Seminar in Analysis: Applications to Number Theory

Calendar

SES #	Topics	KEY DATES
1	Infinitude of The Primes Formulas Producing Primes?
2	Summing Powers of Integers, Bernoulli Polynomials
3	Generating Function for Bernoulli Polynomials The Sine Product Formula and \(\zeta(2n)\)	Assignment 1 due
4	A Summary of the Properties of Bernoulli Polynomials and more on Computing \(\zeta(2n)\)
5	Infinite Products, Basic Properties, Examples
6	Fermat’s Little Theorem and Applications
7	Fermat’s Great Theorem	Assignment 2 due
8	Applications of Fermat’s Little Theorem to Cryptography: The RSA Algorithm
9	Averages of Arithmetic Functions
10	The Arithmetic-geometric Mean; Gauss’ Theorem	Assignment 3 due
11	Wallis’s Formula and Applications I	Topic proposal and full outline of the paper due
12	Wallis’s Formula and Applications II (The Probability Integral) Stirling’s Formula
13	Stirling’s Formula (cont.)
14	Elementary Proof of The Prime Number Theorem I
15	Elementary Proof of The Prime Number Theorem II: Mertens’ Theorem, Selberg’s Formula, Erdos’ Result
16	Short Analytic Proof of The Prime Number Theorem I (After D. J. Newman and D. Zagier)
17	Short Analytic Proof of The Prime Number Theorem II: The Connection between PNT and Riemann’s Hypothesis	First draft of paper due
18	Discussion on the First Draft of the Papers and some Hints on how to Improve the Exposition and use of Latex
19	Euler’s Proof of Infinitude of Primes Density of Prime Numbers
20	Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm Binet’s Formula	Second draft of paper due
21	Golden Ratio Spira Mirabilus
22	Final Paper Presentations I
23	Final Paper Presentations II
24	Final Paper Presentations III	Final version paper due