SIMUW 2012 -- The Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture

Resources

• The book What is Riemann's Hypothesis?
• SIMUW Sage Notebook Server
• Sage Website, Tutorial

Course Goals

• Learn the powerful free open source math software system Sage.
• Learn what the Riemann Hypothesis really is, and much interesting mathematics along the way: prime numbers, Fourier series, some advanced Calculus, and more.
• Learn what the Birch and Swinnerton-Dyer conjecture states, and something about what is known about it.

NOTES

• You must be logged into simuw.sagenb.org in order for clicking on the worksheet links below create a worksheet!
• Clicking on the link creates a brand '''new''' worksheet. If you then edit it, you'll find your modified worksheet in your home screen.

Day 1: Prime Numbers

• Introduction to Sage (click to create worksheet)
• Prime Numbers (click to create worksheet)
• Infinitely Many Prime Numbers (click to create worksheet)
• Finding all prime numbers up to a given bound (click to create worksheet)
• Ask your own question about primes (click to create worksheet)

Day 2: A Precise Statement of the Riemann Hypothesis

• Handout a prime number to everybody
• Discuss the questions you came up with about primes numbers
• Prime gaps (click to create worksheet)
• Multiplicative parity (click to create worksheet)
• What Proportion of Numbers are Prime? (click to create worksheet)
• [Counting Primes - Riemann Hypothesis (first formulation) (click to create worksheet)] (http://simuw.sagenb.org/upload_worksheet?url=https://raw.github.com/williamstein/simuw12/master/day02/rh1.txt)

Day 3: Fourier Analysis and Prime Numbers

• Review (click to create worksheet)
• Tinkering with the staircase of primes (click to create worksheet)
• MP3 Files and Prime Numbers (click to create worksheet)
• Trigonometric sums and the staircase of primes (click to create worksheet)
• Handout some functions to differentiate to everybody
• What can you do in Sage? (click to create worksheet)

Day 4: The Riemann Spectrum

• Handout the Sage Quickreference Card
• A Crash Course in Distributions (click to create worksheet)
• Fourier Transform (click to create worksheet)
• The Spectrum of the Prime Numbers (click to create worksheet)
• Fourier Transform and the Riemann Spectrum (click to create worksheet)
• Building pi(x) from the Spectrum (click to create worksheet)

Day 5: Elliptic Curves and the Congruent Number Problem

• Handout number theory and general Sage quickrefs
• Introduction to the Congruent Number Problem (click to create worksheet)
• Congruent Numbers and Elliptic Curves (click to create worksheet)
• Deriving the bijection (click to create worksheet)
• The Birch and Swinnerton-Dyer Conjecture (click to create worksheet)

Day 6: Modular forms, elliptic curves, and the Congruent number problem

• If the TV/VCR works... watch the Fermat movie!
• Perimeter Numbers (click to create worksheet)
• BSD and the Congruent Number Problem (click to create worksheet)
• Tunnell's criterion (click to create worksheet)
• Handout: prove that 1 is not a congruent number

Day 7: Congruent Numbers

• Discussion and questions about the handout
• What are epsilon and N for the congruent number curves? (click to create worksheet)
• Heuristic idea behind the BSD Conjecture (click to create worksheet)
• Using elliptic curves to share secrets (click to create worksheet)

Day 8: Further Connections

• Elliptic curve Diffie-Hellman -- step-by-step tutorial (click to create worksheet)
• How are the number of points modulo p distributed (click to create worksheet)
• Golden and Gaussian Congruent Numbers (click to create worksheet)
• Partial results toward the BSD conjecture (click to create worksheet)
• The Generalized Riemann Hypothesis (click to create worksheet)