Lecture 26

Question 1

Give A Contour Integral Representation for ψ1(x)/x^2

Answer 1

If c > 1 and x ≥ 1 we have

• ψ1(x)/x^2 = (1/2πi) ∫^(c+i∞)_(c−i∞) x^(s−1)/(s(s + 1)) (−ζ′(s)/ζ(s)) ds.

Question 2

If c > 1 and x ≥ 1, find

• ψ1(x)/x^2 − (1/2)(1 −1/x)^2

Answer 2

• (1/2πi) ∫^(c+i∞)_(c−i∞) x^(s−1) h(s)ds

where,
* h(s) = 1/(s(s + 1)) (−ζ′(s)/ζ(s) −1/(s − 1))

Question 3

Upper bounds for ∣ζ(s)∣ and ∣ζ
′(s)∣ near the line σ = 1

Answer 3

For every A > 0, there exists a constant M (depending on A) such that

• ∣ζ(s)∣ ≤ M log t and ∣ζ′(s)∣ ≤ M log^2(t)

for all s with σ ≥ 1/2 satisfying

• σ > 1 −A/log t and t ≥ e