RIEMANN’S FIRST PROOF OF THE ANALYTIC CONTINUATION OF ζ(s) AND L(s, χ)

In this chapter, we will see a proof of the analytic continuation of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) via the Hurwitz zeta function. This then gives rise to a functional equation for ζ(s) and a direct computation for the value of this function at negative integer points. 1. The Hurwitz zeta function We have already seen the definition of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) as series: ζ(s) = ∞ ∑ n=1 1 ns and L(s, χ) = ∞ ∑ n=1 χ(n) ns where s = σ + it ∈ C, σ > 1, t ∈ R and χ is a Dirichlet character modulo k, k ∈ N. These notations will be used throughout this chapter. We can unify the treatment of the above two functions by introducing the Hurwitz zeta function: Definition 1.1. For σ > 1, we define ζ(s, a) = ∑∞ n=0 1 (n+a)s , 0 < a ≤ 1, the Hurwitz zeta function. Remark 1.2. (1) For a = 1, ζ(s, 1) = ζ(s). (2) L(s, χ) = k−s ∑k r=1 χ(r)ζ(s, r k ), if χ is a Dirichlet character modulo k. The second statement in the remark follows from the following calculation: Write n = qk + r, 1 ≤ r ≤ k, q = 0, 1, 2, .... We then have: L(s, χ) = ∞ ∑