Readings

Readings are in the required course textbook:

Amazon logo 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.


SES # Topics READINGS
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. (This resource may not render correctly in a screen reader.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". (This resource may not render correctly in a screen reader.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. (This resource may not render correctly in a screen reader.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