HOMEWORK AND READINGS

Please spend at least 5 hours a week reading the material/looking at the proofs/making sure you understand the details. Below is a reading list and homework assignments. The topics can be done in any order.

• For those taking this course as Math 308, you should do at least 3 topics, for those taking as Math 406 you should do at least 4. Note some of the homework problems have hints in the back of the book.
• Every group will write a review for one paper for Mathemetical Reviews; if you are interested I can also get you papers to referee for journals (the first one has come in, and I'll look at it later today to see if it is appropriate). You can view other reviews online at MathSciNet. If you click on Free Tools there and then Collaboration Distance, you can calculate people's Erdos' number (it lists me as three, but I have a paper in press that'll bring me down to two).
• Go to the math arxiv (http://arxiv.org/list/math/new) at least once or twice a week, though ideally every day. You should read the titles of every post. Whenever you find something that looks interesting, read the abstract. If it looks really interesting, skim the introduction. If you're still interested, let me know! This is a good habit to get into; journals have a very long lag time from receipt of articles to publication. This is a great way to get a sense of what areas of mathematics are active, who's doing what, ... (especially if you're considering graduate study). If you don't want to read / skim all the new posts, you can look at what is new in just the fields you care about, such as http://arxiv.org/list/math.NT/recent (number theory, of course!).
• Every group will look at either a paper posted on the math arxiv or submitted to me for comments. These papers prove important results such as the Riemann hypothesis, Fermat's Last Theorem, the infinitude of twin primes, Goldbach's conjecture, and so on. I have posted these papers here. It's a good skill to look at these and try and get a sense of where the mistake lies. A particularly friendly one to look at is the one by Ghosh, but feel free to choose any.
• Below is a guideline of readings and exercises for nine topics (I'm of course open to adding other items to the list): (1) The Riemann zeta function; (2) The Circle Method; (3) Random Matrix Theory; (4) Additive Number Theory; (5) Poisson Behavior and Fourier Analysis; (6) The 3x+1 Problem; (7) Benford's Law; (8) Linear Programming; (9) Sphere Packing and Error Detecting / Correcting Codes; (10) Continued Fractions.

Videos of the lectures are available after the title of each topic, or go to the end of the page for all of them listed together.

For all of these topics, supplemental reading is available here.

• Topic 1: The Riemann Zeta Function: Chapter 3 of the textbook:    Lecture Explaining Zeta Function
  • General notes: The Riemann zeta function is one of the most important functions in mathematics and physics. It allows us to pass from knowledge of the integers to knowledge of the primes. Transfers like this are common in mathematical physics; another example is in random matrix theory. The zeta function can be massively generalized to yield information about other arithmetically interesting objects, such as primes in arithmetic progressions and the group of solutions to an elliptic curve, as well as properties of graphs!
  • Background: For much of this chapter, no background is needed. There are some parts where complex analysis is useful (so if you are taking Math 302, you may want to wait till later in the course to do this, or just read the complex analysis summary in this book and be well prepared for 302!), and for Dirichlet L-functions and primes in arithmetic progression it's very useful to know elementary group theory (mostly just the notion of a group and some simple examples, nothing beyond what is in Chapter 1 of the book).
  • Reading: You should read: Section 3.1.1 (the Euler product is one of the most important properties of the zeta function), 3.1.2 (this gives the analytic continuation), 3.1.3 (this briefly discusses special values). Proving these results requires some advanced material elsewhere in the book. Section 3.2 is significantly more advanced and is best if you know complex analysis (or are willing to learn a bit now), and can be safely skipped or skimmed. Section 3.3 requires some group theory and introduces the first generalization of the zeta function, Dirichlet L-functions, and culminates in a sketch of the proof of the infinitude of primes in arithmetic progressions.
  • Homework problems:  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 problems worth looking at: 3.1.5, 3.1.7, 3.1.8, 3.1.18, 3.1.22, 3.2.19. You should do at least FOUR problems (including 3.1.9); one learns best by working out exercises; however, I will leave it to you to choose which problems you find particularly interesting. If you would prefer me to choose problems, let me know.  You may write up the problems or orally present them.

   

• Topic 2: The Circle Method: Chapters 13 and 14 of the textbook:  Lecture Explaining Circle Method
  • General notes: The Circle Method is one of our best tools for proving results in additive number theory. While the method isn't that bad to describe, the error analysis becomes unbearably technical and involved in many problems (and, sadly, is often beyond our ability!). That said, in a multitude of situations it can quickly lead to very accurate, testable predictions. It's early successes were in Goldbach type problems (every large odd number is the sum of three primes) and Waring's problem (for any k, there is an s(k) such that each integer is a sum of s(k) perfect kth powers).
  • Background: You should read the cookie problem (pages 11 to 13) before reading this chapter. Other needed material includes variants of the prime number theorem (which are stated and can be used as a black box), and why it's convenient to weight primes with a logarithmic factor (see Sections 2.3.4 and 3.2.2, culminating in Remark 3.2.18). It is also useful to know some of the elementary functions of number theory (Euler's phi or totient function, the Mobius function, the Lambda function). These are defined in Chapters 1 and 2.
  • Reading: You should read all of Chapter 13, which describes the general setting and method. Chapter 14 goes through the details of the major arc calculation for Germain primes (counting the number of these has applications in cryptography, in particular, in primality testing). You should not worry about all the technical details and all the different evaluations of the sums. Unless you're working on a problem and need to do these calculations, there's not much gain to seeing all the details. Note that the book does NOT cover the minor arcs; if you are interested in the estimates there let me know.
  • Homework problems: 13.1.1, 13.1.12, 13.1.13, 13.3.10, 13.3.11, 13.3.16, 13.3.20.  Additional: 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. You should do at least FOUR problems (including 13.1.13); one learns best by working out exercises; however, I will leave it to you to choose which problems you find particularly interesting. If you would prefer me to choose problems, let me know.  You may write up the problems or orally present them.

   

• Topic 3: Random Matrix Theory: Chapters 15 to 18 of the textbook (Chapters 15 and 16 are one topic; adding Chapter 17 or 18 is a second topic): Lecture Explaining RMT
  • General notes:  Random matrix theory is an extremely active area of research in both mathematics and physics. It models everything from L-functions to nuclear physics to networks to bus routes in Mexico! The background needed is a bit higher, but the payoff is huge. Similar to the Riemann zeta function, we see another instance of transfer (passing from knowledge of matrix elements to knowledge of the eigenvalues through the eigenvalue trace lemma).
  • Background: You'll need basic linear algebra (eigenvalues, diagonalizing matrices) and basic probability (reviewed in Chapter 8), as well as some combinatorics.Chapter 17 requires some knowledge of differential equations (second order linear equations, separation of variables). Chapter 18 requires knowledge of L-functions (at the level of Chapter 3) and contour integration from complex analysis (and perhaps a bit of Fourier analysis in a few places, at the level of Chapter 11).
  • Reading: You should read all of Chapters 15 and 16, as well as the survey article I wrote with my undergraduate mentor, Professor Firk (which is an expanded version of Chapters 15 and 16). If you do the second topic, you should read either all of Chapter 17 or all of Chapter 18.
  • Homework:  15.1.1, 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, 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. You should do at least FOUR problems (including 15.1.1); one learns best by working out exercises; however, I will leave it to you to choose which problems you find particularly interesting. If you would prefer me to choose problems, let me know.  You may write up the problems or orally present them. If you elect to do Chapter 17, do three out of 17.1.4, 17.1.6, 17.2.2, 17.2.5, 17.2.11, 17.2.3, 17.3.6, 17.3.9 (fun!). If you do Chapter 18, do three out of 18.1.2, 18.1.3, 18.1.4, 18.1.9, 18.1.10, 18.1.12, 18.1.14, 18.2.4, 18.2.9.

     

• Topic 4: Additive Number Theory: Lecture Explaining Additive Number Theory
  • General notes: Additive number theory is a vast subject. We frequently have elementarily stateable problems that degenerate into interesting and sometimes difficult combinatorics. The sequence of papers leads to a lovely result involving phase transition in behavior as a certain parameter passes through a critical value. Phase transitions are extremely important, and arise everywhere from pure math to physics to graph theory (a fun problem is the birth of the giant component). The main problem is given a finite set of integers A, form A+A and A-A, where A+A = {a1 + a2: a_i in A} and A-A = {a1 - a2: a_i in A}. As addition is commutative but subtraction isn't, a generic pair gives two differences and only one sum. Thus expect A+A to be smaller than A-A. Amazingly, Martin and O'Bryant showed that a positive percent of the time A+A is larger! If you use a difference model of choosing sets randomly, however, Hegarty and I showed almost all sets are difference dominated. In the other direction, I and some students constructed the world record for densest families of sum-dominated sets
  • Background: basic probability (at the level of Chapter 8) and elementary combinatorics.
  • Reading:.
    • Many sets have more sums than differences (Martin and O'Bryant)
    • Problems in Additive Number Theory I (Nathanson)
    • When almost all sets are difference dominated (Steven Miller and Peter Hegarty)
    • Miller-Roman-Sinnott: y^2= n(n+1)(n+2)(n+3): A nice proof when four consecutive integers can and cannot be a square; can you generalize to products of six consecutive integers? This could make a really nice Monthly article.
  • Homework: TBD, but this is a great topic to try your hand at a small research project.

   

• Topic 5: Poissonian Behavior and Fourier Analysis (Chapters 11 and 12):  Lecture Explaining Poissonian Behavior & Fourier Analysis
  • General notes: Fourier analysis is not just interesting in its own right, but it's used to prove a multitude of results, ranging from the functional equation of the Riemann zeta function to the uncertainty principle in Quantum Mechanics! This unit looks at the spacings between certain special sequences. In some sense, there is a lot of similarity between these questions and those in random matrix theory. The general question is: let alpha be irrational and k fixed. Look at the sequence with x_n = n^k alpha mod 1. If we take n up to N, order these fractional parts in increasing order. Look at how they fall in the interval [0,1], look at gaps between adjacent spacings.
  • Background: Some basic probability (at the level of Chapter 8).
  • Reading: All of Chapter 12, and up to and including Section 11.3.1 of Chapter 11.
  • Homework: 12.1.8, 12.1.10, A.4.4, A.4.5, 12.2.6, 12.3.8, 12.6.3 (hard!). Additional: 12.1.11, A.4.8, 12.2.7, 12.3.6, 12.3.7. You should do at least FOUR problems (including 12.2.6); one learns best by working out exercises; however, I will leave it to you to choose which problems you find particularly interesting. If you would prefer me to choose problems, let me know. 

   

• Topic 6: The 3x+1 Problem:   Lecture Explaining 3x+1
  • General notes: This problem has been described as a Soviet conspiracy to slow down American mathematics due to all the time people have spent on it without solving it! Enough said!
  • Background: Some basic probability (at the level of Chapter 8).
  • Reading:
    • The 3x+1 problem and its generalizations (J. Lagarias). Pages 305-334 in Organic mathematics (Burnaby, BC, 1995), CMS Conf. Proc., vol. 20, AMS, Providence, RI, 1997.
    • The 3x+1 problem: An annotated bibliography (J. Lagarias), preprint.
    • Benford's Law, Values of L-Functions and the 3x+1 Problem (Alex Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297.
    • Benford's Law for the 3x+1 function (J. Lagarias and K. Soundararajan),  J. London Math. Soc. (2) 74 (2006), no. 2, 289--303.
    • Stochastic models of the 3x+1 and 5x+1 Problems (Alex Kontorovich and Jeff Lagarias). Jeff Lagarias is writing a book on the 3x+1 Problem; if you are interested I can get you chapters to read.
  • Homework: TBD, but this is a great topic to try your hand at research.

   

• Topic 7: Benford's Law:  Lecture Explaining Benford's Law
  • General notes: Benford's law of digit bias says that, for many `natural' data sets, the probability of observing a first digit of d base 10 is not 1/9 but rather log₁₀(1 + 1/d); my favorite applications are to detecting tax fraud for the IRS.
  • Background: Some basic probability (at the level of Chapter 8) and depending on how deeply you go, some Fourier analysis (at the level of Chapter 11).
  • Reading:
    • Chapter 9 of the book.

    • The first-digit phenomenon (T. Hill), American Scientist 86 (1996), 358--363.

    • A statistical derivation of the significant-digit law (T. Hill), Statistical Science 10 (1996), 354--363.

    • The first digit problem (R. A. Raimi), Amer. Math. Monthly 83 (1976), no. 7, 521--538.

    • Applications:
      • I've got your number (M. Nigrini), Journal of Accountacy (1999).
    • Examples
      • Benford's Law, Values of L-Functions and the 3x+1 Problem (Alex Kontorovich, Steven J. Miller), Acta Arithmetica 120 (2005), 269--297.
      • Benford's Law for the 3x+1 function (J. Lagarias and K. Soundararajan),  J. London Math. Soc. (2) 74 (2006), no. 2, 289--303.
      • Order Statistics and Shifted Almost Benford Behavior (Steven J. Miller and Mark J. Nigrini),  International Journal of Mathematics and Mathematical Sciences, Volume 2008 (2008), Article ID 382948, 19 pages, doi:10.1155/2008/382948.
    • Numerics
      • Analysis of Benford's law applied to the 3x+1 problem (S. Minteer), Number Theory Working Group, The Ohio State University, 2004.
    • I'm currently editing a book on Benford's law -- if you are interested in this topic, let me know and I can share chapters.
  • Homework: TBD, but this is a great topic to try your hand at research.

   

• Topic 8: Linear Programming:  Lecture Explaining Linear Programming
  • General notes: This is a nice application of advanced linear algebra. Linear programming is a great way of solving or approximately solving many optimization problems.
    It's used in designing schedules for MLB (minimize travel time, have Sox - Yankees games at good times, have lots of division games at the end of the season). It's also used to correctly compute elimination numbers, taking into account who has games against whom (MLB does not correctly calculate elimination numbers!). It helps airlines optimize schedules....
  • Background: linear algebra.
  • Reading:
    • My notes on Linear Programming as a guide (these are based on the excellent book by Joel Franklin on Mathematical Methods in Economics).
    • Here is a paper that implements linear programming to very efficiently solve this problem (click here for more information).
    • Silver Scheduler: Demand-Driven Scheduling of Movies in a Multiplex (Jehoshua Eliashberg, Quintus Hegie, Jason Ho, Dennis Huisman, Steven J. Miller, Sanjeev Swami, Charles B. Weinberg and Berend Wierenga), lead article International Journal of Market Research and won award for best paper of 2009 (doi:10.1016/j.ijresmar.2008.09.004).
  • Homework: TBD, but I have two people who have requested solutions to problems (one involves scheduling play in bridge matches, the other involves creating athletic conferences for the State of Maine).

   

• Topic 9: Sphere Packing and Error Correcting / Detecting Codes:  Lecture Explaining Sphere Packing and Codes    Supplement (answer to a question)
  • General notes: These are some of the most important topics in modern number theory. The applications are more than just cryptography, as the ability to detect and correct errors in data streams.
  • Background: basic number theory and basic combinatorics.
  • Reading:
    • I am currently writing a book on cryptography; the chapters on error detecting and correcting, as well as factorization, is available here (note this is a very rough draft!).
    • Cannonballs and honeycombs (Hales)
    • Lexicographic Codes and Error Correcting Codes From Game Theory (Conway and Sloane)
  • Homework: TBD

   

• Topic 10: Continued Fractions:  Lecture Explaining Continued Fractions
  • General notes: Continued fractions are useful for a variety of purposes, mostly in approximating irrationals with rationals (which allow us to measure how `irrational' a number is). Unlike base expansions (such as binary or decimal), there is no base in a continued fraction expansion.
  • Background: elementary recurrence relations
  • Reading: Chapter 7: Sections 7.1, 7.2, 7.3, 7.4, 7.5, 7.8, and then either 7.6 or 7.7 or 7.9 or Chapter 10.
  • Homework: 7.1.1, 7.1.3, 7.2.7, 7.2.8, 7.2.9, 7.2.10, 7.3.7, 7.3.10, 7.3.11, 7.3.15, 7.3.17, 7.3.22, 7.5.2, rest TBD. You should do at least FOUR problems (including 7.3.15); one learns best by working out exercises; however, I will leave it to you to choose which problems you find particularly interesting. If you would prefer me to choose problems, let me know. 

   

Lectures: These are the introductory remarks I've made about the problems.

1. Lecture Explaining Zeta Function
2. Lecture Explaining Circle Method
3. Lecture Explaining RMT
4. Lecture Explaining Additive Number Theory
5. Lecture Explaining Poissonian Behavior & Fourier Analysis
6. Lecture Explaining 3x+1
7. Lecture Explaining Benford's Law
8. Lecture Explaining Linear Programming
9. Lecture Explaining Sphere Packing and Codes    Supplement (answer to a question)
10. Lecture Explaining Continued Fractions