18.104 | Fall 2006 | Undergraduate

Seminar in Analysis: Applications to Number Theory



Infinitude of The Primes

Formulas Producing Primes?

2 Summing Powers of Integers, Bernoulli Polynomials  

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

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  

Euler’s Proof of Infinitude of Primes

Density of Prime Numbers


Definition and Elementary Properties of Fibonacci Numbers, Application to the Euclidean Algorithm

Binet’s Formula

Second draft of paper due

Golden Ratio

Spira Mirabilus

22 Final Paper Presentations I  
23 Final Paper Presentations II  
24 Final Paper Presentations III Final version paper due

Course Info

As Taught In
Fall 2006
Learning Resource Types
Problem Sets
Presentation Assignments with Examples