• Lecture 1: Overview
• Lecture 2: Factorization
• Lecture 3: Arithmetic functions and Dirichlet series
• Lecture 4: Congruence | Large prime numbers | Ruby code
• Lecture 5: The Sun-Ze Theorem
• Lecture 6: Hensel's Lemma | Taylor's theorem for polynomials | Completing a metric space | Zp is compact
• Lecture 7: Unit group structure | Finitely-generated Abelian groups | The exponential and logarithmic power series are formal inverses
• Lecture 8: Quadratic reciprocity | L^2-synthesis of a continuous periodic function (beamer talk) (beamer handout) | Zolotarev's proof of QR | Coin flips by telephone
• Lecture 9: Algebraic numbers and algebraic integers
• Lecture 10: Finite fields
• Lecture 11: Characters, Gauss sums, Jacobi sums | Whence Gauss sums? | The Gauss sum norm via Fourier analysis
• Lecture 12: The Eisenstein integers and cubic reciprocity | An easy case of Fermat's Last Theorem

• Assignment A | B | C | Some study topics for the first quiz | (First quiz unavailable online) D | E | (Second quiz unavailable online) | F | G | H

Further in-class topics are chosen to introduce analytic and algebraic number theory. Not all the writeups posted here are covered.

• Primes in arithmetic progressions (because the argument in this writeup implicitly uses cyclotomic field ideas, some of these accompanying writeups explicate them)
  • Nonvanishing of Dirichlet L-functions at s=1
  • The cyclotomic zeta function
  • Cyclotomic polynomials
  • Irreducibility of cyclotomic polynomials via Dirichlet's arithmetic progression theorem
  • The prime-power cyclotomic integer ring by traces and norms
  • The cyclotomic integer ring in general
  • Sums of two squares via a quadratic zeta function
  • Primes in sectors via a Grossencharakter (sketch)

• Continuations and functional equations | (Mathematica notebook)
  • The Fourier transform and the Mellin transform
  • A basic theta function
  • The Bernoulli numbers, power sums, and zeta values
  • Dirichlet L values at nonpositive integers

• The ideal class number formula for an imaginary quadratic field | (Mathematica notebook)

• The unit group of a real quadratic field | (Mathematica notebook)

• The Prime Number Theorem
  • Weierstrass/Hadamard products
  • Sketch of the Riemann-von Mangoldt explicit formula | (Mathematica notebook)

• Cyclotomic-intermediate fields via Gauss sums

• Hecke characters, classically and idelically
• Quadratic reciprocity, after Weil | Self-dualities and the adelic dual (by Paul Garrett)
• Self-duality of Q_p and R via the solenoid
• Fujisaki's Lemma, after Weil | Classical consequences of Fujisaki's Lemma (by Paul Garrett)
• Zeta integrals

• The Hasse-Davenport relation
• The L-function of an elliptic curve
• The zeta function of an algebraic set

• The simplest nonabelian example of modularity

Lectures | Assignments | Back to my home page