Analytic continuation, functional equation: examples

1. L(s, χ) for even Dirichlet characters 2. L(s, χ) for odd Dirichlet characters 3. Dedekind zeta function ζ o (s) for Gaussian integers o 4. Grossencharacter L-functions L-functions for Gaussian integers We try to imitate the argument used by Riemann for proving the analytic continuation of ζ(s) and its functional equation π −s/2 Γ(s 2) ζ(s) = π −(1−s)/2 Γ(1−s 2) ζ(1 − s) from the integral representation π −s/2 Γ(s 2) ζ(s) = ∞ 0 y −s/2 θ(iy) − 1 2 dy y in terms of the basic theta series [1] θ(iy) = n∈Z e −πn 2 y whose functional equation θ(iy) = 1 √ y · θ i y is proven by Poisson summation. The discussion of L-functions L(s, χ) for Dirichlet characters (over Q) bifurcates into two families, depending upon the parity of χ, that is, whether χ(−1) = +1 or χ(−1) = −1. Analogous discussion of the zeta function of the Gaussian integers extends this discussion in a new direction. Treatment of the grossencharacter L-functions [2] for the Gaussian integers have no counterpart among Dirichlet L-functions for Z. L-functions of ideal class group characters deserve parallel treatment, but would require more background, concerning rings of algebraic integers that fail to be principal ideal domains, and other background. This background is important, but is not the immediate point, so we defer this part of the discussion. These examples are simple, but the book-keeping quickly becomes fragile. This should motivate receptiveness to the more abstract, but very clear, Iwasawa-Tate viewpoint, which we consider soon. Let χ be a non-trivial Dirichlet character mod N > 1, with L-function L(s, χ) = n≥1 χ(n) n s [1] The reason that the argument is given as iy rather than y is that iy can be replaced by any z in the complex upper half-plane H. This aspect is important later. [2] The grossencharacters and their L-functions are also called Hecke characters and L-functions, as Hecke first studied them, about 1920.