7. L-functions

Question 1

Define the Riemann-zeta function ζ(s)

Answer 1

ζ(s) = SUM 1,∞ n^-s

Question 2

Where does ζ(s) converge absolutely and uniformly? What does this imply?

Answer 2

Vertical strips of Re(s) >1

Defines a holomorphic function in this region.

Question 3

Define the Euler-product of ζ(s)

Answer 3

PROD_p 1/1-p^-s

Question 4

Define the completed zeta function Z(s)

Answer 4

Z(s) = π^-s/2 Γ(s/2) ζ(s)

Question 5

Define the gamma function Γ(s)

Answer 5

Γ(s) = INT_0,∞ e^-t t^s dt/t

Question 6

Describe the meromorphic continuation of Z(s)

Answer 6

Simple poles at s = 0,1 with residues -1,1

Question 7

Describe the meromorphic continuation of Γ(s)

Answer 7

Simple poles at s = -n for n=0,1,2,…. with residues (-1)^n/n!

Question 8

Describe the meromorphic continuation of ζ(s)

Answer 8

One simple pole at s=1 with residue 1

Question 9

State the functional equation which Z(s) satsifies

Answer 9

Z(1-s) = Z(s)

Question 10

Define the Hecke L-function

Answer 10

For some function f(τ) = SUM_0,∞ a_nq^n which is a modular form of weight k, we have

L(f,s) = SUM_1,∞ a_n n^-s

Question 11

Where is the Hecke L-function defined?

Answer 11

Converging abs and unif for Re(s) > k

If f is cusp form converge for Re(s) > k/2 + 1

Question 12

Define the completed L-function Λ(f,s)

Answer 12

Λ(f,s) = (2π)^-s Γ(s) L(f,s)

Question 13

Given f∈M_k, define F(t) := f(it). Give the transformation property of F(1/t)

Answer 13

F(1/t) = (it)^kF(t)

Question 14

Give the Mellin form of the completed L-function Λ(f,s)

Answer 14

= INT_0,∞ (F(t) - a_0)t^s dt/t