Seminar in Analysis: Applications to Number Theory

Readings

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 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