Contents

xin

1.1.

1.2.

1.3.

1.4.

1.5.

ife =

fc =

fc =

k =

k =

= 2

= 3

= 5

= 7

= 11

11.2. The level n congruences for the prime 2 426

11.3. The level n congruences for the prime 5 428

11.4. The level two congruences for the prime 11 430

§12. Taylor series expansions for infinite products 430

Chapter 6. Identities related to partition functions 439

§1. Some more identities related to covering maps 439

440

440

442

443

444

§2. The j-function and generalizations of the discriminant A 447

§3. Congruences for the Laurent coefficients of the j-function 453

3.1. Averaging fk 457

3.2. Completion of the proof of Theorem 3.6 for k = 5 458

3.3. Proof of Theorem 3.6 for k = 11, n = 1 459

3.4. A further analysis of the k = 2 case 461

Chapter 7. Combinatorial and number theoretic applications 463

§1. Generalities on partitions 464

1.1. Euler series and some old identities 467

1.2. Partitions and sums of divisors 472

1.3. Lambert series 473

§2. Identities among partitions 480

2.1. A curious property of 8 481

2.2. A curious property of 3 481

2.3. A curious property of 7 482

§3. Partitions, divisors, and sums of triangular numbers 482

3.1. Sums of 4 squares 486

3.2. A remarkable formula 489

3.3. Weighted sums of triangular numbers 493

§4. Counting points on conic sections 495

§5. Continued fractions and partitions 499