Lecture 27

Question 1

If σ > 1, what is the lower bound of
ζ^3(σ)∣ζ(σ + it)∣4∣ζ(σ + 2it)∣

Answer 1

ζ^3(σ)∣ζ(σ + it)∣4∣ζ(σ + 2it)∣ ≥ 1

Question 2

We have ζ(1 + it) ≠ 0 for what t

Answer 2

for all real t

Question 3

Inequalities for ∣1/ζ(s)∣ and ∣ζ′(s)/ζ(s)∣

Answer 3

There is a constant M > 0 such that ∣1/ζ(s)∣ < M log^7t and ∣ζ′(s)/ζ(s)∣ < M log^9t whenever σ ≥ 1 and t ≥ e