Number Theory: An Introduction to Mathematics

544 XIII Connections with Number Theory

it follows that

θ 002 ( 0 )= 1 + 4

∑

n≥ 1

∑

k≥ 0

{q(^4 k+^1 )n−q(^4 k+^3 )n}

= 1 + 4

∑

m≥ 1

{d 1 (m)−d 3 (m)}qm,

whered 1 (m)and d 3 (m)are respectively the number of positive divisors ofm
congruent to 1 and 3 mod 4. Hence

r 2 (m)= 4 {d 1 (m)−d 3 (m)}. 

From Proposition 2 we immediately obtain again that any primep≡1mod4may
be represented as a sum of 2 squares and that the representation is essentially unique.
Proposition II.39 may also be rederived.
The numberrs(m)of representations of a positive integermas a sum ofssquares
has been expressed by explicit formulas for many other values ofsbesides 2 and 4.
Systematic ways of attacking the problem are provided by the theory of modular forms
and the circle method of Hardy, Ramanujan and Littlewood.

2 Partitions..................................................

Apartitionof a positive integernis a set of positive integers with sumn. For example,
{ 2 , 1 , 1 }is a partition of 4. We denote the number of distinct partitions ofnbyp(n).
For example,p( 4 )=5, since all partitions of 4 are given by

{ 4 },{ 3 , 1 },{ 2 , 2 },{ 2 , 1 , 1 },{ 1 , 1 , 1 , 1 }.

It was shown by Euler (1748) that the sequencep(n)has a simplegenerating
function:

Proposition 3If|x|< 1 ,then

1 /( 1 −x)( 1 −x^2 )( 1 −x^3 )···= 1 +

∑

n≥ 1

p(n)xn.

Proof If|x|<1, then the infinite product

∏

m≥ 1 (^1 −x

m)converges and its recip-

rocal has a convergent power series expansion. To determine the coefficients of this
expansion note that, since

( 1 −xm)−^1 =

∑

k≥ 0

xkm,

the coefficient ofxn(n≥ 1 )in the product

∏

m≥ 1 (^1 −x

m)− (^1) is the number of repre-
sentations ofnin the form
n= 1 k 1 + 2 k 2 +···,
where thekjare non-negative integers. But this number is preciselyp(n), since any
partition is determined by the number of 1’s, 2’s,...that it contains.