The Critical Values of Generalizations

of the Hurwitz Zeta Function

Goro Shimura

Received: November 10, 2009

Communicated by Don Blasius

Abstract. We investigate a few types of generalizations of the Hurwitz zeta function, written Z(s, a) in this abstract, where s is a complex variable and a is a parameter in the domain that depends on the type. In the easiest case we take a ∈ R, and one of our main results is that Z(−m, a) is a constant times E_{m}(a) for 0 ≤ m ∈ Z, where E_{m} is the generalized Euler polynomial of degree n. In another case, a is a positive definite real symmetric matrix of size n, and Z(−m, a) for 0 ≤ m ∈ Z is a polynomial function of the entries of a of degree ≤ mn. We will also define Z with a totally real number field as the base field, and will show that Z(−m, a) ∈ Q in a typical case.

2010 Mathematics Subject Classification: 11B68, 11M06, 30B50, 33E05.

Introduction

This paper is divided into four parts. In the first part we consider a generalization of Hurwitz zeta function given by

(0.1) ζ(s; a, γ) = ∑_{n=0}^{∞} γ^{n}(n + a)^{−s},

where s ∈ C, 0 < a ∈ R, and γ ∈ C, 0 < |γ| ≤ 1. Clearly the infinite series is convergent for Re(s) > 1. For γ = 1 this becomes ∑_{n=0}^{∞}(n+a)^{−s}, which is the classical Hurwitz zeta function usually denoted by ζ(s, a). This generalization is not new. It was considered by Lerch in [Le], a work five years after the paper [Hu] of Hurwitz in 1882. Its analytic properties can be summarized as follows.

Theorem 0.1. For a and γ as above the product (e^{2πis}−1)Γ (s)ζ(s; a, γ) can be continued to an entire function in s. In addition, there exists a holomorphic function in (s, a, γ)∈C^{3}, defined for Re(a)>0 and γ ∉ {x∈R | x≥1} with no condition on s, that coincides with the product when Re(s) > 1, 0 < a ∈ R, and 0 < |γ| ≤ 1.

The proof will be given in §1.1. To state a more interesting fact, we first put

(0.2)  e(z) = exp(2πiz)  (z ∈ C),

and define a function E_{c,n}(t) in t for c ∈ C and 0 < n ∈ Z, that is called the nth generalized Euler polynomial, by

(0.3) \frac{(1 + c)e^{tz}}{e^{z} + c} = ∑_{n=0}^{∞} \frac{E_{c,n}(t)}{n!}z^{n}.

We assume c = −e(α) with α ∈ R, ∉ Z. The function E_{c,n}(t) was introduced in [S07]. If c = 1, E_{1,n}(t) is the classical Euler polynomial of degree n. In [S07] we showed that E_{c,n}(t) is a polynomial in t of degree n; it is also a polynomial in (1 + c)^{−1}. Its properties are listed in [S07, pp. 25–26]. We mention here only (0.3a) E_{c,n}(1 − t) = (−1)^{n}E_{c^{−1},n}(t),

(see [S07, (4.3f)]), which will become necessary later. Now we have

Theorem 0.2. For 0 < k ∈ Z, Re(a) > 0, and γ ∉ {x ∈ R | x ≥ 1} the value ζ(1 − k, a; γ) is a polynomial function of a and (γ − 1)^{−1}. More precisely, we have

(0.4) ζ(1 − k; a, γ) = E_{c,k−1}(a)/(1 + c^{−1})

for such k, a, and γ, where c = −γ^{−1}.

This will be proven in §1.2.

As for the original Hurwitz function, there is a well known relation

(0.5) ζ(1 − k, a) = −B_{k}(a)/k for 0 < k ∈ Z,

where B_{k} is the k-th Bernoulli polynomial. This is essentially due to Hurwitz; see [Hu, p. 92]; cf. also [E, p.27, (11)] and [WW, p. 267, 13.14].

In [S07] and [S08] we investigated the critical values of the L-function L(s, χ) with a Dirichlet character χ, and proved especially (see [S07, Theorem 4.14] and [S08, Theorem 1.4])

Theorem 0.3. Let χ be a nontrivial primitive Dirichlet character modulo a positive integer d, and let k be a positive integer such that χ(−1) = (−1)^{k}.

(i) If d = 2q + 1 with 0 < q ∈ Z, then

(0.6) L(1 − k, χ) = \frac{d^{k−1}}{2^{k}χ(2) − 1} ∑_{b=1}^{q}(−1)^{b}χ(b)E_{1,k−1}(b/d).

(ii) If d = 4d_{0} with 1 < d_{0} ∈ Z, then

(0.7) L(1 − k, χ) = (2d_{0})^{k−1} ∑_{a=1}^{d_{0}−1} χ(a)E_{1,k−1}(2a/d).

In §1.4 we will give a shorter proof for these formulas by means of (0.4), and in Section 2 we will prove a functional equation for ζ(s; a, γ) by producing an expression for ζ(1 − s; a, γ).

The second part of the paper concerns the analogue of (0.1) defined when the base field is a totally real algebraic number field F. If F ≠ Q, there are nontrivial units, which cause considerable difficulties, and for this reason we cannot give a full generalization of which (0.1) is a special case. However, taking such an F as the base field, we will present a function of a complex variable s and two parameters a and p in F, that includes as a special case at least ζ(s; a, γ) with a ∈ Q and γ a root of unity. We then prove in Theorem 3.4 a rationality result on its critical values.

The third part is a kind of interlude. Observing that the formula for L(k, χ) (not L(1 − k, χ)) involves the Gauss sum G(χ) of χ, we will give in Section 4 a formula for G(χλ)/G(χ) for certain Dirichlet characters χ and λ.

The fourth and final part of the paper, which has a potential of future development, concerns the analogue of (0.1) defined for a complex variable s, with nonnegative and positive definite symmetric matrices of size n in place of n and a. We will show in Section 5 that it is an entire function of s and also that its value at s = −m for 0 ≤ m ∈ Z is a polynomial function of the variable symmetric matrix of degree ≤ mn.

1. Proof of Theorems 0.1, 0.2, and 0.3

1.1. To prove Theorem 0.1, assuming that 0 < a ∈ R and 0 < |γ| ≤ 1, we start from an easy equality Γ (s)(n + a)^{−s} = ∫_{0}^{∞} x^{s−1}e^{−(n+a)x}dx. Therefore

Γ (s)ζ(s; a, γ) = ∑_{n=0}^{∞} Γ (s)γ^{n}(n + a)^{−s} = ∑_{n=0}^{∞} ∫_{0}^{∞} x^{s−1}γ^{n}e^{−(n+a)x}dx

= ∫_{0}^{∞} ∑_{n=0}^{∞} x^{s−1}γ^{n}e^{−(n+a)x}dx = ∫_{0}^{∞} \frac{x^{s−1}e^{−ax}}{1 − γe^{−x}} dx.

Our calculation is justified for σ = Re(s) > 1, since

∑_{n=0}^{∞} ∫_{0}^{∞} |x^{s−1}γ^{n}e^{−(n+a)x}|dx ≤ ∑_{n=0}^{∞} Γ (σ)(n + a)^{−σ} < ∞.

Thus we obtain

(1.1) Γ (s)ζ(s; a, γ) = ∫_{0}^{∞} \frac{x^{s−1}e^{x(1−a)}}{e^{x} − γ} dx for Re(s) > 1.

We now consider

∫_{∞}^{0+} \frac{z^{s−1}e^{z(1−a)}}{e^{z} − γ} dz

with the standard symbol ∫_{∞}^{0+} of contour integration. The integral is the sum of three integrals: ∫_{∞}^{δ}, ∮ on the circle |z| = δ, and ∫_{δ}^{∞}, where 0 < δ ∈ R; we naturally take the limit as δ tends to 0. We take z^{s−1} = exp ((s − 1) log z) for the first integral ∫_{∞}^{δ} with log z ∈ R for 0 < z ∈ R; for the evaluation of the other integrals we continue z^{s−1} analytically without passing through the positive real axis. Then the first and third integrals produce

(e(s) − 1) ∫_{δ}^{∞} \frac{x^{s−1}e^{x(1−a)}}{e^{x} − γ} dx,

which is meaningful for every s ∈ C and every (a, γ) ∈ C^{2} such that (1.1a) Re(a) > 0 and γ ∉ {x ∈ R | x > 1}.

As for ∮ , we first observe that given γ ∈ C, we can find a small δ_{0} ∈ R, > 0, such that e^{z} ≠ γ for 0 < |z| ≤ δ_{0}, since the map z ↦ w = e^{z} sends the punctured disc 0 < |z| ≤ δ_{0} into a punctured disc 0 < |w − 1| < ε that does not contain γ. (This is clearly so even for γ = 1.) Therefore ∮ is meaningful for every s ∈ C and sufficiently small δ in both cases γ ≠ 1 and γ = 1; the integral is independent of δ because of Cauchy’s theorem. Now put z = δe^{iθ} with δ such that 0<δ < δ_{0} and 0 ≤ θ < 2π. Then z^{s−1} = exp{(s−1)(log δ +iθ)}, and so for s = σ +iτ with real σ and τ, we have |z^{s−1}| = δ^{σ−1}|e^{−θτ} | ≤ δ^{σ−1}e^{2π|τ|}. If γ ≠ 1, we see that Min_{|z|≤δ_{0}}|e^{z} −γ| > 0, and so |e^{z(1−a)}/(e^{z} −γ)| is bounded for |z| ≤ δ_{0}. If γ = 1, the function e^{z(1−a)}/(e^{z} − γ) is 1/z plus a holomorphic function at z = 0. Thus for 0 < δ ≤ δ_{0} we see that | ∮ | ≤ Mδ^{σ} if γ ≠ 1 and | ∮ | ≤ Mδ^{σ−1} if γ = 1 with a constant M that depends on a, γ, and δ_{0}, and so ∮ tends to 0 as δ → 0 if Re(s) > 1, and we obtain

(1.2) (e(s) − 1)Γ (s)ζ(s; a, γ) = ∫_{∞}^{0+} \frac{z^{s−1}e^{z(1−a)}}{e^{z} − γ} dz

for 0 < |γ| ≤ 1, 0 < a ∈ R, and Re(s) > 1. (If γ ≠ 1, the condition Re(s) > 0 instead of Re(s) > 1 is sufficient.) Now the right-hand side of (1.2) is meaningful for every (s, a, γ) ∈ C^{3} under condition (1.1a) and so it establishes the product on the left-hand side as an entire function of s. If γ ≠ 1, the contour integral can define the product for the variables s, a, γ as described in Theorem 0.1. The integrand as a function of (z, s, a, γ) is not finite in a domain including γ = 1, and so the result for γ = 1 must be stated separately.

Remark. If 0 < |γ| ≤ 1 and γ ≠ 1, then the series of (0.1) is convergent for Re(s) > 0 and defines a holomorphic function of s there. This is clear if |γ| < 1. For |γ| = 1 and γ ≠ 1, this follows from [S07, Lemma 4.3].

We are interested in the value of ζ(s; a, γ) at s = −m with 0 ≤ m ∈ Z. We first note

{(e(s) − 1)Γ (s)}_{s=−m} = 2πi(−1)^{m}/m!.

For s = −m the function z^{s−1} in the integrand is a one-valued function, and so ∫_{∞}^{δ} + ∫_{δ}^{∞} = 0. Therefore

(1.3) ∫_{∞}^{0+} = ∮ = 2πi · Res_{z=0} \frac{z^{−m−1}e^{(1−a)z}}{e^{z} − γ} .

By (0.3), for γ ≠ 1 we have

\frac{z^{−m−1}e^{(1−a)z}}{e^{z} − γ} = \frac{1}{1 − γ} ∑_{n=0}^{∞} \frac{E_{−γ,n}(1 − a)}{n!}z^{n−m−1},

and so the residue in question is (1 − γ)^{−1}E_{−γ,m}(1 − a)/m!. Combining this with (0.3a) and (1.2), we obtain

(1.4) ζ(−m; a, γ) = E_{c,m}(a)/(1 + c^{−1}) for 0 ≤ m ∈ Z, where c = −γ^{−1}. This proves (0.4). In [S07, p. 26] we showed that E_{c, n}(t) is a polynomial in t and (1+c)^{−1}. For c = −γ^{−1} we have (1+c)^{−1} = 1+(γ −1)^{−1}, and so we obtain Theorem 0.2.

For the reader’s convenience, we give a proof of (0.5) here. If γ = 1, instead of (0.2) we use ze^{tz}/(e^{z} − 1) = ∑_{n=0}^{∞} B_{n}(t)z^{n}/n!. Then we obtain (0.5) in the same manner as in the case γ ≠ 1.

1.3. Let us insert here a historical remark. Fixing a positive integer m and an integer a such that 0 ≤ a ≤ m, Hurwitz considered in [Hu] an infinite series

(1.5) f(s, a) = ∑_{n=0}^{∞}(mn + a)^{−s}.

Since this depends on m, he also denoted it by f(s, a|m). He proved analytic continuation of these functions and stated a functional equation for f(1−s, a), basically following Riemann’s methods for the investigation of ζ(s) in [R]. At that time not much was known about Dirichlet’s L-function beyond his formulas for the class number of a binary quadratic form and his theorem about prime numbers in an arithmetic progression. Employing the results on f(s, a), Hurwitz was able to prove that the L-function for a quadratic character has analytic continuation and satisfies a functional equation. Using the standard notation ζ(s, a) = ∑_{n=0}^{∞}(n + a)^{−s} employed at present, we have f(s, a|m) = m^{−s}ζ(s, a/m), and so he considered ζ(s, a) only for a ∈ Q. It is noticeable however that he proved essentially (0.5) as we already said.

As noted at the beginning of the paper, Lerch investigated the series of (0.1); one can also find an exposition of this topic in [E, p. 27, §1.11]. The paper [Li] of Lipschitz may be mentioned in this connection.

1.4. Let the symbols be as in Theorem 0.3(i). Put

Λ(s) = ∑_{n=1}^{∞}(−1)^{n}χ(n)n^{−s}.

Then Λ(s) + L(s, χ) = 2 ∑_{n=1}^{∞} χ(2n)(2n)^{−s} = χ(2)2^{1−s}L(s, χ), and so

(1.6) Λ(s) = L(s, χ){χ(2)2^{1−s} − 1}.

Since {a | 1 ≤ a < d} = {a | 1 ≤ a ≤ q} ⊔ {d − a | 1 ≤ a ≤ q}, we have

Λ(s)=∑_{a=1}^{q} ∑_{n=0}^{∞}(−1)^{nd+a}χ(a)(nd+a)^{−s}+∑_{a=1}^{q} ∑_{n=0}^{∞}(−1)^{nd+d−a}χ(−a)(nd+d−a)^{−s},

and so d^{s}Λ(s) equals

∑_{a=1}^{q}(−1)^{a}χ(a)\{ ∑_{n=0}^{∞}(−1)^{n}(n + \frac{a}{d})^{−s} − χ(−1) ∑_{n=0}^{∞}(−1)^{n}(n + \frac{d − a}{d})^{−s}\}

= ∑_{a=1}^{q}(−1)^{a}χ(a){ζ(s; a/d, −1) − (−1)^{k}ζ(s; 1 − a/d, −1)}.

Putting s = 1 − k and employing (0.4) and (0.3a), we obtain (0.6).

Our next task is to prove (0.7). Let d = 4d_{0} with 1 < d_{0} ∈ Z as in Theorem 1.3(ii). We note an easy fact(see [S08, Lemma 1.3]):

494 Goro Shimura

(1.7) χ(2d_{0} + a) = −χ(a) for every a ∈ Z. Observe that {x ∈ Z | x > 0, d_{0} ∤x} is a disjoint union (∗) {nd + a | 0<a<d_{0}, 0 ≤ n∈Z} ⊔ {nd + 2d_{0} + a | 0<a<d_{0}, 0 ≤ n∈Z}

⊔ {nd − a | 0<a<d_{0}, 0<n∈Z} ⊔ {nd + 2d_{0} − a | 0<a<d_{0}, 0 ≤ n∈Z}. The sum of ∑ χ(x)x^{−s} for x belonging to the first two sets equals

∑_{a=1}^{d_{0}−1} { ∑_{ν=0}^{∞} χ(a)(4νd_{0} + a)^{−s} + ∑_{ν=0}^{∞} χ(2d_{0} + a)(2(2ν + 1)d_{0} + a)^{−s}}.

Employing (1.7), we see that this equals

(1.8) ∑_{a=1}^{d_{0}−1} ∑_{m=1}^{∞}(−1)^{m}χ(a)(2md_{0} + a)^{−s} = (2d_{0})^{−s} ∑_{a=1}^{d_{0}−1} χ(a)ζ(s; 2a/d, −1).

Similarly, from the last two sets of (∗) we obtain

∑_{a=1}^{d_{0}−1} { ∑_{ν=1}^{∞} χ(−a)(4νd_{0} − a)^{−s} + ∑_{ν=0}^{∞} χ(2d_{0} − a)(2(2ν + 1)d_{0} − a)^{−s}}

= − ∑_{a=1}^{d_{0}−1} ∑_{m=0}^{∞} χ(−a)(−1)^{m}(2md_{0} + 2d_{0} − a)^{−s}

= −(2d_{0})^{−s} ∑_{a=1}^{d_{0}−1} χ(−a)ζ(s; 1 − 2a/d, −1)

by (1.7). Thus, adding (1.8) to this and putting s = 1−k, from (0.4) we obtain

(2d_{0})^{1−k}L(1 − k, χ) = 2^{−1} ∑_{a=1}^{d_{0}−1} χ(a){E_{1,k−1}(2a/d) − χ(−1)E_{1,k−1}(1 − 2a/d)}.

Suppose χ(−1) = (−1)^{k}; then applying (0.3a) to E_{1,k−1}(1 − 2a/d), we obtain (0.7). The proof of Theorem 0.3 is now complete

1.5. The case d = 4 is excluded in Theorem 0.3(ii). In this case, however, the matter is simpler. Indeed, for μ_{4}(n) = (\frac{−1}{n}) we have

L(s, μ_{4}) = ∑_{m=0}^{∞}(−1)^{m}(2m + 1)^{−s}

= 2^{−s} ∑_{m=0}^{∞}(−1)^{m}(m + 1/2)^{−s} = 2^{−s}ζ(s; 1/2, −1),

and so by (0.4) we obtain (1.9) L(1 − k, μ_{4}) = 2^{k−2}E_{1,k−1}(1/2) = E_{1,k−1}/2 for every odd positive integer k, where E_{1,n} denotes the nth Euler number. This is classical, except that the result is usually given in terms of L(k, μ_{4}) instead of L(1 − k, μ_{4}).

Documenta Mathematica 15 (2010) 489–506

1.6. Let us now show that a special case of Theorem 0.3(ii) can be given in a somewhat different way. Let ψ be a primitive character whose conductor d is odd, μ_{4} the primitive character modulo 4 as above, and k a positive integer such that ψ(−1) = (−1)^{k+1}; put m = (d − 1)/2. Then

(1.10) L(1 − k, ψμ_{4}) = (−1)^{m}(2d)^{k−1} ∑_{j=1}^{m}(−1)^{j}ψ(2j)E_{1,k−1}(\frac{1}{2} + \frac{j}{d}).

This was given in [S07, (6.2)], if in terms of L(k, ψμ_{4}), but we can derive it also from Theorem 0.3(ii) as follows. Take χ = ψμ_{4} in (0.7). Then the sum on the right-hand side of (0.7) is ∑_{a=1}^{2m} χ(a)E_{1,k−1}(a/2d), which equals ∑_{b=1}^{2m} χ(d − b)E_{1,k−1}((d − b)/2d). Since the bth term is nonvanishing only for even b, employing (0.3a), we see that the last sum equals ψ(−1) times ∑_{j=1}^{m}(ψμ_{4})(d − 2j)E_{1,k−1}((d + 2j)/2d). Since d − 2j = 2(m − j) + 1, we have μ_{4}(d − 2j) = (−1)^{m−j}, and so we obtain (1.10).

1.7. We can show that ζ(−m; a, γ) for 0 ≤ m ∈ Z is a polynomial in a by a formal calculation as follows. Assuming that |γ| < 1, we have

ζ(−m; a, γ) = a^{m} + ∑_{n=1}^{∞} γ^{n}(n + a)^{m} = a^{m} + ∑_{ν=0}^{m} ( \begin{matrix}m \\ ν \end{matrix} ) a^{m−ν}c_{ν}

with c_{ν} = ∑_{n=1}^{∞} n^{ν}γ^{n}, and so ζ(−m; a, γ) is a polynomial in a at least for |γ| < 1, and so Theorem 0.1 guarantees the same in a larger domain as described in that theorem

We have c_{ν−1} = γ(1 − γ)^{−ν}P_{ν}(γ) for 1 ≤ ν ∈ Z with a polynomial P_{ν} introduced in [S07, (2.16)]; see also [S08, (4.3)]. We showed that P_{ν+1}(γ) = (γ− 1)^{ν}E_{−γ,ν}(0) for ν > 0 in [S07, (4.6)] and that γ^{n−2}P_{n}(γ^{−1}) = P_{n}(γ) in [S07,

(2.19)]; also, E_{c,n}(t) = ∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) E_{c,k}(0)t^{n−k} by [S08, (1.15)]. Combining

these together, we obtain (0.4).

Though clearly this is not the best way to prove (0.4), at least it explains an elementary aspect of the nature of the problem. In Section 5, we will return to this idea in our discussion in the higher-dimensional case.

2. The functional equation for ζ(s; a, γ)

2.1. For Re(s) > 1 Hurwitz proved (see [Hu, p. 93, 1)] and [WW, p. 269])

(2.1) ζ(1 − s, a)

= \frac{2Γ (s)}{(2π)^{s}}\{ cos(πs/2) ∑_{n=1}^{∞} \frac{cos(2πna)}{n^{s}} + sin(πs/2) ∑_{n=1}^{∞} \frac{sin(2πna)}{n^{s}}\}.

Therefore it is natural to ask if there is a meaningful formula for ζ(1 − s, a; γ). Though this was essentially done by Lerch in [Le], here we take a different approach. For s ∈ C, a ∈ R, p ∈ R, and ν = 0 or 1 we put

(2.2) D^{ν}(s; a, p) = ∑_{−a≠n∈Z}(n + a)^{ν}|n + a|^{−ν−s}e(p(n + a)),

496 Goro Shimura

(2.3) T^{ν}(s; a, p) = g_{ν}(s)D^{ν}(s; a, p), g_{ν}(s) = π^{−(s+ν)/2}Γ ((s + ν)/2). These were introduced in [S08]. In particular, we proved (2.4) T^{ν}(1 − s; a, p) = i^{−ν}e(ap)T^{ν}(s; −p, a).

Let γ = e(p) and b = 1 − a with 0 < a < 1. Then it is easy to verify that (2.5a) D^{0}(s; a, p) = e(ap)ζ(s; a, γ) + e(−bp)ζ(s; b, γ^{−1}), (2.5b) D^{1}(s; a, p) = e(ap)ζ(s; a, γ) − e(−bp)ζ(s; b, γ^{−1}), and so (2.6) 2e(ap)ζ(s; a, γ) = D^{0}(s; a, p) + D^{1}(s; a, p). Employing (2.3) and (2.4), we obtain

(2.7) 2ζ(1 − s; a, γ) = \frac{g_{0}(s)}{g_{0}(1 − s)}D^{0}(s; −p, a) − \frac{ig_{1}(s)}{g_{1}(1 − s)}D^{1}(s; −p, a).

From (2.5a, b) we see that D^{ν}(s; −p, a) is a linear combination of ζ(s; −p, δ) and ζ(s; 1 + p, δ^{−1}), where δ = e(a). Thus, for −1 < p < 0 we have

2ζ(1 − s; a, γ) = e(−ap)Aζ(s; −p, δ) + e(−a − ap)Bζ(s; 1 + p, δ^{−1}) with

(2.8) A = \frac{g_{0}(s)}{g_{0}(1 − s)} − \frac{ig_{1}(s)}{g_{1}(1 − s)}, B = \frac{g_{0}(s)}{g_{0}(1 − s)} + \frac{ig_{1}(s)}{g_{1}(1 − s)}.

Recalling that Γ (s/2)Γ ((s + 1)/2) = 2^{1−s}π^{1/2}Γ (s) and Γ (s)Γ (1 − s) = π/ sin(πs), we find that (2.9) A = 2^{1−s}π^{−s}e(−s/4)Γ (s), B = 2^{1−s}π^{−s}e(s/4)Γ (s), and so we obtain (2.10) ζ(1 − s; a, γ)

= \frac{e(−ap)Γ (s)}{(2π)^{s}}\{e(−s/4)ζ(s; −p, δ) + e(s/4 − a)ζ(s; p + 1, δ^{−1})\}.

at least when −1 < p < 0 and 0 < a < 1, where γ = e(p) and δ = e(a).

This does not apply to the case γ = 1. In this case we have

2ζ(s; a, 1) = D^{0}(s; a, 0) + D^{1}(s; a, 0) for 0 < a ≤ 1,

D^{ν}(s; 0, a) = δζ(s; 1, δ) + (−1)^{ν}δ^{−1}ζ(s; 1, δ^{−1}).

Repeating the same argument as in the case γ ≠ 1, we find that

2ζ(1 − s; a, 1) = δAζ(s; 1, δ) + δ^{−1}Bζ(s; 1, δ^{−1})

with the same A and B as in (2.8) and (2.9), and so (2.11) ζ(1 − s; a, 1)

= \frac{Γ (s)}{(2π)^{s}} \{e(a − \frac{s}{4})ζ(s; 1, δ) + e(\frac{s}{4} − a)ζ(s; 1, δ^{−1})\}.

where δ = e(a), 0 < a ≤ 1. This gives (2.1). Indeed, we have

Documenta Mathematica 15 (2010) 489–506

e(±a)ζ(s; 1, δ^{±1}) = ∑_{m=1}^{∞} \frac{δ^{±m}}{m^{s}} = ∑_{m=1}^{∞} \frac{cos(2πma) ± i sin(2πma)}{m^{s}},

and so a simple calculation transforms (2.11) into (2.1).

Returning to (2.10) in which we assumed −1 < p < 0, put q = −p. Then 0 < q < 1 and

(2.12) \frac{(2π)^{s}}{Γ (s)} ζ(1 − s; a, e(q)^{−1})

= e(−s/4) ∑_{h=0}^{∞} \frac{e(a(h + q))}{(h + q)^{s}} + e(s/4) ∑_{h=0}^{∞} \frac{e(a(q − h − 1))}{(h + 1 − q)^{s}}.

Then employing (0.4), for 0 < s = k ∈ Z we obtain

(2.13) \frac{(2πi)^{k}}{(k − 1)!(1 + c^{−1})}E_{c,k−1}(a) = ∑_{h∈Z} \frac{e(a(h + q))}{(h + q)^{k}},

where c = −e(q). This formula was given in [S07, (4.5)]. Thus we have given a proof of (2.13) different from that of [S07]. (The case k = 1 must be handled carefully; see [S07, pp. 26–27].) In [S08] we asked the question whether the parameter k in (2.13) can be extended to a complex variable, and presented D^{ν}(s; a, p) as an answer. Since (2.13) is a special case of (2.12), we can now say that (2.12) is another answer to that question.

3. The case of a totally real number field

3.1. Let F be a totally real algebraic number field. We ask whether we can define a function similar to ζ(s; a, γ) by taking the totally positive integers in F in place of n in (0.1). We are going to give a partially affirmative answer to this question. We let 𝔤 denote the maximal order of F, 𝔡 the different of F relative to Q, and a the set of all archimedean primes of F. For each v ∈ a we denote by F_{v} the v-completion of F, identified with R. In other words, v defines an injection of F into R, and for ξ ∈ F we denote by ξ_{v} the image of ξ under this injection. We put [F : Q] = g, F_{a} = ∏_{v∈a} F_{v}, and F_{a}^{×} = ∏_{v∈a} F_{v}^{×}. Then F_{a} can be identified with R^{g}, and for ξ ∈ F the map ξ ↦ (ξ_{v})_{v∈a} defines an injection of F into F_{a}. We then put

e_{a}(ξ) = e( ∑_{v∈a} ξ_{v}) (ξ ∈ F_{a}),
ξ^{k} = ∏_{v∈a} ξ_{v}^{k_{v}},  ξ^{a} = ∏_{v∈a} ξ_{v},  (k = (k_{v})_{v∈a} ∈ Z^{a}, ξ ∈ F_{a}^{×}).

We also put N(ξ) = N_{F/Q}(ξ) for ξ ∈ F. Then N(ξ𝔤) = |N(ξ)| = |ξ^{a}| for ξ ∈ F^{×}. We write ξ ≫ 0 if ξ_{v} > 0 for every v ∈ a and put 𝔤_{+}^{×} = {u ∈ 𝔤^{×} | u ≫ 0}.

Now for s ∈ C, a fractional ideal 𝔟, and a, p ∈ F we put

(3.1)  ζ(s; 𝔟, a, p) = [𝔤_{+}^{×} : U]^{−1} ∑_{0≪ξ∈𝔟+a (mod U)} |N(ξ)|^{−s}e_{a}(pξ).

Here the sum is taken over F^{×}/U under the condition that ξ ≫ 0 and ξ − a ∈ 𝔟. We take a subgroup U of 𝔤_{+}^{×} of finite index such that u − 1 ∈ a^{−1}𝔟 ∩ p^{−1}(𝔟^{−1}𝔡^{−1} ∩ a^{−1}𝔡^{−1}) for every u ∈ U. Clearly such a U exists, the sum of (3.1) is meaningful, and ζ(s; 𝔟, a, p) is well defined independently of the choice of U.

This is an analogue of (0.1), but it should be remembered that here both a and p belong to F. Thus (1.5) is a special case of (3.1).

Next, let k ∈ Z^{a}. With s, 𝔟, a, and p as above, we put

(3.2)  D_{k}(s; 𝔟, a, p) = [𝔤_{+}^{×} : U]^{−1} ∑_{0≠ξ∈𝔟+a (mod U)} ξ^{−k}|ξ|^{k−sa}e_{a}(pξ),

where the summation is the same as in (3.1) except that this time we do not impose the condition ξ ≫ 0. This is well defined and both (3.1) and (3.2) are convergent for Re(s) > 1.

Lemma 3.2. Let k ∈ Z^{a} with k_{v} = 0 or 1 for every v ∈ a. Then the product

D_{k}(s; 𝔟, a, p) ∏_{v∈a} Γ ((s + k_{v})/2)

can be continued to a meromorphic function of s on the whole s-plane that is holomorphic except for possible simple poles at s = 0 and 1, which occur only when k = 0. The pole at s = 0 occurs if and only if k = 0 and a ∈ 𝔟.

This is included in [S00, Lemma 18.2], as D_{k}(s; 𝔟, a, p) is a special case of the series D_{k}(s, κ) of [S00, (18.1)]. Notice that D_{k}(s; 𝔟, a, p) is finite at s = 0 for every k.

Lemma 3.3. For 0 < μ ∈ Z and k ∈ Z^{a} with k_{v} = 0 or 1 the following assertions hold:

(i) D_{k}(1 − μ; 𝔟, a, p) = 0 if k_{v} − μ ∉ 2Z for some v ∈ a.

(ii) Suppose k_{v} − μ ∈ 2Z for every v ∈ a; then D_{k}(1 − μ; 𝔟, a, p) ∈ Q_{ab}, where Q_{ab} denotes the maximal abelian extension of Q; in particular, D_{k}(1 − μ; 𝔟, a, 0) ∈ Q.

Proof. Suppose k_{v} − μ ∉ 2Z for one particular v. Then Γ ((s + k_{v})/2) has a pole at s = 1 − μ, and so (i) follows from Lemma 3.2. As for (ii), that D_{k}(1−μ; 𝔟, a, 0) ∈ Q is given in Proposition 18.10(ii) of [S00]. For any p ∈ F, we see that D_{k}(s; 𝔟, a, p) is a finite Q_{ab}-linear combination of D_{k}(s; 𝔟′, a′, 0) with several (𝔟′, a′), and so D_{k}(1 − μ; 𝔟, a, p) ∈ Q_{ab}.

Now our principal result on ζ(s; 𝔟, a, p) can be stated as follows.

Theorem 3.4. (i) ζ(s; 𝔟, a, p) can be continued to a meromorphic function of s on the whole s-plane, which is holomorphic except for a possible simple pole at s = 1.

(ii) Let 0 < μ ∈ Z. Then ζ(1 − μ; 𝔟, a, p) ∈ Q_{ab}, and in particular, ζ(1 − μ; 𝔟, a, 0) ∈ Q.

Proof. For ξ ∈ F^{×} we note an easy fact

∑_{k∈Z^{a}/2Z^{a}} ξ^{−k}|ξ|^{k} = \{ \begin{matrix}2^{g} if ξ ≫ 0, \\ 0 otherwise, \end{matrix}

where g = [F : Q]. Therefore we obtain

(3.3) 2^{g}ζ(s; 𝔟, a, p) = ∑_{k∈Z^{a}/2Z^{a}} D_{k}(s; 𝔟, a, p),

and so assertion (i) follows immediately from Lemma 3.2. Take μ as in (ii). By Lemma 3.3(i), the terms on the right-hand side of (3.3) vanish except for the term with k such that k_{v} − μ ∈ 2Z for every v. Therefore we obtain the desired result from Lemma 3.3(ii), and our proof is complete.

4. Some explicit expressions for Gauss sums

4.1. There are two kinds of formulas for the critical values of L(s, χ): one is for L(k, χ) and the other for L(1−k, χ). The former involves π and the Gauss sum of χ, whereas the latter does not. In a sense L(1 − k, χ) is conceptually more natural than L(k, χ), but there is an interesting aspect in the computation of L(k, χ), since it allows us to find an explicit expression for a certain Gauss sum. This can be achieved by computing L(k, χ) in two different ways, which involve two different Gauss sums. Let us begin with the definition of a Gauss sum and an easy lemma.

Given a primitive or an imprimitive Dirichlet character χ′ modulo a positive integer, we take the primitive character χ associated with χ′, and define the Gauss sum G(χ′) to be the same as the Gauss sum G(χ) of χ, given by

(4.1) G(χ) = ∑_{a=1}^{d} χ(a)e(a/d),

where d is the conductor of χ. Now we have an elementary

Lemma 4.2. (i) Let χ_{1}, …, χ_{m} be Dirichlet characters. Then the number G(χ_{1}) ⋯ G(χ_{m})/G(χ_{1} ⋯ χ_{m}) belongs to the field generated by the values of χ_{1}, …, χ_{m} over Q.

(ii) Let ψ and χ be primitive characters of conductor c and d, respectively. If c and d are relatively prime, then

(4.2) G(ψχ) = ψ(d)χ(c)G(ψ)G(χ).

Proof. For the proof of (i) see [S78, Proposition 4.12], which generalizes [S76, Lemma 8]. In the setting of (ii) take r, s ∈ Z so that cr + ds = 1. Then for x, y ∈ Z the map (x, y) ↦ xds+ycr gives a bijection of (Z/cZ)×(Z/dZ) onto Z/cdZ, and so

G(ψχ) = ∑_{x=1}^{c} ∑_{y=1}^{d} e((xds + ycr)/cd)ψ(xds)χ(ycr)

= ∑_{x=1}^{c} ψ(xds)e(xs/c) ∑_{y=1}^{d} χ(ycr)e(yr/d),

which proves (4.2).

Theorem 4.3. Let χ be a primitive character of conductor d such that χ(−1) = −1, and let λ(m) = (\frac{3}{m}). Suppose d is odd and 0 < d/9 ∈ Z. Then

(4.3) \frac{G(χλ)}{G(χ)} = \{2\sqrt{3}χ(2) ∑_{a=1}^{d−1}(χλ)(a)\}/\{ ∑_{j=1−g}^{d−g}(−1)^{j}χ(j)\},

where g = [d/6] + 1.

Proof. In [S07, Theorem 6.3(v)] we gave a formula for L(k, χλ). Taking k = 1 and χ̄ in place of χ, we obtain

2\sqrt{3}χ(2)(πi)^{−1}G(χ)L(1, χ̄λ) = ∑_{j=1−g}^{d−g}(−1)^{j}χ(j),

since E_{1,0}(t) = 1. Now χ̄λ is primitive and has conductor 4d, and so the formula [S07, (4.34)] applied to χ̄λ produces

(πi)^{−1}G(χλ)L(1, χ̄λ) = ∑_{a=1}^{d−1}(χλ)(a).

Taking the quotient of these two formulas, we obtain (4.3).

In [S07, Theorem 6.3] we gave eight formulas for L(k, χλ), where λ is a “constant” character and χ is a “variable” character. In the above theorem we employed only one of those formulas. We can actually state results about G(χλ)/G(χ) in the other seven cases, but they are not so interesting, since we can apply (4.2) to χλ in those cases, and the case we employed in the above theorem is the only case to which (4.2) is not applicable. Even in that case, the significance of (4.3) is rather obscure. Still, the formula is clear-cut and nontrivial, and we state it here with the hope that future researchers will be able to clarify its nature in a better perspective.

We end our discussion of this subject by showing the quantity of (4.3) can be determined in a different way. We begin with some preliminary results.

Lemma 4.4. Let χ and ψ be primitive characters of conductor p^{m} and p^{n}, respectively, where p is a prime number and m, n are positive integers. Suppose m ≥ n and χψ has conductor p^{m}. Then

(4.4) G(χ)G(ψ) = G(χψ) ∑_{a=1}^{p^{n}} χ(1 − p^{m−n}a)ψ(a).

This was given in [S76, (4.2)].

Lemma 4.5. Let χ be a primitive character of conductor c, where c = 3^{m} with m > 1 or c = 2^{m} with m > 3, and let μ_{3} and μ_{4} denote the primitive characters of conductor 3 and 4, respectively. Let χ′ = χμ_{3} if c = 3^{m} and χ′ = χμ_{4} if c = 2^{m}. Then G(χ′) = εG(χ) with ε = ±1 determined by χ(1 − 3^{m−1}) = e(ε/3) if c = 3^{m} and χ(1 − 2^{m−2}) = e(ε/4) if c = 2^{m}.

Proof. We first consider the case c = 2^{m}. By (4.4), G(χ)G(μ_{4})/G(χ′) = β − γ with β = χ(1−2^{m−2}) and γ = χ(1+2^{m−2}). Since m > 3, we have (1−2^{m−2})^{2} ≡ 1−2^{m−1} (mod 2^{m}), (1−2^{m−2})(1+2^{m−2}) ≡ 1 (mod 2^{m}), and (1−2^{m−2})^{4} ≡ 1 (mod 2^{m}), and so β^{4} = βγ = 1. Suppose β = ±1. Then χ(1 − 2^{m−1}) = 1, and so χ(1 − 2^{m−1}a) = χ(1 − 2^{m−1})^{a} = 1 for every a ∈ Z, which means that χ has conductor ≤ 2^{m−1}, a contradiction. Thus β ≠ ±1, and so β = εi with ε = ±1. Since G(μ_{4}) = 2i, we have G(χ)/G(χ′) = (β − β^{−1})/2i = ε, which proves the case c = 2^{m}.

If c = 3^{m}, we have similarly G(χ)G(μ_{3})/G(χ′) = β−γ with β = χ(1−3^{m−1}) and γ = χ(1 + 3^{m−1}). We easily see that βγ = β^{3} = 1 and β ≠ 1. Since G(μ_{3}) = \sqrt{3}i, we have G(χ)/G(χ′) = (β − β^{−1})/\sqrt{3}i = ε, and ε is determined by β = e(ε/3). This completes the proof.

4.6. Returning to the setting of Theorem 4.3, put d = 3^{m}f with m ∈ Z and 0 < f ∈ Z, 3 ∤ f; put also χ = χ_{0}χ_{1} with characters χ_{0} and χ_{1} of conductor 3^{m} and f, respectively. Since λ = μ_{3}μ_{4}, we have χλ = χ_{0}μ_{3}χ_{1}μ_{4}. By (4.2) we have G(χ_{1}μ_{4}) = 2iχ_{1}(4)μ_{4}(f)G(χ_{1}), and so

G(χλ) = (χ_{1}μ_{4})(3^{m})(χ_{0}μ_{3})(4f)G(χ_{0}μ_{3})G(χ_{1}μ_{4})
= 2i(−1)^{m}χ_{0}(f)χ(4)χ_{1}(3^{m})λ(f)G(χ_{0}μ_{3})G(χ_{1}).

Also, G(χ) = χ_{0}(f)χ_{1}(3^{m})G(χ_{0})G(χ_{1}). Therefore

G(χλ)/G(χ) = 2i(−1)^{m}λ(f)χ(4)G(χ_{0}μ_{3})/G(χ_{0}).

Applying Lemma 4.5 to G(χ_{0}μ_{3})/G(χ_{0}), we thus obtain

(4.5) G(χλ)/G(χ) = 2i(−1)^{m}λ(f)χ(4)ε,

where ε = ±1 is determined by χ_{0}(1 − 3^{m−1}) = e(ε/3).

5. The case of a domain of positivity

5.1. There is a natural analogue of ζ(s; a, γ) defined on the space of symmetric matrices. To be explicit, with a positive integer n we denote by V the set of all real symmetric matrices of size n, and write h > 0 resp. h ≥ 0 for h ∈ V when h is positive definite resp. nonnegative. We put V_{C} = V ⊗_{R} C,

κ = (n + 1)/2, P = {h ∈ V | h > 0}, and
ℌ = {x + iy ∈ V_{C} | x ∈ V, y ∈ P }.

For h ∈ V we denote by λ(h) and μ(h) the maximum and minimum absolute value of eigenvalues of h, respectively. We easily see that V is a normed space with λ(h) as the norm of h, that is, λ(ch) = |c|λ(h) for c ∈ R and λ(h + k) ≤ λ(h) + λ(k). For 0 ≤ d ∈ Z we denote by S_{d} the space of all C-valued homogeneous polynomial functions on V of degree d. Here are two easy facts:

(5.1) tr(gh) ≥ λ(h)μ(g) if g, h ∈ P. (5.2) |ξ(h)| ≤ c_{ξ}λ(h)^{d} for every ξ ∈ S_{d} and h ∈ V with a positive constant

c_{ξ} that depends only on ξ.

To prove (5.1), we may assume that g is diagonal. For g = diag[μ_{1}, …, μ_{n}] we have tr(gh) = ∑_{i=1}^{n} μ_{i}h_{ii} ≥ μ(g)tr(h) ≥ μ(g)λ(h), since tr(h) is the sum of all eigenvalues of h. As for (5.2), given h ∈ V, take an orthogonal matrix p so that h = ^{t}p · diag[κ_{1}, …, κ_{n}]p with κ_{i} ∈ R. Since |p_{ij}| ≤ 1 and |κ_{i}| ≤ λ(h), we see that |h_{ij}| ≤ nλ(h), from which we obtain (5.2).

We now consider three types of infinite series:

(5.3) Φ(s; L, a, z) = ∑_{h∈L} det (2πi(h − z))^{−s}e(tr(a(h − z))),

(5.4) F (s; L, a, z) = ∑_{h∈L, h+a>0} det(h + a)^{−s}e(tr(hz)),

(5.5) F_{0}(s; L, a, z) = ∑_{0≤h∈L} det(h + a)^{−s}e(tr(hz)).

Here s ∈ C, L is a lattice in V, a ∈ V, z ∈ ℌ, and the sum of (5.4) is extended over all h ∈ L such that h + a ∈ P; the sum of (5.3) is simply over all h ∈ L; in (5.5) we assume that a > 0 and the sum is extended over all nonnegative h in L. For z ∈ ℌ and s ∈ C we define det(−2πiz)^{s} so that it coincides with det(2πp)^{s} if z = ip with p ∈ P.

Lemma 5.2. For every ξ ∈ S_{d} the infinite series

(5.6)  ∑_{h∈L, h+a>0} det(h + a)^{s}ξ(h + a)e(tr(hz))

converges absolutely and locally uniformly for (s, a, z) ∈ C × V × ℌ.

Proof. For a fixed positive number α the number of h ∈ L such that λ(h)≤α is finite, as {h∈V | λ(h)≤α} is compact and L is discrete in V. Thus, to prove the convergence of (5.6), we can restrict h to those that satisfy λ(h) > λ(a)+1 for every a ∈ A, where A is a fixed compact subset of V. For such an h we have 1 ≤ λ(h + a) < 2λ(h) and det(h + a) ≤ λ(h + a)^{n} ≤ 2^{n}λ(h)^{n}. Also, by (5.2), |ξ(h + a)| ≤ c_{ξ}λ(h + a)^{d} ≤ c_{ξ}2^{d}λ(h)^{d}. Thus for Re(s) = σ we have | det(h + a)^{s}ξ(h + a)| ≤ 2^{nσ+d}c_{ξ}λ(h)^{nσ+d}. On the other hand, for 0 < N ∈ Z the number of h ∈ L such that N ≤ λ(h) < N + 1 is less than CN^{b} with positive constants C and b. Put g = 2πIm(z). Then |e(tr(hz))| = e^{−tr(gh)}. Since −tr(gh) ≤ −λ(h)μ(g) by (5.1), our partial sum can be majorized by 2^{nσ+d}c_{ξ}C ∑_{N=1}^{∞}(N + 1)^{nσ+d+b}e^{−Nμ(g)}, which proves our lemma.

Theorem 5.3. (i) The infinite series of (5.3) converges absolutely and locally uniformly in (s, a, z) ∈ {s ∈ C | Re(s) > n} × V × ℌ. Thus it defines a holomorphic function in s for Re(s) > n.

(ii) The infinite series of (5.4) and (5.5) converge absolutely and locally uniformly in (s, a, z) ∈ C × V × ℌ, and so they define entire functions of s.

(iii) For Re(s) > n we have

(5.7) vol(V/L)F (κ − s; L, a, z) = Γ_{n}(s)Φ(s; L′, a, z),

where L′ = {x ∈ V | tr(xL) ⊂ Z} and

Γ_{n}(s) = π^{n(n−1)/4} ∏_{k=0}^{n−1} Γ (s − (k/2)).

(iv) For 0 ≤ m ∈ Z the values F (−m; L, a, z), F_{0}(−m; L, a, z), and Φ(κ + m; L, a, z) are polynomial functions of a of degree ≤ mn, whose coefficients depend on L and z.

Statement (iv) for F_{0} can be taken literally, but the cases of F and Φ require some clarifications, which will be given in the proof.

Proof. Assertion (i) is included in [S82, Lemma 1.3]; (ii) follows from Lemma 5.2. To prove (iii), given s ∈ C and p ∈ P, we consider a function g on V defined by

g(u) = \{ \begin{matrix}e^{−tr(up)} det(u)^{s−κ} (u ∈ P), \\ 0 (u ∉ P), \end{matrix}

and define its Fourier transform ĝ by

ĝ(t) = ∫_{V} e( − tr(ut))g(u)du (t ∈ V),

where du = ∏_{i≤j} du_{ij}. As shown in [S82, (1.22)], ĝ(t) = Γ_{n}(s) det(p + 2πit)^{−s},

provided Re(s) > κ − 1. Then the Poisson summation formula establishes equality (5.7) with both sides multiplied by e(tr(az)) for z = (−2πi)^{−1}p when the series of (5.3) and (5.4) are convergent, which is the case at least when Re(s) > n. Since both sides of (5.7) are holomorphic in z if Re(s) > n, we obtain (iii) as stated.

To prove (iv), take a C-basis B of ∑_{d=0}^{mn} S_{d}. For 0 ≤ m ∈ Z we have

(5.8) F_{0}(−m; L, a, z) = ∑_{0≤h∈L} det(h + a)^{m}e(tr(hz)).

We can put det(h+a)^{m} = ∑_{β∈B} β(a)f_{β}(h) with polynomial functions f_{β}, and so

(5.8a) F_{0}(−m; L, a, z) = ∑_{β∈B} β(a)G_{β}(z) with

(5.8b) G_{β}(z) = ∑_{0≤h∈L} f_{β}(h)e(tr(hz)).

Thus F_{0}(−m; L, a, z) is a polynomial in a as stated in (iv). This argument is basically valid for F in place of F_{0}, but the functions corresponding to G_{β} in that case may depend on a. To avoid that difficulty, we first take a compact subset A of V and restrict a to A. As shown in the proof of Lemma 5.2, we can find a subset M of L independent of a such that

{h ∈ L | h > 0, h + a > 0} = M ⊔ K_{a},

with a finite set K_{a} for each a ∈ A. Then, taking F in place of F_{0}, we obtain, for a ∈ A,

(5.9a) F (−m; L, a, z) = ∑_{β∈B} β(a)H_{β}(z) with

(5.9b) H_{β}(z) = ∑_{h∈K_{a}} f_{β}(h)e(tr(hz)) + ∑_{h∈M} f_{β}(h)e(tr(hz)).

Thus the statement about F (−m; L, a, z) in (iv) must be understood in the sense of (5.9a, b). It is a polynomial in a whose coefficients H_{β}(z) is the sum of a “principal part” that is independent of a and a finite sum depending on a.

As for the value of Φ, from (5.7) we see that Φ is an entire functions of s. Also Γ_{n}(s)^{−1} is nonzero for (n − 1)/2 < s ∈ R. Thus Φ(κ + m; L′, a, z) is a nonzero constant times F (−m; L, a, z), and so is a polynomial in a in the sense explained above.

5.4. Let us add some remarks. Clearly F_{0} of (5.5) is a natural generalization of (0.1), but we introduced F and Φ as in (5.4) and (5.3), as we think they are natural objects of study closely related to (5.5). It must be remembered, however, that (5.5) includes (0.1) as a special case only if |γ| < 1. To define something like (5.5) that includes (0.1) with |γ| = 1 is one of the open problems in this area.

Next, there are four classical types of domains of positivity associated with tube domains discussed in [S82]. Our V, P, and ℌ in this section belong to the easiest type. We can in fact define the analogues of (5.3), (5.4), and (5.5) for all three other types of domains, and prove the results similar to Theorem 5.3 in those cases.

As to the nature of the polynomials in the variable a obtained in Theorem 5.3(iv), we do not have their description as explicit as what we know about E_{c,m}(t). Still, we can show that they are of a rather special kind. For that purpose we need a matrix of differential operators ∂_{a} = (∂_{ij})_{i,j=1}^{n} on V as folows. Taking a variable symmetric matrix a = (a_{ij}) on V, we put ∂_{ii} = ∂/∂a_{ii} and ∂_{ij} = 2^{−1}∂/∂a_{ij} for i ≠ j. Then for every φ ∈ S_{d} we can define a diffrential operator φ(∂_{a}). In particular, taking φ(a) = det(a), we put

(5.10) Δ_{a} = det(∂_{a}) = ∑_{σ} sgn(σ)∂_{1σ(1)} ⋯ ∂_{nσ(n)},

where σ runs over all permutations of {1, …, n}. It is well known that

(5.11) Δ_{a}( det(a)^{s}) = ∏_{k=0}^{n−1}(s + k/2) · det(a)^{s−1}.

This is a special case of a general formula on φ(∂_{a}) det(z)^{s} for φ ∈ S_{d} given in [S84].

Fixing L and z, for 0 ≤ m ∈ Z put

(5.12) ℰ_{m}(a) = F_{0}(−m; L, a, z).

We have shown that ℰ_{m} is a polynomial of degree ≤ mn. Notice that ℰ_{0}(a) = ∑_{0≤h∈L} e(tr(hz)). From (5.8) and (5.11) we obtain

(5.13) Δ_{a}ℰ_{m}(a) = ∏_{k=0}^{n−1}(m + k/2) · ℰ_{m−1}(a).

This is a generalization of the formula (d/dt)E_{c,m}(t) = mE_{c,m−1}(t), noted in [S07, (4.3c)].

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[R] B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichten der Berliner Akademie 1859 (= Gesammelte Mathematische Werke, 145–153).

[S76] G. Shimura, The special values of the zeta functions associated with cusp forms, Communications on pure and applied Mathematics, 29 (1976), 783–804 (=Collected Papers II, 740–761).

[S78] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Mathematical Journal 45 (1978), 637–679. (Corrections to this paper were given in Duke Mathematical Journal 48 (1981),

697. A better revised version of the paper can be found in Collected Papers III, 75–114.)

[S82] G. Shimura, Confluent hypergeometric functions on tube domains, Mathematische Annalen, 260 (1982), 269–302 (=Collected Papers III, 388– 421).

[S84] G. Shimura, On differential operators attached to certain representations of classical groups, Inventiones mathematicae, 77 (1984), 463–488 (=Collected Papers III, 584–609).

[S00] G. Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82, American Mathematical Society, 2000.

[S07] G. Shimura, Elementary Dirichlet series and modular forms, Springer Monographs in Mathematics, Springer, 2007.

[S08] G. Shimura, The critical values of certain Dirichlet series, Documenta Mathematika 13, (2008), 775–794.

[WW] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed., Cambridge, 1927.

Goro Shimura Department of Mathematics Princeton University Princeton NJ 08544–1000 USA goro@Math.Princeton.EDU