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# 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