Course outline
Module 1
Module 1: Introduction to the Riemann Hypothesis
Introduction to the Riemann Hypothesis
Knowledge check
Q1. Who proposed the Riemann Hypothesis?
- A. Bernhard Riemann ✓
- B. Leonhard Euler
- C. Carl Friedrich Gauss
- D. Johann Peter Gustav Lejeune Dirichlet
Why: The Riemann Hypothesis was proposed by Bernhard Riemann in 1859.
Q2. What does the Riemann Hypothesis concern?
- A. The distribution of prime numbers ✓
- B. The solution to the Navier-Stokes equations
- C. The properties of elliptic curves
- D. The behavior of quantum particles
Why: The Riemann Hypothesis concerns the distribution of prime numbers, specifically the zeros of the Riemann zeta function.
Q3. Where is the Riemann zeta function defined?
- A. On the real line
- B. On the complex plane ✓
- C. On the unit circle
- D. On the sphere
Why: The Riemann zeta function is defined on the complex plane, and the Riemann Hypothesis makes a claim about its zeros.
Q4. What is the critical line in the context of the Riemann Hypothesis?
- A. The line Re(s) = 0
- B. The line Re(s) = 1
- C. The line Re(s) = 1/2 ✓
- D. The line Im(s) = 0
Why: The critical line in the context of the Riemann Hypothesis is the line Re(s) = 1/2, on which the hypothesis claims all non-trivial zeros of the zeta function lie.
Q5. What is the significance of the Riemann Hypothesis?
- A. It is a proven theorem
- B. It is one of the most important unsolved problems in mathematics ✓
- C. It has no mathematical significance
- D. It is only relevant in number theory
Why: The Riemann Hypothesis is one of the most important unsolved problems in mathematics, with profound implications for number theory and other areas.
Module 2
Module 2: Complex Analysis and the Zeta Function
Complex Analysis and the Zeta Function
Knowledge check
Q1. What is the domain of the Riemann zeta function?
- A. All real numbers
- B. All complex numbers except 1 ✓
- C. Only positive integers
- D. Only negative integers
Why: The Riemann zeta function is defined for all complex numbers except 1, where it has a simple pole.
Q2. What is the value of the Riemann zeta function at s = 2?
Why: The value of the Riemann zeta function at s = 2 is known to be π²/6.
Q3. Which of the following is a property of analytic functions?
- A. They are not differentiable
- B. They are differentiable everywhere in their domain ✓
- C. They are only defined on the real line
- D. They have discontinuities everywhere
Why: Analytic functions are differentiable everywhere in their domain of definition.
Q4. What is the residue of the Riemann zeta function at s = 1?
Why: The Riemann zeta function has a simple pole at s = 1, so its residue there is infinite.
Q5. Which of the following is the correct definition of the Riemann zeta function for Re(s) > 1?
- A. ζ(s) = ∑_{n=1}^∞ n^s
- B. ζ(s) = ∑_{n=1}^∞ 1/n^s ✓
- C. ζ(s) = ∏_{p prime} (1 - 1/p^s)^{-1}
- D. ζ(s) = ∫_0^∞ x^s / e^x - 1 dx
Why: For Re(s) > 1, the Riemann zeta function is defined as the sum of 1/n^s over all positive integers n.
Module 3
Module 3: History and Motivation
History and Motivation
Knowledge check
Q1. Who proposed the Riemann hypothesis?
- A. Leonhard Euler
- B. Bernhard Riemann ✓
- C. Carl Friedrich Gauss
- D. Gottfried Wilhelm Leibniz
Why: Bernhard Riemann proposed the hypothesis in 1859.
Q2. What does the Riemann hypothesis concern?
- A. Prime numbers
- B. The distribution of prime numbers ✓
- C. The sum of reciprocals of primes
- D. The behavior of the zeta function
Why: The Riemann hypothesis concerns the distribution of prime numbers.
Q3. What is the Riemann zeta function defined as?
- A. ζ(s) = ∑ 1/n^s for n=1 to ∞
- B. ζ(s) = ∑ n^s for n=1 to ∞
- C. ζ(s) = ∏ (1 - 1/p^s) for primes p
- D. Both A and C ✓
Why: The Riemann zeta function is defined as both a Dirichlet series and an Euler product.
Q4. Where is the Riemann hypothesis stated?
- A. In a paper on number theory
- B. In Riemann's 1859 paper on the distribution of prime numbers ✓
- C. In Euler's work on infinite series
- D. In Gauss's unpublished notes
Why: The Riemann hypothesis was stated in Riemann's 1859 paper on the distribution of prime numbers.
Q5. What is the significance of the Riemann hypothesis?
- A. It is a conjecture with no practical applications
- B. It is one of the most important unsolved problems in mathematics ✓
- C. It has been proven and is not a conjecture
- D. It is only relevant to theoretical physics
Why: The Riemann hypothesis is one of the most important unsolved problems in mathematics.
Module 4
Module 4: Zeros of the Zeta Function
Zeros of the Zeta Function
Module 5
Module 5: Connections to Prime Numbers
Connections to Prime Numbers
Module 6
Module 6: Numerical Evidence and Computations
Numerical Evidence and Computations
Module 7
Module 7: Modern Approaches and Open Problems
Modern Approaches and Open Problems