18.104 | Fall 2006 | Undergraduate

# 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

## Course Info

Fall 2006
##### Learning Resource Types
Problem Sets
Presentation Assignments with Examples