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 Arithmeticgeometric 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 