Absolute zeta functions

Shin-ya Koyama and Nobushige Kurokawa

Abstract. We first survey the absolute tensor product. In particular we find that zeta functions which are meromorphic of order r are expected to have Euler factors expressed in terms of the elliptic gamma function of order r−1. We give some observation and evidence for this phenomenon. Secondly, we generalize the absolute Weil type zeta function of Deitmar-Koyama-Kurokawa to noncommutative versions. We obtain the determinant expressions and give many examples.

1 Zeta Functions of Tensor Products over F_{1}

We recall the construction of the absolute tensor product, which is called the Kurokawa tensor product by Manin [23]. We refer to [18] and [22] for details.

Let

Z_{j}(s) = ∏ ∐_{ρ∈C}(s − ρ)^{m_{j}(ρ)}

= exp (−\frac{∂}{∂w}|_{w=0} ∑_{ρ∈C} \frac{m_{j}(ρ)}{(s − ρ)^{w}} )

be “zeta functions” expressed as regularized product, where

m_{j} : C → Z

denotes the multiplicity function for j = 1, ..., r. The absolute tensor product (Z_{1} ⊗ ⋯ ⊗ Z_{r})(s) is defined as

(Z_{1} ⊗ ⋯ ⊗ Z_{r})(s) = ∏ ∐_{ρ_{1},··· ,ρ_{r}∈C}(s − (ρ_{1} + ⋯ + ρ_{r}))^{m(ρ_{1},··· ,ρ_{r})}

with

m(ρ_{1}, ⋯, ρ_{r}) = m_{1}(ρ_{1}) ⋯ m_{r}(ρ_{r}) × \{ \begin{matrix}1 & Im(ρ_{j}) ≥ 0, & (j = 1, ..., r) \\ (−1)^{r−1} & Im(ρ_{j}) < 0, & (j = 1, ..., r) \\ 0 & otherwise. \end{matrix}

This definition originates from [15]. We refer to the excellent survey of Manin [23]. The notation of the regularized product is due to Deninger [9]. See [11] concerning the needed regularized products. The absolute tensor product was studied by Schröter [24] in the name of the “Kurokawa tensor product.”

We are especially interested in the case of Hasse zeta functions Z_{j}(s) = ζ(s, A_{j}) for commutative rings A_{1}, ..., A_{r} of finite type over Z. We recall that the Hasse zeta function ζ(s, A) of a commutative ring A is defined to be

ζ(s, A) = ∏_{m}(1 − N(m)^{−s})^{−1}

= exp (∑_{m} ∑_{k=1}^{∞} \frac{1}{k}N(m)^{−ks}) ,

where m runs over maximal ideals of A and N(m) = #(A/m). This is also written as

ζ(s, A) = exp ( ∑_{p:primes} ∑_{m=1}^{∞} \frac{|Hom_{ring}(A, F_{p^{m}})|}{m}p^{−ms}) .

For simplicity we write

ζ(s, A_{1} ⊗ ⋯ ⊗ A_{r}) = ζ(s, A_{1}) ⊗ ⋯ ⊗ ζ(s, A_{r}).

It holds that ζ(s, A_{1} ⊗ ⋯ ⊗ A_{r}) has the following additive structure on zeros and poles: if ζ(s_{j}, A_{j}) = 0 or ∞ and Im(s_{j}) (j = 1, ..., r) have the same signature, then

ζ(s_{1} + ⋯ + s_{r}, A_{1} ⊗ ⋯ ⊗ A_{r}) = 0 or ∞.

Such an additive structure was crucial in the study of Hasse zeta functions of positive characteristic (congruence zeta functions) pursued by Grothendieck [12] and Deligne [6], where Euler products were important to restrict the region of zeros and poles for our reaching to the analogue of the Riemann Hypothesis.

We expect that our multiple zeta functions also have Euler products of the following form:

ζ(s, A_{1}) ⊗ ⋯ ⊗ ζ(s, A_{r}) = ∏_{(m_{1},...,m_{r})} H_{(m_{1},...,m_{r})}(N(m_{1})^{−s}, ..., N(m_{1})^{−s})

where m_{i} runs over the maximal ideals of A_{i} and H_{(m_{1},...,m_{r})}(T_{1}, ..., T_{r}) is a power series in T_{1},...,T_{r} of the constant term 1 with a possible degeneration at (m_{1}, ..., m_{r}), where N(m_{i}) = N(m_{j}) for some i ≠ j. More generally we expect that the multiple zeta function Z_{1}(s) ⊗ ⋯ ⊗ Z_{r}(s) has an Euler product

Z_{1}(s) ⊗ ⋯ ⊗ Z_{r}(s) = ∏_{(p_{1},...,p_{r})∈P_{1}×···×P_{r}} H_{(p_{1},...,p_{r})}(N(p_{1})^{−s}, ..., N(p_{r})^{−s})

when each zeta function Z_{j}(s) has an Euler product

Z_{j}(s) = ∏_{p∈P_{j}} H_{p}^{j}(N(p)^{−s})

and a functional equation; here H_{p}^{j}(T) is a power series in T and H_{(p_{1},...,p_{r})}(T_{1}, ..., T_{r}) is a power series in (T_{1}, ..., T_{r}) with a possible degeneration at (p_{1}, ..., p_{r}), where N(p_{i}) = N(p_{j}) for some i ≠ j.

In [18] we investigated the absolute tensor product ζ(s, F_{p}) ⊗ ζ(s, F_{q}) for primes p and q by using a signed double Poisson summation formula, where ζ(s, F_{p}) = (1 − p^{−s})^{−1}. In other words we constructed a zeta function having zeros (or poles) at sums of poles of ζ(s, F_{p}) and those of ζ(s, F_{q}). We state the results as follows. Their proofs and some refinements were in [18] and [19]. For simplicity we use the notation F (s) ≅ G(s) for functions F (s) and G(s) to indicate that F (s) = e^{Q(s)}G(s) for some polynomial Q(s).

Theorem 1.1 Let p and q be distinct prime numbers. Define the function ζ_{p,q}(s) in Re(s) > 0 as follows:

ζ_{p,q}(s) := exp (−\frac{i}{2} ∑_{n=1}^{∞} \frac{cot (πn\frac{log p}{log q})}{n}p^{−ns} − \frac{i}{2} ∑_{n=1}^{∞} \frac{cot (πn\frac{log q}{log p})}{n}q^{−ns}

−\frac{1}{2} ∑_{n=1}^{∞} \frac{1}{n}p^{−ns} − \frac{1}{2} ∑_{n=1}^{∞} \frac{1}{n}q^{−ns}) .

Then the function ζ_{p,q}(s) has the following properties:

(0) It converges absolutely in Re(s) > 0.

(1) The function ζ_{p,q}(s) has an analytic continuation to all s ∈ C as a meromorphic function of order two.

(2) All zeros and poles of ζ_{p,q}(s) are simple and located at

s = 2πi (\frac{m}{log p} + \frac{n}{log q} ) ,

where (m, n) is either a pair of nonnegative integers or a pair of negative integers. Indeed it gives a zero or pole according as they are nonnegative or negative.

(3) We have the identification

ζ_{p,q}(s) ≅ ζ(s, F_{p}) ⊗ ζ(s, F_{q}).

(4) The function ζ_{p,q}(s) satisfies a functional equation:

ζ_{p,q}(−s) = ζ_{p,q}(s)^{−1}(pq)^{\frac{s}{2}} (1 − p^{−s})(1 − q^{−s})

× exp (\frac{i log p log q}{4π} s^{2} − \frac{πi}{6} (\frac{log q}{log p} + \frac{log p}{log q} + 3)) .

When p = q the result is as follows:

Theorem 1.2 Let

ζ_{p,p}(s) := exp (\frac{i}{2π} ∑_{n=1}^{∞} \frac{1}{n^{2}} p^{−ns} − (1 − \frac{i log p}{2π}s) ∑_{n=1}^{∞} \frac{1}{n}p^{−ns})

in Re(s) > 0. Then the function ζ_{p,p}(s) has the following properties:

(0) It converges absolutely in Re(s) > 0.

(1) The function ζ_{p,p}(s) has an analytic continuation to all s ∈ C as a meromorphic function of order two.

(2) All zeros and poles of ζ_{p,p}(s) are located at

s = \frac{2πin}{log p},

which gives a zero or pole of order |n + 1|, according as n is a nonnegative or negative integer.

(3) We have the identification

ζ_{p,p}(s) ≅ ζ(s, F_{p}) ⊗ ζ(s, F_{p}).

(4) The function ζ_{p,p}(s) satisfies a functional equation:

ζ_{p,p}(−s) = ζ_{p,p}(s)^{−1}p^{s}(1 − p^{−s})^{2} exp (\frac{i(log p)^{2}}{4π} s^{2} − \frac{5πi}{6}) .

The proof of Theorems 1.1 and 1.2 were done by constructing the signed double Poisson summation formula, and the theory of multiple sine functions developed in [17] are essential.

From our viewpoint, it is very interesting to see the nature of ∏_{p,q} ζ_{p,q}(s). Unfortunately, however, it does not converge even for sufficiently large Re(s). Our “α-version” ζ_{p,q}^{α}(s) treated below remedies the situation. In passing we find the analyticity of the diagonal Euler product:

Theorem 1.3 Let

Z(s) = ∏_{p} ζ_{p,p}(s).

Then, Z(s) is absolutely convergent in Re(s) > 1, and it has an analytic continuation with singularities to Re(s) > 0 with the natural boundary Re(s) = 0.

Later in [19] we constructed “the double Riemann zeta function” ζ(s, Z) ⊗ ζ(s, Z) by establishing the signed double explicit formula which is stated in the following theorem. It generalizes the signed double Poisson summation formula used in the proof of Theorems 1.1 and 1.2.

For simplicity put ξ(s) = ζ̂(s + \frac{1}{2}) = ζ̂(s + \frac{1}{2}, Z) for ζ̂(s) = Γ_{R}(s)ζ(s) with Γ_{R}(s) = π^{−s/2}Γ(s/2). The functional equation of ζ(s) is written as ξ(s) = ξ(−s). We recall that nontrivial zeros of ζ(s) are zeros in the strip |Re(s − \frac{1}{2})| < 1/2. We denote by \frac{1}{2} + iγ such a zero, where γ is a complex number in −1/2 < Im(γ) < 1/2.

Hereafter let h(t) be an odd regular function in |Re(t)| < 1 satisfying h(t) = O(|t|^{−3}) as |t| → ∞. We put H_{α}(t) := h(2α + it) and

H̃(u) := ∫_{−∞}^{∞} H(t)e^{itu}dt.

Theorem 1.4 Let 1/2 < α < 1. We have

∑_{Re(γ_{1}),Re(γ_{2})>0} H_{0}(γ_{1} + γ_{2}) = ∑_{{p,q \atop p≠q}} H_{p,q}^{α} + ∑_{p} H_{p,p}^{α} + ∑_{p} H_{p,∞}^{α} + H_{∞,∞}^{α} + H_{0}^{α},

where the sum in the left hand side is taken over pairs (\frac{1}{2} + iγ_{1}, \frac{1}{2} + iγ_{2}) of nontrivial zeros of the Riemann zeta function, the sum in the right hand side is taken over pairs of distinct primes p, q or primes p, and we define for pairs of distinct primes p, q as

H_{p,q}^{α} = \frac{i}{4π^{2}} ∑_{m,n} \frac{log p log q}{log(p^{m}q^{n})} \frac{1}{p^{m(α+ \frac{1}{2} )}q^{n(α+ \frac{1}{2} )}} (H̃_{0}(−m log p) + H̃_{0}(−n log q))  (1.1)

+\frac{i}{4π^{2}} ∑_{m,n} \frac{log p log q}{log(\frac{p^{m}}{q^{n}})} \frac{1}{p^{m(α+ \frac{1}{2} )}q^{n(α+ \frac{1}{2} )}} (H̃_{α}(−m log p) − H̃_{α}(−n log q)) , (1.2)

and for a prime p,

H_{p,p}^{α} = \frac{i}{4π^{2}} ∑_{m,n} \frac{log p}{(m + n)p^{(m+n)(α+ \frac{1}{2} )}} (H̃_{0}(−m log p) + H̃_{0}(−n log p))  (1.3)

+\frac{i}{4π^{2}} ∑_{m≠n} \frac{log p}{(m − n)p^{(m+n)(α+ \frac{1}{2} )}} (H̃_{α}(−m log p) − H̃_{α}(−n log p))  (1.4)

+\frac{1}{4π^{2}} (log p)^{2} ∑_{m=1}^{∞} p^{−2m(α+ \frac{1}{2} )}{ \atop tH_{α}(t)}(−m log p),  (1.5)

H_{p,∞}^{α} = −\frac{1}{2π^{2}} ∑_{m=1}^{∞} \frac{log p}{p^{m(α+ \frac{1}{2} )}} ∫_{−∞}^{∞} p^{−imt}H_{α}(t) ∫_{0}^{t} p^{imt′} \frac{Γ′_{R}}{Γ_{R}} (α + \frac{1}{2} + it′) dt′dt  (1.6)

−\frac{1}{2π^{2}} ∑_{m=1}^{∞} \frac{log p}{p^{m(α+ \frac{1}{2} )}} ∫_{−∞}^{∞} H_{0}(t) ∫_{0}^{t} Re (p^{im(t−t′)} \frac{Γ′_{R}}{Γ_{R}} (α + \frac{1}{2} + it′)) dt′dt, (1.7)

H_{∞,∞}^{α} = \frac{1}{4π^{2}} ∫_{−∞}^{∞} H_{α}(t) ∫_{0}^{t} \frac{Γ′_{R}}{Γ_{R}} (α + \frac{1}{2} + it_{1}) \frac{Γ′_{R}}{Γ_{R}} (α + \frac{1}{2} + i(t − t_{1})) dt_{1}dt (1.8)
+\frac{1}{4π^{2}} ∫_{−∞}^{∞} H_{0}(t) ∫_{0}^{t} \frac{Γ′_{R}}{Γ_{R}} (α + \frac{1}{2} + it_{1}) \frac{Γ′_{R}}{Γ_{R}} (α + \frac{1}{2} − i(t − t_{1})) dt_{1}dt, (1.9)

and

H_{0}^{α} = −\frac{α}{π} ∫_{0}^{π} ∑_{Re(γ_{1})>0} h(iγ_{1} + αe^{iθ})\frac{ξ′}{ξ} (αe^{iθ})e^{iθ}dθ  (1.10)

−\frac{α^{2}}{4π^{2}} ∫_{0}^{π} ∫_{0}^{π} h(αe^{iθ_{1}} + αe^{iθ_{2}})\frac{ξ′}{ξ} (αe^{iθ_{1}})\frac{ξ′}{ξ} (αe^{iθ_{2}})e^{i(θ_{1}+θ_{2})}dθ_{1}dθ_{2},  (1.11)

where m, n ∈ Z, m, n ≥ 1.

Notice that only pairs of zeros in the upper (or lower) half plane are counted in the left hand side of Theorem 1.4. The method of Cramér [5] is important in the proof.

For defining the (p, q)-Euler factors, we put

h(t) = \frac{1}{(t + s)^{2}} − \frac{1}{(t − s)^{2}}  (1.12)

in Theorem 1.4. We denote by p, q any (finite or infinite) places. We call ζ_{p,q}^{α}(s) a (p, q)-Euler factor of the double Riemann zeta function ζ(s, Z) ⊗ ζ(s, Z) if and only if it holds that

ζ_{p,q}^{α}(s + 1) = exp (∫∫ H_{p,q}^{α}(s)dsds) .

We also denote the remainder factor ζ_{0}^{α}(s) by

ζ_{0}^{α}(s + 1) = exp (∫∫ H_{0}^{α}(s)dsds) .

Note that these definitions implies some ambiguity emerged from the integral constants, that is the factor exp(Q(s)) with Q(s) a polynomial with deg(Q) ≤ 2. The double Riemann zeta function ζ(s, Z) ⊗ ζ(s, Z) is expressed by an Euler product over the pairs of places (p, q). In the following theorem we denote the dilogarithm of order r by Li_{r}(u) = ∑_{n=1}^{∞} \frac{u^{n}}{n^{r}} (r = 1, 2, 3, ...).

Theorem 1.5 The (p, q)-Euler factors of the double Riemann zeta function ζ(s, Z)⊗ζ(s, Z) are described as follows:

(1) For distinct prime numbers p and q, we put

ζ_{p,q}^{α}(s) = exp (\frac{1}{πi} ∑_{m,n} \frac{(log p)(log q)}{(m log p)^{2} − (n log q)^{2}}

(\frac{cosh(mα log p)}{q^{n(α+ \frac{1}{2} )}}p^{−m(s− \frac{1}{2} )} + \frac{n log q}{m log p} \frac{sinh(mα log p)}{q^{n(α+ \frac{1}{2} )}}p^{−m(s− \frac{1}{2} )}

−\frac{m log p}{n log q} \frac{sinh(nα log q)}{p^{m(α+ \frac{1}{2} )}}q^{−n(s− \frac{1}{2} )} − \frac{cosh(nα log q)}{p^{m(α+ \frac{1}{2} )}}q^{−n(s− \frac{1}{2} )}) )

in Re(s) > α + \frac{1}{2}, where the sum is taken over all pairs of all positive integers m and n. Then it is a (p, q)-Euler factor of the double Riemann zeta function, and it has an analytic continuation to the entire plane.

(2) For a prime number p, a (p, p)-Euler factor is given as follows in Re(s) > α + \frac{1}{2}:

ζ_{p,p}^{α}(s) = exp (\frac{2}{πi} ∑_{m≠n} \frac{p^{−m(s− \frac{1}{2} )−n(α+ \frac{1}{2} )}}{m^{2} − n^{2}} (cosh(mα log p) + \frac{n}{m} sinh(mα log p)) (1.13)

+\frac{1}{2πi} ((log p)(s − 1 − 2α) log(1 − p^{−s}) − Li_{2} (p^{−s}) + Li_{2} (p^{−s−2α}))).(1.14)

It has an analytic continuation to the entire plane.

(3) The (p, ∞)-factor ζ_{p,∞}^{α}(s) of the double Riemann zeta function has an analytic continuation to the entire plane, and moreover ∏_{p} ζ_{p,∞}^{α}(s) has an analytic continuation to the entire plane with possible singularities at s = \frac{1}{2} − 2k ± α (k ≥ 0), 1 − 2k (k ≥ 0), −2k (k ≥ 1), ρ − 2k (k ≥ 0), \frac{1}{2} + ρ ± α, \frac{3}{2} ± α with ρ any nontrivial zero of ζ(s).

(4) The (∞, ∞)-factor ζ_{∞,∞}^{α}(s) of the double Riemann zeta function is analytic with possible singularities at s = −2n, −2n + α + \frac{1}{2} with n = 0, 1, 2, ...

Theorem 1.6 The remaining factor ζ_{0}^{α}(s) of the double Riemann zeta function is an analytic function on C with possible singularities at s = ρ + \frac{1}{2} + sgn(Im(ρ))αe^{iθ} with 0 ≤ θ ≤ π for any nontrivial zero ρ of ζ(s) and at s belonging to |s − 1| ≤ 2α.

In the next theorem we use the half Riemann zeta function ζ_{+}(s) studied in [11] and the multiple gamma function Γ_{r}(s) of Barnes [4].

Theorem 1.7 The Euler product for the double Riemann zeta function

ζ(s, Z) ⊗ ζ(s, Z) ≅ (∏_{p,q} ζ_{p,q}^{α}(s)) ζ_{0}^{α}(s) (\frac{∏_{m=1}^{∞}ζ_{+}(s + 2m)}{ζ_{+}(s − 1)})^{2} Γ_{2} (\frac{s}{2})^{−1} Γ_{1}(s)^{2}s(s − 2)

is absolutely convergent in Re(s) > α + \frac{3}{2}, where (p, q) runs through pairs of all (finite or infinite) places. It has an analytic continuation (with singularities) to the entire plane and satisfies a functional equation between s and 2 − s.

Later Akatsuka [1] successfully eliminated the parameter α from the double Riemann zeta function. He obtained the (p, q)-Euler factor

ζ_{p,q}(s) = exp(\frac{1}{πi} ∑_{p} ∑_{m=1}^{∞} ∑_{q} ∑_{\begin{matrix}n=1 \\ q^{n}≠p^{m} \end{matrix}}^{∞} \frac{p^{−m(s−1)}q^{−n} log p}{n(m log p − n log q)}

− \frac{1}{πi} ∑_{p} ∑_{m=1}^{∞} ∑_{q} ∑_{n=1}^{∞} \frac{p^{−ms}q^{−n} log p}{n(m log p + n log q)}),

and proved that the double Euler product

ζ^{⊗2}(s) = ∏_{p,q} ζ_{p,q}(s)

satisfies the similar property as in Theorem 1.5.

For r ≥ 0 and x, q_{j} ∈ C (j = 1, ..., r) we define the elliptic gamma function of order r by

G_{r}(x; q_{1}, ..., q_{r}) := ∏_{n_{1},...,n_{r}≥0}(1 − xq_{1}^{n_{1}} ⋯ q_{r}^{n_{r}}).  (1.15)

We conventionally put G_{0}(x) = 1 − x. The product in (1.15) is absolutely convergent when |q_{j}| < 1 for j = 1, ..., r.

The function (1.15) was originally dealt with by Appell [2]. He actually considered

O_{q}(x; ω_{1}, ..., ω_{r}) = ∏_{n_{1},...,n_{r}≥0}(1 − q^{n_{1}ω_{1}+···+n_{r}ω_{r}+x})
= G_{r}(q^{x}; q^{ω_{1}}, ..., q^{ω_{r}})

with Re(ω_{j}) > 0 (j = 1, ..., r) and 0 < q < 1.

The elliptic gamma function (1.15) has another expression (Kurokawa-Wakayama [21, Proposition 2.1]).

G_{r}(x; q_{1}, ..., q_{r}) = exp (− ∑_{m=1}^{∞} \frac{x^{m}}{m(1 − q_{1}^{m}) ⋯ (1 − q_{r}^{m})}) .  (1.16)

The sum in (1.16) is absolutely convergent for suitable q_{j} ∈ C outside of |q_{j}| < 1. For example, it is valid for q_{j} = e^{2πiα_{j}} with α_{j} ∈ R Z such that either α_{j} ∈ (\overline{Q} \ Q) ∩ R (Roth) or α_{j} = log p/ log q with p and q distinct primes (Baker).

In [21, Theorem 1.1] Kurokawa and Wakayama proved the following theorem.

Theorem 1.8 Let p_{1}, ..., p_{r} be distinct prime numbers. Then for Re(s) > 0,

ζ(s, F_{p_{1}}) ⊗ ⋯ ⊗ ζ(s, F_{p_{r}})

= (G_{r−1} (p_{1}^{−s}; exp (\frac{2πi log p_{1}}{log p_{2}} ) , ⋯, exp(\frac{2πi log p_{1}}{log p_{r}} )) × ⋯

× G_{r−1} (p_{r}^{−s}; exp (\frac{2πi log p_{r}}{log p_{1}} ) , ⋯, exp (\frac{2πi log p_{r}}{log p_{r−1}} )))^{(−1)^{r}}.

Here, G_{r−1} is considered under the expression (1.16) and the convergence comes from a delicate transcendency result of Baker [3, Theorem 3.1]

In particular, when r = 2, we have for distinct prime numbers p and q

ζ(s, F_{p}) ⊗ ζ(s, F_{q}) = G_{1} (p^{−s}; exp (\frac{2πi log p}{log q})) G_{1} (q^{−s}; exp (\frac{2πi log q}{log p} )) .

As ζ(s, F_{p_{1}}) ⊗ ⋯ ⊗ ζ(s, F_{p_{r}}) is a meromorphic function of order r, we reach the following expectation.

Expectation.

Zeta functions which are meromorphic of order r have Euler products with Euler factors being expressed in terms of G_{r−1}.

Example. 1.1 (Dedekind zeta functions (r = 1))

Let K be a number field. The Dedekind zeta function of K is

ζ_{K}(s) = ∏_{p}(1 − N(p)^{−s})^{−1} = ∏_{p} G_{0}(N(p))^{−1},

where p runs through all maximal ideals of O_{K} and N(p) = |O_{K}/(p)|.

Example. 1.2 (Selberg zeta functions for SL(2, R) (r = 2))

Let Γ ⊂ SL(2, R) be a discrete subgroup. The Selberg zeta function of Γ is

Z_{Γ}(s) = ∏_{n=0}^{∞} ∏_{p}(1 − N(p)^{−s−n}) = ∏_{p} G_{1}(N(p)^{−s}; N(p)^{−1}),

where p runs through all primitive hyperbolic conjugacy classes of Γ, and N(p) is the larger square of the eigenvalues of p.

Example. 1.3 (Selberg zeta functions for SL(2, C) (r = 3))

Let Γ ⊂ SL(2, C) be a discrete subgroup. The Selberg zeta function of Γ is

Z_{Γ}(s) = ∏_{p} ∏_{{k≥0, l≥0 \atop k≡l (mod νp)}}(1 − a(p)^{−2k}\overline{a(p)}^{−2l}N(p)^{−s}),

where p runs through certain hyperbolic conjugacy classes of Γ, the eigenvalues of p are denoted by a(p) and \overline{a(p)} with |a(p)| > 1, and N(p) = |a(p)|^{2}. The symbol ν_{p} is defined as ν_{p} = |(Γ_{p})_{tor}|, the order of the torsion of the centralizer of p in Γ. For such hyperbolic classes p that ν_{p} = 1, the Euler factor has the form

∏_{l=0}^{∞} ∏_{k=0}^{∞}(1 − a(p)^{−2k}\overline{a(p)}^{−2l}N(p)^{−s}) = G_{2}(N(p)^{−s}; a(p)^{−2}, \overline{a(p)}^{−2}).

2 Absolute Weil Zeta Functions

For a group G we denote F_{1}[G] = G ∪ {0}, where 0 is an element satisfying 0 · g = g · 0 = 0 for any x ∈ G. For a positive integer m ≥ 1, we denote

F_{1^{m}} = F_{1}[μ μ μ_{m}]
= {0} ∪ μ μ μ_{m}
= {0} ∪ {z ∈ C | z^{m} = 1}.

We define an F_{1}-algebra as a multiplicative monoid with 0.

Let 1 ≤ r ∈ Z. For a Z-algebra A, we recall that the noncommutative Hasse zeta function of A as

ζ_{Z}^{r}(s, A) = exp ( ∑_{p: prime} ∑_{m=1}^{∞} \frac{|Hom(A, M_{r}(F_{p^{m}}))|}{m}p^{−ms}) ,  (2.1)

where M_{r}(F_{p^{m}}) is the ring of matrices over F_{p^{m}} of size r, and Hom is the set of ring homomorphisms.

When r = 1 and A is a commutative finitely generated Z-algebra, the definition (2.1) coincides with the usual Hasse zeta function. Namely, we have an expression

ζ_{Z}^{1}(s, A) = ∏_{M⊂A maximal ideal}(1 − N(M)^{−s})^{−1}

with N(M) = |A/M|. When r ≥ 2 and A is a commutative finitely generated Z-algebra, Kurokawa [14] and Fukaya [10] study some analogous versions of (2.1).

When A is an F_{1}-algebra, we analogously define

ζ_{F_{1}}^{r}(s, A) = exp (∑_{m=1}^{∞} \frac{|Hom(A, M_{r}(F_{1^{m}}))|}{m}e^{−ms}) ,  (2.2)

where Hom is the set of monoid homomorphisms.

When r = 1 and A is a finitely generated abelian monoid, which is an F_{1}-algebra, (2.2) was studied in [8] and [13]. The explicit form and the functional equation of ζ_{F_{1}}(s, A) are given by the following proposition and theorem.

Proposition 2.1 ([8],Proposition 2.1) Assume A = F_{1}[G] = G ∪ {0} with G a finitely generated multiplicative abelian group of rank r. Put

G ≅ Z^{r} × μ μ μ_{n_{1}} × μ μ μ_{n_{2}} × ⋯ × μ μ μ_{n_{k}}

with n_{1}|n_{2}| ⋯ |n_{k}. Then

ζ_{F_{1}}(s, A) = \begin{cases}∏_{\underline{d}|\underline{n}} (1 − e^{−|\underline{d}|s})^{−φ(\underline{d})d_{1}^{k−2}···d_{k}^{−1}} for r = 0, \\ ∏_{\underline{d}|\underline{n}} exp (\frac{g_{r}(e^{−|\underline{d}|s})φ(\underline{d})d_{1}^{r+k−1}···d_{k}^{r−1}}{(1−e^{−|\underline{d}|s})^{r}} ) for r ≥ 1, \end{cases}

where the notation \underline{d}|\underline{n} means that the product is over all tuples

\underline{d} = (d_{1}, …, d_{k}) ∈ N^{k}

such that d_{1}|n_{1}, d_{2}|\frac{n_{2}}{d_{1}} ,..., d_{k}|\frac{n_{k}}{d_{1}···d_{k−1}} . Further, we put

|\underline{d}| = d_{1} ⋯ d_{k}

and

φ(\underline{d}) = φ(d_{1}) ⋯ φ(d_{k}).

Theorem 2.1 ([13],Theorem 2) (i) Assume A = F_{1}[G] = G ∪ {0} with G a finite abelian group of order n. Put

G ≅ μ μ μ_{n_{1}} × μ μ μ_{n_{2}} × ⋯ × μ μ μ_{n_{k}}

with n_{1}|n_{2}| ⋯ |n_{k} and n = n_{1} ⋯ n_{k}. Then

ζ_{F_{1}}(s, A) = det (1 − Φ_{A}e^{−s})^{−1/n}  (2.3)

and the following functional equation holds:

ζ_{F_{1}}(−s, A) = w(A)e^{−ns}ζ_{F_{1}}(s, A),

where w(A) is a complex number of modulus 1 satisfying

w(A) = (−1)^{n} det(Φ_{A})^{\frac{1}{n}} ,

and where we define the absolute Frobenius operator Φ_{A} on the group

G^{(2)} = μ μ μ_{n_{1}^{2}} × μ μ μ_{n_{2}^{2}} × ⋯ × μ μ μ_{n_{k}^{2}}

as

Φ_{A}(α_{1}, ..., α_{k}) = (α_{1}^{n_{1}+1}, ..., α_{k}^{n_{k}+1}).  (2.4)

(ii) Assume A = F_{1}[G] = G ∪ {0} with G a finitely generated free abelian group of rank 1.

Put

G ≅ Z × μ μ μ_{n_{1}} × μ μ μ_{n_{2}} × ⋯ × μ μ μ_{n_{k}}

with n_{1}|n_{2}| ⋯ |n_{k}. Then the following functional equations hold:

ζ_{F_{1}}(−s, A) = ζ_{F_{1}}(s, A)^{−1} ∏_{\underline{d}|\underline{n}} e^{−φ(\underline{d})d_{1}^{k−1}···d_{k}}.

(iii) If A = F_{1}[G] = G ∪ {0} with G is a finitely generated abelian group of rank r ≥ 2, the following functional equation holds:

ζ_{F_{1}}(−s, A) = ζ_{F_{1}}(s, A)^{(−1)^{r}}.

Above all the determinant expression (2.3) was crutial. By this we proved an analogue of the Riemann hypothesis as well as a tensor structure of the zeta functions ([13]). We generalize (2.3) to r ≥ 2 in the following theorem.

Theorem 2.2 Assume all prime divisors of n are bigger than r. In other words, we assume that

p | n  ⟹  p > r.

Let A = F_{1}[G] = G ∪ {0} with G a finite abelian group of order n. Then ζ_{F_{1}}^{r}(s, A) has an Euler product

ζ_{F_{1}}^{r}(s, A) = ∏_{d|n}(1 − e^{−ds})^{− \frac{1}{d} (d^{r}− ∑_{d′|d} φ(d′)^{r})}  (2.5)

and a determinant expression

ζ_{F_{1}}^{r}(s, A) = det (1 − Φ_{A}^{⊗r}e^{−s})^{−1/n^{r}} ,  (2.6)

where Φ_{A} is defined by (2.4) which is a square matrix of size n^{2}, and Φ_{A}^{⊗r} denotes the Kronecker tensor power, which is a square matrix of size n^{2r}.

Proof. Put

G ≅ μ μ μ_{n_{1}} × μ μ μ_{n_{2}} × ⋯ × μ μ μ_{n_{k}}

with n_{1}|n_{2}| ⋯ |n_{k} and n = n_{1} ⋯ n_{k}. We first claim

Hom(A, M_{r}(F_{1^{m}})) = (n_{1}, m)^{r} ⋯ (n_{k}, m)^{r}.  (2.7)

Then (2.5) follows from a direct calculation. For proving (2.6), it suffices to prove that

Hom(μ μ μ_{l}, M_{r}(F_{1^{m}})) = (l, m)^{r},

by our considering the image of a generator ζ_{l} of each μ μ μ_{l} (l = n_{1}, ..., n_{k}).

Since the image of ζ_{l} has to be invertible, it holds that

|Hom(μ μ μ_{l}, M_{r}(F_{1^{m}}))| = |Hom(μ μ μ_{l}, GL_{r}(F_{1^{m}}))|.

Here GL_{r}(F_{1^{m}}) is the group of invertible matrices in the form of a permutation matrix with the entry 1 replaced by any element in μ μ μ_{m}.

Denote by α the image of ζ_{l} by a homomorphism. By the assumption, α is a diagonal matrix with diagonal entries in μ μ μ_{m}, because the order of α must be equal to l, not divisible

by any positive integer less than l. Any element α ∈ (\begin{matrix}μ μ μ_{m} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & μ μ μ_{m} \end{matrix}) with α^{l} = 1 can be

an image. Thus

|Hom(μ μ μ_{l}, M_{2}(F_{1^{m}}))| = |\{α ∈ (\begin{matrix}μ μ μ_{m} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & μ μ μ_{m} \end{matrix}) : α^{l} = 1\}| = (l, m)^{r}.

This proves (2.7). Then we have

ζ_{F_{1}}^{r}(s, A) = exp (∑_{m=1}^{∞} \frac{|Hom(A, M_{r}(F_{1^{m}}))|}{m}e^{−ms})

= exp (∑_{m=1}^{∞} \frac{(n_{1}, m)^{r} ⋯ (n_{k}, m)^{r}}{m} e^{−ms}) .

For proving (2.6), it suffices to show that

∑_{m=1}^{∞} \frac{n^{r}(n_{1}, m)^{r} ⋯ (n_{k}, m)^{r}}{m} e^{−ms} = ∑_{m=1}^{∞} \frac{tr((Φ_{A}^{⊗r})^{m})}{m}e^{−ms}  (2.8)

by taking the logarithm of (2.6). From a general identity tr((A ⊗ B)^{m}) = tr(A^{m} ⊗ B^{m}) = tr(A^{m})tr(B^{m}), we have

tr((Φ_{A}^{⊗r})^{m}) = tr((Φ_{A}^{m})^{r}).

On the other hand, it holds by [13, Lemma 1] that

tr(Φ_{A}^{m}) = n |Hom(A,μ μ μ_{m})| = n(n_{1}, m) ⋯ (n_{k}, m).

These identities lead to (2.8).

Example. 2.1 For any prime p > r, it holds that

ζ_{F_{1}}^{r}(s, F_{1^{p}}) = (1 − e^{−s})^{−1}(1 − e^{−ps})^{− \frac{p^{r}−1}{p}} .

By this theorem, it is easily shown that ζ_{F_{1}}^{r}(s, A) satisfies an analog of the Riemann hypothesis and the tensor structure concerning zeros and poles. The proofs are the same as in [13, Theorem 3]. The functional equation is also obtained as follows.

Corollary 1 The zeta function in the preceding theorem satisfies the funcitonal equation

ζ_{F_{1}}^{r}(−s, A) = ζ_{F_{1}}^{r}(s, A)(−1)^{n^{r}}det(Φ_{A})^{rn^{r−2}}e^{−n^{r}s}.

Proof. In the previous paper [13, Proposition 1] we obtained the functional equation of the zeta function ζ_{σ}(s) = det(1 − σe^{−s})^{−1} of a finite permutation matrix σ over N elements as

ζ_{σ}(−s) = (−1)^{N}det(σ)e^{−Ns}ζ_{σ}(s).

Identifying Φ_{A}^{⊗r} with a permutation matrix over n^{2r} elements, we compute by the theorem that

ζ_{F_{1}}^{r}(−s, A) = ζ_{Φ_{A}^{⊗r}}(−s)^{\frac{1}{n^{r}}}
= (ζ_{Φ_{A}^{⊗r}}(s)(−1)^{n^{2r}}det(Φ_{A}^{⊗r})e^{−n^{2r}s})^{\frac{1}{n^{r}}}
= (ζ_{Φ_{A}^{⊗r}}(s)(−1)^{n^{2r}}det(Φ_{A})^{rn^{2(r−1)}}e^{−n^{2r}s})^{\frac{1}{n^{r}}}
= ζ_{F_{1}}^{r}(s, A)^{\frac{1}{n^{r}}} (−1)^{n^{r}}det(Φ_{A})^{rn^{r−2}}e^{−n^{r}s}.

Investigating the cases when n has small prime divisors is an interesting problem. It seems that the situation is much more complicated. We give some examples below.

Example. 2.2 When A = F_{1}[μ μ μ_{2} × μ μ μ_{2}], we have

ζ_{F_{1}}^{2}(s, A) = (1 − e^{−s})^{−25} exp (\frac{10 − 36e^{−s} + 9e^{−2s} + 18e^{−3s}}{(1 − e^{−s})^{2}} ) .

Proof. The result follows from

|Hom(A, M_{2}(F_{1^{m}}))| = \begin{cases}(m + 5)^{2} & m ≥ 3 \\ (m + 2)^{2} & m = 1, 2 \end{cases} .

Example. 2.3 When A = F_{1^{3}}, we have

ζ_{F_{1}}^{3}(s, A) = (1 − e^{−s})^{−1}(1 − e^{−3s})^{− \frac{20}{3}} exp (3\frac{e^{−3s}(1 + 2e^{−3s})}{(1 − e^{−3s})^{2}}) .

Proof. The result follows from

|Hom(A, M_{3}(F_{1^{m}}))| = \begin{cases}1 & 3 |̸ m \\ 27 + 2^{2} & 3 | m \end{cases} .

Example. 2.4 When A = F_{1^{2}}, we have

ζ_{F_{1}}^{2}(s, A) = (1 − e^{−s})^{−5} exp (\frac{−4 + 3e^{−s} + 3e^{−2s}}{2(e^{−s} − 1)} ) .

Proof. The result follows from

|Hom(A, M_{2}(F_{1^{m}}))| = \begin{cases}m + 5 & m ≥ 3 \\ m + 2 & m = 1, 2 \end{cases} .

Example. 2.5 For A = F_{1^{2}}, we have

ζ_{F_{1}}^{r}(s, A) = exp (∑_{l=1}^{∞} \frac{r!}{(2l − 1)!} ∑_{k=0}^{[r/2]} \frac{e^{−2ls}}{2^{k}(r − 2k)!} (\frac{(2l)^{k}}{2l − 1}e^{s} + \frac{(2l + 1)^{k}}{(2l)^{2}} 2^{2(l−k)}))

Proof. The result follows from

|Hom(A, M_{r}(F_{1^{m}}))| = \begin{cases}∑_{k=0}^{[r/2]} \frac{r!(m + 1)^{k}}{2^{k}(r − 2k)!} & 2 |̸ m \\ ∑_{k=0}^{[r/2]} \frac{r!(m + 1)^{k}}{2^{k}(r − 2k)!}2^{m−2k} & 2 | m \end{cases} .

In what follows we give some examples for the case when A is infinite. We define the Euler polynomials g_{r}(T) ∈ ℤ[T] (r = 1, 2, 3, ...) by

g_{1}(T) = T and g_{r+1}(T) = ∑_{k=1}^{r} \binom{r}{k − 1}(T − 1)^{r−k}g_{k}(T).  (2.9)

For example,

g_{1}(T) = T,
g_{2}(T) = T,
g_{3}(T) = T^{2} + T,
g_{4}(T) = T^{3} + 4T^{2} + T.

In [8, Lemma 2.2], we proved the following lemma. Lemma 2.1 For r = 1, 2, 3, ..., we have

∑_{ν=1}^{∞} ν^{r−1}T^{ν} = \frac{g_{r}(T)}{(1 − T)^{r}} ,

where g_{r}(T) ∈ ℤ[T] is defined by (2.9). By using this lemma, we prove the following. Example. 2.6 When A = F_{1}[T^{±}], the explicit form of ζ_{F_{1}}^{r}(s, A) is given by

ζ_{F_{1}}^{r}(s, A) = exp (\frac{g_{r}(e^{−s})r!}{(1 − e^{−s})^{r}} ) .

It satisfies the functional equation ζ_{F_{1}}^{r}(−s, A) = ζ_{F_{1}}^{r}(s, A)^{(−1)^{r}}.

Proof. Since an element in Hom(A, M_{r}(F_{1^{m}})) is determined by the image of T, which must be invertible. Thus

|Hom(A, M_{r}(F_{1^{m}}))| = |GL_{r}(F_{1^{m}})| = m^{r}r!.

Then we compute

ζ_{F_{1}}^{r}(s, A) = exp (∑_{m=1}^{∞} \frac{m^{r}r!}{m}e^{−ms})

= exp (∑_{m=1}^{∞} m^{r−1}e^{−ms})^{r!}

= exp (\frac{g_{r}(e^{−s})r!}{(1 − e^{−s})^{r}} ) .

Example. 2.7 When A = F_{1}[T], the explicit form of ζ_{F_{1}}^{r}(s, A) is given by

ζ_{F_{1}}^{r}(s, A) == (1 − e^{−s})^{−1} exp (∑_{k=1}^{r} \binom{r}{k}r^{k} \frac{g_{k}(e^{−s})}{(1 − e^{−s})^{k}} ) .

Proof. The element in Hom(A, M_{r}(F_{1^{m}})) is determined by the image of T, which must belong to M_{r}(F_{1^{m}}) = End(F_{1^{m}}^{r}). Thus

|Hom(A, M_{r}(F_{1^{m}}))| = |End(F_{1^{m}}^{r})| = (rm + 1)^{r}.

Then

ζ_{F_{1}}^{r}(s, A) = exp (∑_{m=1}^{∞} \frac{(rm + 1)^{r}}{m} e^{−ms})

= exp (∑_{m=1}^{∞} \frac{1}{m} ∑_{k=0}^{r} \binom{r}{k}(rm)^{k}e^{−ms})

= exp (∑_{m=1}^{∞} (\frac{1}{m} + ∑_{k=1}^{r} \binom{r}{k}r^{k}m^{k−1}) e^{−ms})

= (1 − e^{−s})^{−1} exp (∑_{k=1}^{r} \binom{r}{k}r^{k} ∑_{m=1}^{∞} m^{k−1}e^{−ms})

= (1 − e^{−s})^{−1} exp (∑_{k=1}^{r} \binom{r}{k}r^{k} \frac{g_{k}(e^{−s})}{(1 − e^{−s})^{k}} )

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Shin-ya Koyama: The Institute of Mathematical Sciences, Ewha Womans University, Daehyun-dong 11-1, Sedaemoon-ku, 120-750, Seoul, South Korea. Department of Biomedical Engineering, Toyo University, 2100 Kujirai, Kawagoe-shi 350-8585, Japan. koyama@toyonet.toyo.ac.jp

Nobushige Kurokawa: Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan. kurokawa@math.titech.ac.jp