Lectures for Mathematics 361, Spring 2023-24

M 1/22: Overview: Z[i], sums of two squares
- Lecture 1: Overview
- Assignment A - due 1/31
T 1/23: Pythagorean triples; start Euclidean rings
- Lecture 2: Factorization
W 1/24: More Euclidean rings
F 1/26: Finish Euclidean rings; start Euler's proof that \sum 1/p diverges
M 1/29: Arithmetic functions and Dirichlet series
- Lecture 3: Arithmetic functions and Dirichlet series
T 1/30: More arithmetic functions and Dirichlet series
W 1/31: [Assignment A due] Euclid's lemma, lcm, (Z/nZ,+,.), (Z/nZ)^x, Euler's rule, Fermat's Little Theorem
- Lecture 4: Congruence
- Large prime numbers
- Ruby code
- Assignment B - due 2/7
F 2/2: [Add/section-change/unit reduction deadline] ax+ny=b, ax=b(n); algorithms, Ruby code
M 2/5: Sun-Ze Theorem
- Lecture 5: The Sun-Ze Theorem
T 2/6: Hensel's Lemma, start finitely generated abelian groups
- Lecture 6: Hensel's Lemma
- Taylor's theorem for polynomials
- The exponential and logarithmic power series are formal inverses
- Completing a metric space
- Zp is compact
W 2/7: [Assignment B due] Finish finitely generated abelian groups, (Z/pZ)^\times, Z/p^eZ ~ Zp/p^eZp
- Finitely-generated Abelian groups
- Lecture 7: Unit group structure
- Assignment C - due 2/14
F 2/9: (Z/p^eZ)^\times for p odd, (Z/2^eZ)^\times, when is (Z/nZ)^\times cyclic?
M 2/12: The Bernoulli numbers, power sums, and zeta values
- The Bernoulli numbers, power sums, and zeta values
- A series representation of the cotangent
- Zeta at negative odd integers, a la Euler
- Heuristic derivation of Stirling's formula
T 2/13: The Bernoulli numbers, power sums, and zeta values
W 2/14: [Assignment C due] Start quadratic reciprocity--Euler's criterion, Gauss's lemma, (-5/p) by Gauss
- Lecture 8: Quadratic reciprocity
- Zolotarev's proof of quadratic reciprocity
- Coin flips by telephone
- Some study topics for the first quiz
- First quiz out - due 2/21
F 2/16: Legendre's formulation of QR, Euler QR <=> Legendre QR
M 2/19: Jacobi symbol, start algebraic numbers and algebraic integers
- Lecture 9: Algebraic numbers and algebraic integers
- The resultant
T 2/20: QR by Gauss sums: (2/p) and (p^*/q)=(q/p)
W 2/21: [First quiz due] Review QR by Gauss sums, gloss the polyomial of the sum or product of algebraic numbers by resultants, gloss sign of the Gauss sum by Fourier analysis
- L^2-synthesis of a continuous periodic function (beamer talk) | (beamer handout)
- Assignment D - due 2/28
F 3/23: Zolotarev's proof of QR, start finite fields
- Lecture 10: Finite fields
M 2/26: [No-W drop deadline] Finish most of the finite fields writeup
T 2/27: Finite fields loose ends, start characters
- Lecture 11: Characters, Gauss sums, Jacobi sums | (Mathematica notebook)
- Whence Gauss sums?
- The Gauss sum norm via Fourier analysis
W 2/28: [Assignment D due] Image and kernel, N(x^e=u), orthogonality of characters, start Gauss sums
- Assignment E - due 3/6
F 3/1: Quick review of Gauss sums, Jacobi sum definitions and diagonal counting formula, quadratic example, most of the table of Jacobi sum values
M 3/4: Finish Jacobi sum values, start cubic example and its modularity
T 3/5: Finish cubic example and its modularity
W 3/6: [Assignment E due] Start arithmetic of D=Z[\omega]: unique factorization, units, primes, factorization of rational primes, primary primes
- Lecture 12: The Eisenstein integers and cubic reciprocity
- An easy case of Fermat's Last Theorem
- Second quiz out - due 3/22
F 3/8: (Informal meeting)
Spring break week
M 3/18: Review arithmetic of D, cubic character
T 3/19: Prove properties of the cubic character, state cubic reciprocity, review proof of quadratic reciprocity
W 3/20: Prove cubic reciprocity, examples in Z, sketch examples beyond Z
F 3/22: [Second quiz due] Fermat for n=3, start Dirichlet's theorem on primes in arithmetic progressions
- Assignment F - due 3/27
M 3/25: Continue Dirichlet's theorem
- Primes in arithmetic progressions
- The Fourier transform and the Mellin transform
- Fourier transform of the Gaussian
- Continuations and functional equations | (Mathematica notebook)
- Sketch of the Riemann-von Mangoldt Explicit Formula
- Local factors of zeta functions
- The prime-power cyclotomic integer ring by traces and norms
- The cyclotomic integer ring in general
- The cyclotomic zeta function
- Sums of two squares via a quadratic zeta function
- Nonvanishing of Dirichlet L-functions at s=1
- Primes in sectors via a Grossencharakter (sketch)
T 3/26: Continue Dirichlet's theorem
W 3/27: [Assignment F due] Continue Dirichlet's theorem
- Assignment G - due 4/3
F 3/29: (No meeting - Qual break)
M 4/1: [Withdraw/leave deadline] Finish Dirichlet's theorem on a.p.'s, preview rest of semester
T 4/2: Start continuations and functional equations: Fourier transform and the Gaussian, transformation law of theta
W 4/3: [Assignment G due] Finish continuation and functional equation of zeta
- Assignment H - due 4/10
F 4/5: Start continuation and functional equation for Dirichlet L-functions
M 4/8: zeta as product of local integrals, review Dirichlet L and continue onward through most of the argument
T 4/9: Finish continuation and functional equation for Dirichlet L-functions, skim Riemann-von Mangoldt
W 4/10: [Assignment H due] Quadratic fields, through norm
- The ideal class number formula for an imaginary quadratic field | (Mathematica notebook)
F 4/12: More on quadratic fields, through quadratic character
M 4/15: Review through quadratic character, start factorization of ideals
T 4/16: Factorization of ideals
W 4/17: More on quadratic fields, through the geometry section
F 4/19: More on quadratic fields, through L(1,\chi)
M 4/22: Quadratic fields: lattice points in a disk, estimate A_n for Dedekind zeta
T 4/23: Finish imaginary quadratic field class number formula
- The unit group of a real quadratic field | (Mathematica notebook) | (continued fractions Mathematica notebook)
W 4/24: (No meeting)
F 4/26: (No meeting)
M 5/6: [Term project due at noon]

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