Readings are in the required course textbook:

Young, Robert M. Excursions In Calculus: An Interplay of The Continuous and The Discrete. Washington, DC: Mathematical Association of America, 1992. ISBN: 0883853175. Some additional readings are linked from this page.

1

Infinitude of The Primes

Formulas Producing Primes?

### Infinitude of The Primes

Text, chapter II (1a), pp. 58-63, possibly complemented by exercise 2; p. 34, exercise 2 (maybe also 1); p. 70.

### Formulas Producing Primes?

Text, chapter II (1b), pp. 64-69, possibly complemented by exercise 6 (maybe 4,5); p. 70, exercises 13 and 15; p. 71.

2 Summing Powers of Integers, Bernoulli Polynomials Text, chapter II (2), pp. 74-93; possibly complemented by exercises 2 and 3; p. 95, exercise 11; p. 97, exercises 19 and 20; p. 99.
3

Generating Function for Bernoulli Polynomials

The Sine Product Formula and $\zeta(2n)$

### Generating Function for Bernoulli Polynomials

Text, pp. 160-161.

### The Sine Product Formula and $\zeta(2n)$

Text, pp. 345-348.

4 A Summary of the Properties of Bernoulli Polynomials and More on Computing $\zeta(2n)$
5 Infinite Products, Basic Properties, Examples (Following Knopp, Theory and Applications of Infinite Series)
6 Fermat's Little Theorem and Applications Text, pp. 100-110 (without Mersenne Primes) and exercises 13 and 14; p. 117.
7 Fermat's Great Theorem Text, pp. 110-114, exercise 24; p. 119.
8 Applications of Fermat's Little Theorem to Cryptography: The RSA Algorithm Reference: Trappe, Washington. Introduction to Cryptography with Coding Theory. Section 6.1, a little of 6.3
9 Averages of Arithmetic Functions Text, pp. 219-225 with exercises 11, 12 and 13; p. 241.
10 The Arithmetic-geometric Mean; Gauss' Theorem Text, pp. 231-238; maybe supplemented by some material from Cox, David A. Notices 32, no. 2 (1985) (QA.A5135) and Enseignment Math 30, no. 3-4 (1984).
11 Wallis's Formula and Applications I Text, pp. 248-254, exercises 9 and 10; p. 263, maybe also exercise 11; p. 264.
12

Wallis's Formula and Applications II (The Probability Integral)

Stirling's Formula

### Wallis's Formula and Applications II (The Probability Integral)

Exercise 1; pp. 272-273, and the "usual" proof, also consult section 5.2, pp. 267-272 if needed.

### Stirling's Formula

Exercises 13 and 14; pp. 264-267.

13 Stirling's Formula (cont.) Exercises 13 and 14; pp. 264-267.
14 Elementary Proof of The Prime Number Theorem I Following M. Nathanson's "Elementary methods in number theory.": Chebyshev's Functions and Theorems. For a historical account, see D. Goldfeld's Note. (PDF)
15 Elementary Proof of The Prime Number Theorem II: Mertens' theorem, Selberg's Formula, Erdos' Result

The original papers can be found on JSTOR:

Selberg, A. "An Elementary Proof of the Prime-Number Theorem."

Erdos, P. "On a New Method in Elementary Number Theory Which Leads to an Elementary Proof of the Prime Number Theorem."

16 Short Analytic Proof of The Prime Number Theorem I (After D. J. Newman and D. Zagier)

The original papers are on JSTOR:

Newman, D. J. "Simple Analytic Proof of the Prime Number Theorem."

Zagier, D. "Newman's Short Proof of the Prime Number Theorem."

17 Short Analytic Proof of The Prime Number Theorem II: The Connection between PNT and Riemann's Hypothesis

An Expository Paper:

Conrey, J. Brian. The Riemann Hypothesis in the "Notices of the AMS". (PDF)

18 Discussion on the First Draft of the Papers and Some Hints on How to Improve the Exposition and Use of Latex References: Knuth, Larrabee, and S. Kleiman Roberts. (PDF)
19

Euler's Proof of Infinitude of Primes

Density of Prime Numbers

Text, pp. 287-292, 296-306, and 299-301 (especially Euler's Theorem, pp. 299-301). Also p. 351 in reference [211] (Hardy-Wright) and exercise 4; p. 294.
20

Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm

Binet's Formula

### Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm

Text, pp. 124-130. Exercises 6, 9, and 24, pp. 134-140.

### Binet's Formula

Morris, pp. 130-132, also the example, "The transmition of information". Exercises 14, 17, and 27, pp. 134-140.

21

Golden Ratio

Spira Mirabilus

### Golden Ratio

Text, pp. 140-144. Exercises 4 and 9; pp. 154-156, exercise 20; p. 136.

### Spira Mirabilus

Text, pp. 148-153. Example 1; pp. 159-160 (The Generating Function for Fibonacci Numbers). Exercise 32; p. 138, 21; p. 136.

22 Final Paper Presentations I
23 Final Paper Presentations II
24 Final Paper Presentations III