Lecture 28

Question 1

If f(s) has a pole of order k at s = α, then the quotient f′(s)/f(s) has apole, state where, the residue and the prder

Answer 1

first order
pole at s = α with residue −k

Question 2

The function F(s) = −ζ′(s)/ζ(s) − 1/(s − 1) is analytic where

Answer 2

at s=1

Question 3

For x ≥ 1, we have ψ1(x)/x^2 − 1/2 (1 −1/x)^2 = (1/2π) ∫^∞_(−∞) h(1 + it)e^(itlog x) dt

where the integral

∫^∞^(−∞) ∣h(1 + it)∣dt converges

What does this imply

Answer 3

Therefore, by the Riemann-Lebesgue Lemma, we have

• ψ1(x) ∼ 1/2 x^2

and hence

• ψ(x) ∼ x as x → ∞.

Question 4

Assume σ ≥ 1/2 . Then, there exists constants A > 0 and C > 0 such that ∣ζ(σ + it)∣ >Clog^7t

When does this hold and what does it imply

Answer 4

whenever 1 −Alog^9t< σ ≤ 1 and t ≥ e (5)

• This implies that ζ(σ + it) ≠ 0 if σ and t satisfy (5).