12.3. Dedekind function The Dedekind function is defined by the infinite product (q = e2iπτ )

η(τ ) = q1/24

∞



n=1

(1−qn).

a. Use the definition of the θi(τ ) functions in terms of the infinite product to prove the identity

η3(τ ) = 1 2

θ2(τ ) θ3(τ ) θ4(τ ).

b. Use the modular transformations of θi(τ ) to prove

η(τ +1) = eiπ/12 η(τ )

η(−1/τ ) =

√

−iτ η(τ).