Lecture 23

Question 1

Riemann Zeta function

Answer 1

For every s ∈ C with σ > 1, the Riemann-zeta function ζ(s) is defined as

• ζ(s) ∶= ∑_(n=1) 1/n^s

Question 2

Euler product representation of Riemann zeta function for σ > 1

Answer 2

ζ(s) = ∏_(p) (1 −1/p^s)^(−1)

Question 3

Suppose that σ > 0, x > 0 and s ≠ 1. Then for N an integer, give ζ(s)

Answer 3

• ζ(s) = ∑_(n≤N) 1/n^s + N^(1−s)/(s − 1) − s∫^∞_N {t}/t^(s+1) dt

Question 4

Where is the pole of the riemann zeta function in the half plane of σ > 0.

Answer 4

s = 1 with residue 1

Question 5

For σ > 0, give the integral representation of the gamma function

Answer 5

Γ(s) = ∫^∞_0 x^(s−1) e^(-x) dx.

Question 6

The Functional Equation for the Riemann-zeta function

Answer 6

For all s we have

• ζ(s) = 2(2π)^(s−1)Γ(1 − s) sin (πs/2) ζ(1 − s).

Question 7

Riemann Hypothesis

Answer 7

If 0 < R(s) < 1 and ζ(s) = 0, then R(s) =1/2