3. The Eisenstein Series and the Discriminant Function

Question 1

Define the Eisenstein series of weight k

Answer 1

The modular form:

G_k (τ) := SUM’ n,m 1/(mτ + n)^k

Question 2

State the power/divisor function σ_k-1

Answer 2

σ_k-1 (n) = SUM d|n d>0 d^k-1

Question 3

Relate the Riemann-zeta function and the Bernoulli number

Answer 3

ζ(k) = -(2πi)^k B_k/ 2(k!)

Question 4

Define the normalised Eisenstein series of weight k

Answer 4

E_k(τ) = 1 - 2k/B_k SUM n=1,∞ σ_k-1(n) q^n

Question 5

Where does E2 fail as a modular form? How do we deal with it?

Answer 5

Not absolutely convergent; have to fix order of summation which prevents the modularity condition.

Defines a holomorphic function on H, not modular.

Question 6

Give how E2(-1/τ) transforms

Answer 6

τ^-2 E2(-1/τ) = E2(τ) + 12/2πiτ

Question 7

Define the Dedekind-eta function η(τ)

Answer 7

η(τ) = q^1/24 PROD n=1,∞ (1-q^n)

Question 8

How does the Dedekind-eta function relate to E2(τ)

Answer 8

∂/∂τ log(η(τ)) = πi/12 E2(τ)

Question 9

Give how η(-1/τ) transforms

Answer 9

sqrt(-iτ) η(τ)

Question 10

Define the discriminant function Δ(τ)

Answer 10

Δ(τ) = η(τ)^24

Question 11

What is special about Δ(τ)?

Answer 11

It is a cusp form of order 12: Δ(τ) != 0 for all τ ∈ H

Question 12

Define p(n). How does this relate to q?

Answer 12

The number of partitions of the the positive integer

SUM n=0,∞ p(n)q^n = PROD k=1,∞ 1/(1-q^n)

Question 13

Define the Ramanujan τ-function; how does this relate to τ?

Answer 13

Δ(τ) = SUM n = 1, τ(n)q^n