. Spaces of Modular Forms
   3. Explicit Examples
   4. Differentiating Modular Forms
   5. Hecke Operators
3. Computing \(\tau(n)\)
   1. Exact Formulas
   2. Recurrences
4. Congruences for \(\tau(n)\)
5. Negative Results
6. Almost Always Divisibility
7. Open Problems

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