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Appendix a Zeta Functions in Number Theory

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Appendix a Zeta Functions in Number Theory Appendix A Zeta Functions in Number Theory In this appendix we collect some basic facts about zeta functions in number theory to which our theory of explicit formulas can be applied. We refer to [Lan, Chapters VIII and XII–XV], [ParSh1, Chapter 4] and [ParSh2, Chapter 1, §6, Chapter 2, §1.13] for more complete information and proofs. A.1 The Dedekind Zeta Function Let K be an algebraic number field of degree d over Q, and let O be the ring of integers of K. The norm of an ideal a of O is defined as the number of elements of the ring O/a: Na =#O/a. (A.1) The norm is multiplicative: N(ab)=Na · Nb. Furthermore, an ideal has a unique factorization into prime ideals, a pa1 pak = 1 ... k , (A.2) where p1,...,pk are prime ideals. We denote by r1 the number of real embeddings K → R and by r2 the number of pairs of complex conjugate embeddings K → C.Thuswe have that r1 +2r2 = d. Further, w stands for the number of roots of unity contained in K. Associated with K, we need the discriminant disc(K), the class number h, and the regulator R. We refer to [Lan, p. 64 and p. 109] for the definition of these notions. 388 Appendix A. Zeta Functions in Number Theory The Dedekind zeta function of K is defined for Re s>1by ∞ −s −s ζK (s)= (Na) = Ann . (A.3) a n=1 Here, a runs over the ideals of O, and, in the second expression, An denotes the number of ideals of O of norm n. This function has a meromorphic con- tinuation to the whole complex plane, with a unique (simple) pole at s =1, with residue 2r1 (2π)r2 hR . (A.4) w disc(K) (See [Lan, Theorem 5 and Corollary, p. 161].) From unique factorization and the multiplicativity of the norm, we de- duce the Euler product for ζK (s): for Re s>1, 1 ζK (s)= −s , (A.5) p 1 − (Np) where p runs over the prime ideals of O. Example A.1 (The case K = Q). Ideals of Z are generated by positive integers, and the norm of the ideal nZ is n, for n ≥ 1. Hence, ∞ −s ζQ(s)= n = ζ(s), (A.6) n=1 so that the Dedekind zeta function of Q is the Riemann zeta function. The Euler product for ζQ(s) is given by 1 ζQ(s)= , (A.7) 1 − p−s p where p runs over the rational prime numbers. A.2 Characters and Hecke L-series Let χ be an ideal-character of O, belonging to the cycle c (see [Lan, Chap- ter VIII, §3 and Chapter VI, §1] for a complete explanation of the terms). The L-series associated with χ is defined by −s Lc(s, χ)= χ(a)(Na) . (A.8) (a,c)=1 A.3 Completion of L-Series, Functional Equation 389 By the multiplicativity of the norm, this function has an Euler product 1 Lc(s, χ)= . (A.9) − p p −s p | c 1 χ( )(N ) This zeta function can be completed with factors corresponding to the divisors of c, to obtain a function L(s, χ), called the Hecke L-series assoc- iated with χ. It is related to the Dedekind zeta function as follows: Let L be the class field associated with an ideal class group of O, and let ζL be the Dedekind zeta function of L. Let χ run over the characters of the ideal class group. Then, ζL(s)= L(s, χ). (A.10) χ Example A.2 (The case K = Q). A multiplicative function χ on the pos- itive integers gives rise to a Dirichlet L-series ∞ L(s, χ)= χ(n)n−s. (A.11) n=1 The value χ(n) only depends on the class of n modulo a certain positive integer c. The minimal such c is called the conductor of χ. A.3 Completion of L-Series, Functional Equation The fundamental property of the Dedekind zeta function and of the L-series 1 is that it can be completed to a function that is symmetric about s = 2 . Let Γ(s) be the gamma function. Let −s/2 −s ζR(s)=π Γ(s/2) and ζC(s)=(2π) Γ(s) . Denote by disc(K) the discriminant of K. Then the function s/2 r1 r2 ξK (s) = disc(K) ζR(s) ζC(s) ζK (s) (A.12) has a meromorphic continuation to the whole complex plane, with simple poles located only at s = 1 and s = 0, and it satisfies the functional equation ξK (1 − s)=ξK (s). (A.13) We deduce from this key fact that the function log |ζK (σ + it)| ψK (σ) := lim sup , (A.14) t→∞ log t 390 Appendix A. Zeta Functions in Number Theory defined for σ ∈ R, is given by the following simple formula for σ/∈ (0, 1): ⎧ ⎨0, for σ ≥ 1, ψK (σ)= d ⎩ (1 − 2σ) , for σ ≤ 0. 2 It is known that ψK is convex on the real line. (The Lindel¨of hypothesis says that ψK (1/2) = 0.) We deduce the following property (see [Lan, Chap- ter XIII, §5]): For every real number σ and ε>0, there exists a constant C, depending on σ and ε, such that for all real numbers t with |t| > 1, ψK (σ)+ε |ζK (σ + it)|≤C|t| . (A.15) Thus ζK (s) satisfies the hypotheses L1 and L2 of Chapter 5. (See also Remark A.5 below.) The formalism required to prove for general L-series the functional equa- tion (A.13) and the estimate (A.15) about the growth along vertical lines was developed in Tate’s thesis [Ta]. We refer to [Lan, Chapter XIV] for the corresponding results. Remark A.3. Our theory also applies to the more general L-series as- sociated with nonabelian representations of Q, such as those considered in [RudSar]. (The abelian case corresponds to the Hecke L-series discussed above.) These L-series can also be completed at infinity, and they sat- isfy a functional equation, relating the zeta function associated to a given representation with the zeta function associated to the contragredient rep- resentation. Moreover, they have an Euler product representation, much like that of L(s, χ), except that the p-th Euler factor may be a polynomial in p−s of degree larger than one. We note that these zeta functions are called primitive L-series in [Rud- Sar]. According to the Langlands Conjectures, they are the building blocks of the most general L-series occurring in number theory. See, for exam- ple, [Gel, KatSar, RudSar]. A.4 Epstein Zeta Functions A natural generalization of the Riemann zeta function is provided by the Epstein zeta functions [Ep].1 Let q = q(x) be a positive definite quadratic form of x ∈ Rd, with d ≥ 1. Then the associated Epstein zeta function is 1See, for example, [Ter, Section 1.4] for detailed information about these functions. However, we use the convention of [Lap2, §4], which is different from the traditional one used in [Ter]. A.5 Two-Variable Zeta Functions 391 defined by −s/2 ζq(s)= q(n) . (A.16) n∈Zd\{0} It can be shown that ζq(s) has a meromorphic continuation to all of C, − with a simple pole at s = d having residue πd/2(det q)−1/2Γ(d/2) 1 . Fur- ther, ζq(s) satisfies a functional equation analogous to that satisfied by ζ(s); namely, for the completed zeta function −s/2 ξq(s)=π Γ(s/2) ζq(s), we have −1/2 ξq(s) = (det q) ξq−1 (d − s), (A.17) where q−1 is the positive definite quadratic form associated with the inverse of the matrix of q. (See [Ter, Theorem 1, p. 59].) 2 ··· 2 In the important special case when q(x)=x1+ +xd, the corresponding Epstein zeta function is denoted ζd(s) in Section 1.4, Equation (1.47), and can be viewed as a natural higher-dimensional analogue of the Riemann zeta function ζ(s). Indeed, we have ζ1(s)=2ζ(s). (See, for example, [Lap2, §4] and the end of Section 1.4 above.) Since in this case, q is associated with the d-dimensional identity matrix, we have q−1 = q and det q =1, so that (A.17) takes the form of a true functional equation relating ζd(s) and ζd(d − s). Moreover, ζd(s) has an Euler product like that of ζ(s)ifand only if d = 1, 2, 4 or 8, corresponding to the real numbers, the complex numbers, the quaternions and the octaves, respectively. (See, e.g., [Bae].) Remark A.4. More generally, one can also consider the Epstein-like zeta functions considered in [Es1]. Such Dirichlet series are associated with suit- able homogeneous polynomials of degree greater than or equal to one. Remark A.5. For all the zeta functions in Sections A.1–A.4, hypothe- ses L1 and L2 (Equations (5.19) and (5.21), page 143) are satisfied with a window W equal to all of C. Moreover, for the example from Section A.6 below, one can take W to be a right half-plane of the form Re s ≥ σ0, for a suitable σ0 > 0. A.5 Two-Variable Zeta Functions Let C be an algebraic curve over a finite field, as in Section 11.5. In that section, we introduced the zeta function of C, ζC (s).
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