Number Theory: An Introduction to Mathematics

4 Theta Functions 519

Moreover, there exists a constantA>0, depending onqbut not onnorN, such that

|cnN|≤A|q|n

2

.

For we can chooseB>0sothat|

∏m

k= 1 (^1 −q

2 k)|≥Bfor allm, we can choose

C>0sothat|

∏m

k= 1 (^1 −q

2 k)|≤Cfor allm, and we can then takeA=C/B (^2) .Since
the series

∑∞

n=−∞q

n^2 znis absolutely convergent, it follows that we can proceed to

the limit term by term in (31) to obtain (30).

In the series

∑∞

n=−∞q

n^2 znwe now put

q=eπiτ, z=e^2 πiv,

so that|q|<1 corresponds toIτ>0, and we define thetheta function

θ(v;τ)=

∑∞

n=−∞

eπiτn

2

e^2 πivn.

The functionθ(v;τ)is holomorphic invandτfor allv∈Candτ∈H (the upper

half-plane). Since initially we will be more interested in the dependence onv, withτ
just a parameter, we will often writeθ(v)in place ofθ(v;τ). Furthermore, we will still
useqas an abbreviation foreπiτ.
Evidently

θ(v+ 1 )=θ(v)=θ(−v).

Moreover,

θ(v+τ)=

∑∞

n=−∞

qn

(^2) + 2 n
e^2 πivn
=q−^1 e−^2 πiv

∑∞

n=−∞

q(n+^1 )

2

e^2 πiv(n+^1 )

=e−πi(^2 v+τ)θ(v).

It may be immediately verified that

∂^2 θ/∂v^2 =− 4 π^2 q∂θ/∂q= 4 πi∂θ/∂τ,

which becomes the partial differential equation of heat conduction in one dimension
on puttingτ= 4 πit.
By Proposition 2, we have also the product representation

θ(v)=

∏∞

n= 1

( 1 +q^2 n−^1 e^2 πiv)( 1 +q^2 n−^1 e−^2 πiv)( 1 −q^2 n).

It follows that the points

v= 1 / 2 +τ/ 2 +m+nτ(m,n∈Z)

are simple zeros ofθ(v), and that these are the only zeros.