**MATH 406: Analysis and Number Theory: MWF 9 - 9:50am Bronfman 104**

(click on the picture for a larger version)

Professor Steven Miller (Steven.J.Miller AT williams.edu), 202 Bronfman Science Center (413-597-3293)

Office hours: M-F 10:05-10:50 and M-F 4-4:30 and whenever I'm in my office (click here for my schedule)

take aways for the course (if you are interested in adding to this, please let me know)

course description syllabus/general objectives course notes / handouts supplemental reading other links additional comments

**
COURSE DESCRIPTION:** Gauss said "Mathematics
is the queen of the sciences and number theory the queen of mathematics"; in
this class we shall meet some of her subjects. We will discuss many of the most
important questions in analytic and additive number theory, with an emphasis on
techniques and open problems. Topics will range from Goldbach's Problem and the
Circle Method to the Riemann Zeta Function and Random Matrix Theory. Other
topics will be chosen by student interest, coming from sum and difference sets,
Poissonian behavior,
Benford's law, the dynamics of the 3x+1 map as well as suggestions from the
class. We will occasionally assume some advanced results for our investigations,
though we will always try to supply heuristics and motivate the material. No
number theory background is assumed, and we will discuss whatever material we
need from probability, statistics or Fourier analysis.

Format:
lecture/discussion and almost surely presentations. Evaluation will be based on
scholarship, discussions, homework and examinations (and if there is student
interest, papers and presentations in place of some of the exams).

Prerequisites: Multivariable calculus, linear algebra, Math 301 or 305, Math 312
or 315.* No
enrollment limit (expected: 21). *

**
SYLLABUS / GENERAL:** The
textbook will be Miller and Takloo-Bighash’s `An
Invitation to Modern Number Theory’ (errata
for the book is here;
additional comments from each day's lecture is
here). On the first day of class we will
describe many of the topics (including
Goldbach's
Problem and the Circle Method, the
Riemann Zeta Function and
Random Matrix
Theory, Benford's Law,
Poissonian behavior,
the 3x+1 map and
sum and difference
sets), and then
determine which topics to explore in detail. Please feel free to swing by
my office or mention before, in or after class any questions or concerns you
have about the course. If you have any suggestions for improvements, ranging
from method of presentation to choice of examples, just let me know. If you
would prefer to make these suggestions anonymously, you can send email from
mathephs@gmail.com (the password
is the first seven Fibonacci
numbers, 11235813). **Grading will
be: 25% homework, 25% projects, 25% midterm, 25% final.**

**
OBJECTIVES:** There
are two main goals to this course: to explore modern number theory and see the
connections between various problems, and to learn
problem solving skills. We will constantly emphasize the techniques we use to
solve problems, as these techniques are applicable to a wide range of problems
in the sciences.

**COURSE NOTES AND HANDOUTS****:** Below
are scanned copies of my lecture notes for the course (click
here for additional comments from each day's lecture). Skimming these notes is a
good way to prepare for lecture and to review the material; however, it is *
not* the case that everything said in lecture will be in these notes
for two reasons: (1) I hope to have a lot of class discussion, and these
comments will undoubtably influence which direction we pursue; (2) detailed
explanations of many arguments are given in the book, so often I have just
jotted down notes to remind myself of what I wanted to mention.

- Errata for the book (if/when you find others, or places where the exposition could be clearer, please let me know)
- Additional comments from each day's lecture
- Complete set of notes (about 7 megs)

- Overview (description of problems, table showing the interconnectedness of the questions and techniques / material)
- n
^{k}α mod 1 and Poissonian behavior (the chapter from the book is available online: click here for Poissonian behavior)- Fourier analysis for n
^{k}α mod 1 (basics of Fourier analysis leading up to Fejer's theorem)- Random matrix theory (the first of the four chapters from the book is available online: click here for Chapter 15, the introduction)
- The Circle Method
- Benford's law
- Riemann Zeta Function and L-functions
- Fourier analysis for Benford's Law and L-functions (Poisson Summation)

**
SUPPLEMENTAL READING:** For
more on each topic, click below to go to a subpage with handouts and reading
recommendations:

**
HOMEWORK PROBLEMS FROM THE TEXTBOOK AND
ASIDES****:**

- n
^{k}α and Poissonian Behavior:

HW: Due Wednesday, Feb 18:12.1.8, 12.1.10, A.4.4, A.4.5, 12.2.6, 12.3.8, 12.6.3 (hard!). Also, write a problem or two based on any of the material or the asides (and, if you can, solve your problem!). The problems you write should be on a separate paper.- Additional: 12.1.11, A.4.8, 12.2.7, 12.3.6, 12.3.7.
- Asides: Algebraic structure of numbers, Hurwitz's theorem, Liouville Numbers, Roth's Theorem, algebraic and transcendental numbers, countable and uncountable sets, Cantor's theorem, Continuum Hypothesis, irrationality and transcendence of numbers (e, π, ...), Monte Carlo integration, measure theory.
- Fourier analysis for n
^{k}α and Poissonian Behavior:

HW: Due Monday, March 2:11.2.2, 11.2.5, A.2.7, 11.2.7, 11.2.10, 11.3.6.- Additional: 11.1.7, 11.2.8, 11.2.12.
- Asides: Dirichlet's Theorem, original version of Weierstrass' Approximation Theorem (and generalizations), convergence of Fourier series (pointwise, L
_{1}, L_{2}, ...).- Random Matrix Theory:

HW: Due Friday, March 11:15.1.4, 15.1.6, 15.2.3, 15.3.5, 15.3.10, 15.3.13, 15.5.8 (note this problem is somewhat open ended); 16.1.16, do one of 16.1.17 or 16.1.18.- Additional: 15.1.5, 15.1.11, 15.1.12, 15.2.3, 15.2.6, 15.2.3, 15.2.6, 15.2.13, 15.3.8, 15.3.9, 15.3.12, 15.3.14, 15.4.7, 15.5.3, 15.5.6; A.1.2, A.1.4, 16.1.13, 16.1.15, 16.1.17 or 16.1.18.
- Asides: Central Limit Theorem, Catalan numbers, McKay's Law for d-regular graphs, Kesten's measure, non-uniqueness of moments, combinatorics / Diophantine equations, matching coefficients and differentiating identities, combinatorics of Toeplitz matrices, GOE and eigenvalues spacings (Chapter 17).
- The Circle Method:

HW: Due Monday, April 20th:13.1.1, 13.1.12, 13.3.10, 13.3.11, 13.3.16, 13.3.20.- Additional: 13.1.3, 13.1.4, 13.1.13, 13.3.4, 13.3.14, 13.3.15, A.6.11 (ie, prove the Cauchy-Schwartz inequality, given in Lemma A.6.9), A.6.13, A.6.14.
- Asides: Prime Number Theorem, generalizing Euclid's theorem, proofs of infinitude of the primes, Chebyshev's theorem (and Bertrand's postulate), Dirichlet's Theorem (and the Siegel-Walfisz theorem), generating functions, differentiating identities and matching coefficients, counting primes with weight log(p) (sections 2.3.4 and 3.2.2), Philosophy of Square-root cancelation, Hasse principle, minor arc contributions, Littlewood problem.
- Benford's Law:

- HW: 9.1.1. TBD
- Additional: 9.3.2, 9.3.3.
- Asides: Lebesgue's theorem on L
_{1}convergence, analytic density, recurrence relations, values of L-functions and Random Matrix Theory, 3x+1 problem, quantified equidistribution, order statistics, Erdos-Turan theorem.- Riemann Zeta Function:

- HW: 3.1.4, 3.1.6, 3.1.9 (important), 3.1.14, 3.1.15, 3.1.23, 3.2.15, 3.2.16, 11.4.9.
- Additional: 3.1.5, 3.1.7, 3.1.8, 3.1.18, 3.1.22, 3.2.19.
- Asides: Properties of Γ(s) and its occurances, relations between roots and coefficients of polynomials and Newton's identities for symmetric polynomials, prove results from Complex Analysis, Hawkins primes, product expansion for ζ(s), Li's constants and the Riemann Hypothesis, Dirichlet characters and L-functions, primes in arithmetic progressions, Fourier transform (sections 11.4.1, 11.4.3, 11.5 and exercise 11.6.4), convergence of Fourier series (pointwise, L
_{1}, L_{2}, ...) (and see exercise 11.6.1 for what can happen at discontinuities), Poisson summation generalizations (what happens if we just use Fejer's theorem).

- Summer Research:

- SMALL (at Williams, deadline Feb 11)
- AMS, MAA and NSF links to summer research programs.
- The Green Chicken / Math Puzzle Night
- LaTeX / Mathematica links
- take aways for the course

This is my first year at Williams; click here for some personal information about me, my family and my research interests.

For extra credit, find the flaw (or flaws) in any of the following papers (or, to be fair, convince me that they're correct or correct them and receive a Fields medal):

- A proof of the Riemann Hypothesis (Cheng, Yuan-You Fu-Rui)
- A proof of Goldbach Conjecture (Jinzhu Han1 Zaizhu Han)
- On a proof of the Goldbach conjecture and the twin prime conjecture (Sze Kui Ng)
- There are infinitely many pairs of twin primes (Zhanle Du and Shouyu Du)

course description syllabus/general objectives course notes / handouts supplemental reading other links (summer research)