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.
SES # | Topics | READINGS |
---|---|---|
1 |
Infinitude of The Primes Formulas Producing Primes? |
Infinitude of The PrimesText, 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 PolynomialsText, 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 FormulaExercises 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 AlgorithmText, pp. 124-130. Exercises 6, 9, and 24, pp. 134-140. Binet’s FormulaMorris, pp. 130-132, also the example, “The transmition of information”. Exercises 14, 17, and 27, pp. 134-140. |
21 |
Golden Ratio Spira Mirabilus |
Golden RatioText, pp. 140-144. Exercises 4 and 9; pp. 154-156, exercise 20; p. 136. Spira MirabilusText, 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 |