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

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