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
Previous Page Next Page