14 1. Introduction

He showed that (a) for every non-trivial Dirichlet character χ modulo

N , we have L(1,χ) = 0 or ∞, and (b) this implies that Ψa,N (1)

diverges to ∞. Part (b) follows from the equality

log(ζ(s)) +

χ mod N

χ=1

χ(a)−1

log(L(s, χ))

= φ(N)

⎛

⎝

p≡a mod N

1

ps

⎞

⎠

+ g(s),

where g(s) is a function with g(1) finite, and φ is the Euler φ-function.

Therefore, there cannot be a finite number of primes of the form

p ≡ a mod N .

Example 1.3.3 (Representations of integers as sums of squares). Is

the number n 0 a sum of three integer squares? In Subsection 1.2,

we saw formulas for the number of representations of an integer as a

sum of k = 2, 4, 6 and 8 integer squares, but we avoided the same

question for odd k. The known formulas for S3(n), S5(n) and S7(n)

involve values of Dirichlet L-functions.

Let us first define the Dirichlet character that we shall use here.

The reader should be familiar with the Legendre symbol

n

p

, which

is equal to 0 if p|n, equal to 1 if n is a square mod p, and equal to

−1 if n is not a square mod p. Let m 0 be a natural number

with prime factorization m =

i

pi (the primes are not necessarily

distinct). First we define

n

2

=

⎧

⎪

⎪

⎨0

⎪

⎪

⎩−1

if n is even,

1 if n ≡ ±1 mod 8,

if n ≡ ±3 mod 8.

Now we are ready to define the Kronecker symbol of n over m 0 by

n

m

=

i

n

pi

.

For any n 0, the symbol

(

−n

)

induces a Dirichlet character χn

defined by χn(a) =

(

−n

a

)

, and we can define the associated L-function