Senior Mathematics Seminar
Math 4513
Fall Semester 2004

• During Final Exam week I will be in my office PHSC 801 on and off, but definitely during the times: Monday, December 13 2-6 PM, Tuesday, December 14, 12-1:30 PM and Thursday, 10-12:00 AM. Stop by during those times (or e-mail me at amiller"at"math.ou.edu to arrange a different time) if you would like discuss progress on the subseries projects.

• Here are the problems that were worked on in class on Friday, November 5. We will explore them further during class in the Computer Lab on Friday, November 12.

• Here is a view of the graph of 1 over the absolute value of the Riemann zeta function "Plot3D[1/Abs[Zeta[x + I*y]^2], {x, -5, 3.5}, {y, -30, 30}, PlotPoints -> 100, PlotRange -> {0, 13}, ViewPoint -> {2.6, -.7, 0.93}]" created by Brian Wilson. The poles in this picture correspond to the first few zeroes on the critical line. Brian also helped to create the movie below, which shows the first few zeroes of the zeta function along a ray emanating from the origin as it sweeps from 80 degrees towards 90 degrees. The black dot represents a point on the critical line.

  

• On Friday October 22 we will meet in the Computer Lab PHSC 230 and work on Zeta graph experiments. We'll also continue to work on the 6th MATHEMATICA assignment (which we started on October 15). Note that many of the questions on this assignment are open-ended and you should have to do quite a bit of experimentation to come up with good windows and viewpoints on the various graphs. See the MATHEMATICA Plotting Commands for Calculus web site for basic information on using the Plot, ParametricPlot, Plot3D and ContourPlot commands.

• By Monday, October 18 you should have a print-out of the list of references for your biographical subject.
  The Biography/History Project for this class will be due on Friday, November 12. Here is a list of the biographical subjects which have been assigned. A useful website for finding information is the St. Andrews History of Mathematics Archive which contains an extensive biography section, and these biographies usually include very helpful lists of additional references. If there seems to be a dearth of information available for your subject, you should discuss it with me as soon as possible.

• On Friday, October 1 we worked on MATHEMATICA Assignment #5 which is due on Wednesday, October 6. This assignment is based on a MATHEMATICA notebook Sketching the function f(z)=z^3, and has to do with representing complex-valued functions graphically. Many such representations can be found in Derbyshire's book; for example: Figures 13-2 (p 208), 13-6 (p 213 and shown at the right), 13-7 (p 218) and 21-3 (p 336).
  Here are the problems on complex numbers and functions worked on in class on Wednesday, September 29. Students will present answers on Monday.

• On Friday, September 24 we worked on MATHEMATICA Assignment #4.
  Here is the list of arithmetic problems for complex numbers that we worked on in class on September 22.

• The third MATHEMATICA assignment should be turned in by 5:00 PM on Wednesday, Septembber 22.

• Note from class: The number 101!+2 only has 160 digits in it so it does not take up more than two screens when printed out (unless you have set a very small window).

• Here is a MATHEMATICA program carrying out the sieve method to find the primes less than or equal to M:

   
    Sieve[M_] := (
      inlist = Range[2, M];
      primelist = {};
      For[k = 1, (p = inlist[[1]]) <= Sqrt[M], k++,
        primelist = Append[primelist, p];
        inlist = Complement[inlist,  p*Range[1, M/p]];
        ];
      primelist = Union[primelist, inlist]
      )
  
  

  This program was written by Nick Brooke and it incorporates a few features that are improvements over the program I gave in problem 2 of MATHEMATICA assignment #2. Two of those improvements are: (i) the for-loop is terminated as soon as the minimum value in the list "inlist" exceeds the square root of M, and (ii) it defines p as the first element of "inlist" rather than the minimum value of "inlist" (this saves on computation time and works because the elements of "inlist" are always in increasing order). Note that Nick has also defined the output as a function of M, so just typing "Sieve[10^6]" (for example) will return a list with all the primes up to one million.

• Assignments:
  • The second MATHEMATICA assignment is due on Wednesday, September 8.
  • Reading: Finish reading Part I of Derbyshire's book by Monday, September 13. Continue reading at least the even chapters in Part II.

• Here is a brief description of the course.

• Bibliography:
  • John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, 2003.
  • Karl Sabbagh, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, Farrar, Strauss and Giroux, 2002.
  • Julian Havil, Gamma: Exploring Euler's Constant, Princeton University Press, 2003.
  • H.M. Edwards, Riemann's Zeta Function, Academic Press, 1974.
  • Marcus du Sautoy, The music of the primes: searching to solve the greatest mystery in mathematics, Harper-Collins, 2003.

• Web Links:
  • http://olimu.com/Riemann/Riemann.htm : home page for Derbyshire's book, including a link to a log of errors in the book.
  • www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf : an English translation of Riemann's original paper delivered to the Berlin Academy in August, 1859.
  • ...more links will be supplied later by the students of this class...

• Mathematica Assignment 1 (due Monday, August 30): Print out the introduction to MATHEMATICA, then take it into the Computer Lab and work through all of the commands. Print your output for this (with mistakes and unnecessary parts removed).
• Homework Assignment 1 (due Wednesday, Sept 1):
  1. Go through the first three chapters of the text and make a list of all of the experimental results that you are able to identify. For example: (on page 7) one-half the sum of the first 999999999999 terms of the harmonic series is a tad less than 2.58868875882; Or (on page 33) Derbyshire lists all of the primes up to 1000; Or the data in Table 3-1, etc.
  2. Come up with what you think would be a good math/historical question to prompt a discussion on the material in the first three chapters of Derbyshire's book. Then write a brief outline response to your question.

• Many of the course handouts for this class (and the pdf files posted at this web site) have been prepared using the mathematical typesetting system TeX, and associated packages known as AMSTeX and LaTeX. This system is extremely well-suited for incorporating mathematical formulas seamlessly into documents, and is universally accepted as the best typesetting system for preparing technical documents. It is not difficult to get started using the software, although mastering it takes a bit of experience. As a student of this course, you will not be required to learn how to use TeX or LaTeX, but if you are serious about pursuing a career in any mathematical science, you should try to familiarize yourself with it as soon as convenient. Beginners should start out with LaTeX. There are a number of elementary starting guides readily available on the web, and the software itself is available in a number of different free implementations (see the links below).
  
  If you want to load LaTeX onto your own PC, I recommend using the TeX/LaTeX package called MikTeX bundled with the ASCII editor TeXnicCenter (which is specially designed to work with MikTeX). Together these provide a very nice platform for working in LaTeX, and both programs are entirely free. You can download these programs:
  • MikTeX at www.miktex.org
  • TeXnicCenter at www.texniccenter.org
  Most TeX packages (such as MikTeX) include a lot of documentation for getting started with LaTeX. For more information, check the TeX Users Group website www.tug.org and CTAN (the Comprehensive TeX Archive Network). There you will find links to manuals for using LaTeX, TeX related software, and all kinds of other information about using TeX. Particularly useful references are an on-line tutorial Formatting Information: A beginner's introduction to typesetting with LaTeX by Peter Flynn, a LaTeX manual The Not So Short Introduction to LaTeX2e by Tobias Oetiker, and a description of using the AMS theorem environment. If you do experiment with LaTeX and have questions, feel free to ask me about them---or if you would like to see how I use TeX on my PC, I would be happy to demonstrate how I create TeX documents for the class.

Math 4513, Fall 2004