Daily Assignment for Monday, March 10

Due Wednesday, March 12 on Canvas.

• Can you find another zero of the Riemann zeta function in the critical strip \( 0 \le \mathrm{Re}(z) \le 1 \) using the Desmos tool ? Is its real part 1/2 ?
• Check your answers on exercise 3 from p. 421 using the numerical integrator from class.
• Explain why the maximum modulus principle implies that any analytic function \( f : \mathbb C \to \mathbb C \) must be unbounded. This is Liouville's theorem : see §7.7.2.
• Do exercise 20 on p. 426.
• Use Liouville's theorem to prove the fundamental theorem of algebra : suppose that \( f \) is a polynomial that never takes the value \( 0 \) in \( \mathbb C \) and apply Liouville's theorem to \( 1/f \). (Hint : to write a rigorous proof, you will need to use the fact that any bounded sequence of points of \( \mathbb C \) has a convergent subsequence, or something equivalent to it.)
• Read §8.3.
• Assign yourself a grade for this assignment :
  1. At least 60 minutes.
  2. About 45 minutes.
  3. About 30 minutes.
  4. About 15 minutes.
  Submit your grade as a submission comment with your assignment. This comment must be the first comment and it must be exactly 1 character long or the grade won't be registered.