Dirichlet L-Functions and the Distribution of Quadratic Residues

Abstract

In Sect. 7.4 of Chap. 4, we saw how the non-vanishing at s = 1 of the L-function L(s, χ) of a non-principal Dirichlet character χ played an essential role in the proof of Dirichlet’s theorem on prime numbers in arithmetic progression (Theorem 4.5). In this chapter, the fact that L(1, χ) is not only nonzero, but positive, when χ is real and non-principal, will be of central importance. The positivity of L(1, χ) comes into play because we are interested in the following problem concerning the distribution of residues and non-residues of a prime p. Suppose that I is an interval of the real line contained in the interval from 1 to p. Are there more residues of p than non-residues in I, or are there more non-residues than residues, or is the number of residues and non-residues in I the same? We will see that this question can be answered if we can determine if certain sums of values of the Legendre symbol of p are positive, and it transpires that the positivity of the sum of these Legendre-symbol values, for certain primes p, are determined precisely by the positivity of L(1, χ) for certain Dirichlet characters χ. We make all of this precise in Sect. 7.1, where the principal theorem of this chapter, Theorem 7.1, is stated and then used to obtain some very interesting answers to our question about the distribution of residues and non-residues. In the next section, the proof of Theorem 7.1 is outlined; in particular we will see how the proof can be reduced to the verification of formulae, stated in Theorems 7.2, 7.3, and 7.4, which express the relevant Legendre-symbol sums in terms of the values of L-functions at s = 1. Sections 7.3–7.7 are devoted to the proof Theorems 7.2–7.4. In Sect. 7.3, the fact that L(1, χ) > 0 for real, non-principal Dirichlet characters is established, and Sects. 7.4–7.6 are devoted to discussing various results concerning Gauss sums, analytic functions of a complex variable, and Fourier series which are required for the arguments we take up in Sect. 7.7. Because it plays such an important role in the results of this chapter, we prove in Sect. 7.8 Dirichlet’s fundamental Lemma 4.8 on the non-vanishing of L(1, χ) for real, non-principal characters. Motivated by the result on the convergence of Fourier series that is proved in Sect. 7.6, we give yet another proof of quadratic reciprocity in Sect. 7.9 that uses finite Fourier series expansions.