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18.104

Seminar in Analysis

3-0-9 units · Undergraduate

Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.

Fall 2006 · Seminar in Analysis: Applications to Number Theory

Topics

  • Summing Powers of Integers, Bernoulli Polynomials
  • A Summary of the Properties of Bernoulli Polynomials and more on Computing \(\zeta(2n)\)
  • Infinite Products, Basic Properties, Examples
  • Fermat’s Little Theorem and Applications
  • Fermat’s Great Theorem
  • Applications of Fermat’s Little Theorem to Cryptography: The RSA Algorithm
  • Averages of Arithmetic Functions
  • The Arithmetic-geometric Mean; Gauss’ Theorem
  • Wallis’s Formula and Applications I
  • Stirling’s Formula (cont.)
  • Elementary Proof of The Prime Number Theorem I
  • Elementary Proof of The Prime Number Theorem II: Mertens’ Theorem, Selberg’s Formula, Erdos’ Result
  • Short Analytic Proof of The Prime Number Theorem I (After D. J. Newman and D. Zagier)
  • Short Analytic Proof of The Prime Number Theorem II: The Connection between PNT and Riemann’s Hypothesis
  • Discussion on the First Draft of the Papers and some Hints on how to Improve the Exposition and use of Latex
  • Final Paper Presentations I
  • Final Paper Presentations II
  • Final Paper Presentations III

Readings

  • Readings are in the required course textbook:
    Listed reading
  • Young, Robert M. Excursions In Calculus: An Interplay of The Continuous and The Discrete . Washington, DC: Mathematical Association of America, 1992. ISBN: 0883853175. Some additional readings are linked from this page.
    Listed reading
  • Text, chapter II (1a), pp. 58-63, possibly complemented by exercise 2; p. 34, exercise 2 (maybe also 1); p. 70.
    Listed reading
  • Text, chapter II (1b), pp. 64-69, possibly complemented by exercise 6 (maybe 4,5); p. 70, exercises 13 and 15; p. 71.
    Listed reading
  • 2 · Summing Powers of Integers, Bernoulli Polynomials · Text, chapter II (2), pp. 74-93; possibly complemented by exercises 2 and 3; p. 95, exercise 11; p. 97, exercises 19 and 20; p. 99.
    Listed reading
  • 8 · Applications of Fermat’s Little Theorem to Cryptography: The RSA Algorithm · Reference: Trappe, Washington. Introduction to Cryptography with Coding Theory . Section 6.1, a little of 6.3
    Listed reading
  • 10 · The Arithmetic-geometric Mean; Gauss’ Theorem · Text, pp. 231-238; maybe supplemented by some material from Cox, David A. Notices 32, no. 2 (1985) (QA.A5135) and Enseignment Math 30, no. 3-4 (1984).
    Listed reading
  • Young, Robert M. Excursions In Calculus: An Interplay of The Continuous and The Discrete. Washington, DC: Mathematical Association of America, 1992. ISBN: 0883853175.
    Listed reading
  • There is also a recently published (2006) book, which I find very suitable for a topics course like ours:
    Listed reading
  • Miller, Steven J., and Ramin Takloo-Bighash. An Invitation to Modern Number Theory. Princeton, NJ: Princeton University Press, March 6, 2006. ISBN-10: 0691120609, ISBN-13: 9780691120607.
    Listed reading
  • Davenport, Harold. The Higher Arithmetic : An Introduction to The Theory of Numbers. 7th ed. Cambridge, UK: Cambridge University Press, 1999. ISBN: 0521632692, 0521634466. (pbk)
    Listed reading
  • Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery. An Introduction to The Theory of Numbers . 5th ed. New York, NY: Wiley, c1991. ISBN: 0471625469.
    Listed reading

Prerequisites & course information

Prereq: 18.100A , 18.100B , 18.100P , or 18.100Q U (Fall) 3-0-9 units

Sources