Normalized Double Sine Functions

Nobushige Kurokawa Shin-ya Koyama

September 3, 2002

Abstract

We express normalized double sine functions of integer periods (N_{1}, N_{2}) via the standard double sine function of period (1, 1). As an application we give an Euler product expression using the di-logarithm for the double zeta function ζ(s, F_{p^{N_{1}}} ) ⊗ ζ(s, F_{p^{N_{2}}} ) for a prime number p and integers N_{1}, N_{2}.

1 Definitions and Results

Normalized multiple sine functions are generalizations of the usual sine function. We studied their basic properties in previous papers [KuKo] [KoKu1] [KoKu2] with some applications.

For ω_{1}, ..., ω_{r} > 0 and x > 0, the multiple Hurwitz zeta function is defined by Barnes [B] as

ζ_{r}(s, x, (ω_{1}, ..., ω_{r})) = ∑_{n_{1},...,n_{r}=0}^{∞}(n_{1}ω_{1} + ⋯ + n_{r}ω_{r} + x)^{−s}

in Re(s) > r. This has the analytic continuation to all s ∈ ℂ as a meromorphic function, and it is holomorphic at s = 0. Then the normalized multiple gamma function is defined as

Γ_{r}(x, (ω_{1}, ..., ω_{r})) = exp ( \frac{∂}{∂s}ζ_{r}(s, x, (ω_{1}, ..., ω_{r}))|_{s=0}) .

This is a constant multiple of the multiple gamma function Γ_{r}^{B}(x, (ω_{1}, ..., ω_{r})) of Barnes [B]:

Γ_{r}(x, (ω_{1}, ..., ω_{r})) = Γ_{r}^{B}(x, (ω_{1}, ..., ω_{r}))/ρ_{r}(ω_{1}, ..., ω_{r}).

Now, the normalized multiple sine function is

S_{r}(x, (ω_{1}, ..., ω_{r})) = Γ_{r}(x, (ω_{1}, ..., ω_{r}))^{−1}Γ_{r}(ω_{1} + ⋯ + ω_{r} − x, (ω_{1}, ..., ω_{r}))^{(−1)^{r}}.

For example

S_{1}(x, ω) = Γ_{1}(x, ω)^{−1}Γ_{1}(ω − x, ω)^{−1}
= 2 sin(πx/ω),

since we have Γ_{1}(x, ω) = (2π)^{−1/2}Γ(x/ω)ω^{\frac{x}{ω} − \frac{1}{2}} from ζ_{1}(s, x, ω) = ω^{−s}ζ(s, x/ω).

To simplify the notation we put S_{r}(x) = S_{r}(x, (1, ..., 1)), Γ_{r}(x) = Γ_{r}(x, (1, ..., 1)) and ζ_{r}(s, x) = ζ_{r}(s, x, (1, ..., 1)). Hence

S_{r}(x) = Γ_{r}(x)^{−1}Γ_{r}(r − x)^{(−1)^{r}}

and

Γ_{r}(x) = exp ( \frac{∂}{∂s}ζ_{r}(s, x)|_{s=0}) .

Here we investigate normalized double sine functions, especially in the rational periods case: S_{2}(x, (ω_{1}, ω_{w})) with ω_{2}/ω_{1} ∈ ℚ. The following theorem expresses them in terms of S_{2}(x):

Theorem 1.1 Let N_{1}, N_{2} be positive integers with the greatest common divisor N_{0}. Then we have

S_{2}(x, (N_{1}, N_{2})) = ∏_{k_{1}=0}^{(N_{2}/N_{0})−1} ∏_{k_{2}=0}^{(N_{1}/N_{0})−1} S_{2} (\frac{x + N_{1}k_{1} + N_{2}k_{2}}{N_{1}N_{2}/N_{0}} ) .  (1.1)

As application of Theorem 1.1, we compute the absolute tensor product of the Hasse zeta functions of finite fields with p^{N_{1}} and p^{N_{2}} elements:

Theorem 1.2 Let N_{1}, N_{2} be positive integers with the greatest common divisor N_{0}. The absolute tensor product of the Hasse zeta functions for finite fields F_{p^{N_{1}}} and F_{p^{N_{2}}} is given as follows:

ζ(s, F_{p^{N_{1}}}) ⊗ ζ(s, F_{p^{N_{2}}})

= exp (−\frac{1}{2πi} \frac{N_{0}^{2}}{N_{1}N_{2}} ∑_{n=1}^{∞} \frac{p^{−snN_{1}N_{2}/N_{0}}}{n^{2}} + (\frac{isN_{0} log p}{2π} − 1) ∑_{n=1}^{∞} \frac{p^{−snN_{1}N_{2}/N_{0}}}{n}

+ ∑_{n=1}^{∞} \frac{p^{−snN_{1}}}{n} f_{1}(n) + ∑_{n=1}^{∞} \frac{p^{−snN_{2}}}{n} f_{2}(n) + Q_{p}(s)) ,

where

f_{1}(n) = \begin{cases}(e^{2πinN_{1}/N_{2}} − 1)^{−1} & (\frac{N_{2}}{N_{0}} |̸n) \\ \frac{N_{2}−N_{0}}{2N_{0}} & (\frac{N_{2}}{N_{0}} |n) \end{cases} , f_{2}(n) = \begin{cases}(e^{2πinN_{2}/N_{1}} − 1)^{−1} & (\frac{N_{1}}{N_{0}} |̸n) \\ \frac{N_{1}−N_{0}}{2N_{0}} & (\frac{N_{1}}{N_{0}} |n) \end{cases}

and Q_{p}(s) is a quadratic polynomial in s.

2 Proof of Theorem 1.1

It suffices to show when N_{0} = 1, since the homogeneity [KuKo, Theorem 2.1(e)] of the multiple sine functions gives

S_{2}(x, (N_{1}, N_{2})) = S_{2} (\frac{x}{N_{0}} , (\frac{N_{1}}{N_{0}} , \frac{N_{2}}{N_{0}} )) .

In case N_{0} = 1, the right hand side of (1.1) is calculated as follows:

∏_{k_{1}=0}^{N_{2}−1} ∏_{k_{2}=0}^{N_{1}−1} S_{2} (\frac{x + N_{1}k_{1} + N_{2}k_{2}}{N_{1}N_{2}} )

= ∏_{k_{1}=0}^{N_{2}−1} ∏_{k_{2}=0}^{N_{1}−1} Γ_{2} (\frac{x + N_{1}k_{1} + N_{2}k_{2}}{N_{1}N_{2}} )^{−1} Γ_{2} (2 − \frac{x + N_{1}k_{1} + N_{2}k_{2}}{N_{1}N_{2}} )
= ∏_{k_{1}=0}^{N_{2}−1} ∏_{k_{2}=0}^{N_{1}−1} exp (\frac{∂}{∂s}|_{s=0} (−ζ_{2} (s, \frac{x+N_{1}k_{1}+N_{2}k_{2}}{N_{1}N_{2}} ) + ζ_{2} (s, 2 − \frac{x+N_{1}k_{1}+N_{2}k_{2}}{N_{1}N_{2}} )))
= exp (\frac{∂}{∂s}|_{s=0} ∑_{k_{1}=0}^{N_{2}−1} ∑_{k_{2}=0}^{N_{1}−1} (−ζ_{2} (s, \frac{x+N_{1}k_{1}+N_{2}k_{2}}{N_{1}N_{2}} ) + ζ_{2} (s, 2 − \frac{x+N_{1}k_{1}+N_{2}k_{2}}{N_{1}N_{2}} )))

(2.1)

The double sum is computed as follows:

∑_{k_{1}=0}^{N_{2}−1} ∑_{k_{2}=0}^{N_{1}−1} (− ∑_{m_{1},m_{2}≥0} (m_{1} + m_{2} + \frac{x+N_{1}k_{1}+N_{2}k_{2}}{N_{1}N_{2}} )^{−s}

+ ∑_{m_{1},m_{2}≥0} (m_{1} + m_{2} + 2 − \frac{x+N_{1}k_{1}+N_{2}k_{2}}{N_{1}N_{2}} )^{−s})

= (N_{1}N_{2})^{s} ∑_{k_{1}=0}^{N_{2}−1} ∑_{k_{2}=0}^{N_{1}−1} (− ∑_{m_{1},m_{2}≥0} ((m_{1}N_{2} + k_{1})N_{1} + (m_{2}N_{1} + k_{2})N_{2} + x)^{−s}

+ ∑_{m_{1},m_{2}≥0} ((m_{1}N_{2} + N_{2} − k_{1} − 1)N_{1} + (m_{2}N_{1} + N_{1} − k_{2} − 1)N_{2} + N_{1} + N_{2} − x)^{−s})

= (N_{1}N_{2})^{s}( − ζ_{2}(s, x, (N_{1}, N_{2})) + ζ_{2}(s, N_{1} + N_{2} − x, (N_{1}, N_{2}))).

We previously obtained in the proof of [KuKo, Theorem 2.1(b)] that the function −ζ_{2}(s, x, (N_{1}, N_{2})) + ζ_{2}(s, N_{1} + N_{2} − x, (N_{1}, N_{2})) has zeros at even nonnegative integers s. In particular it vanishes at s = 0, thus (2.1) equals

exp (\frac{∂}{∂s}|_{s=0} ( − ζ_{2}(s, x, (N_{1}, N_{2})) + ζ_{2}(s, N_{1} + N_{2} − x, (N_{1}, N_{2}))))
= Γ_{2}(x, (N_{1}, N_{2}))^{−1}Γ_{2}(N_{1} + N_{2} − x, (N_{1}, N_{2}))
= S_{2}(x, (N_{1}, N_{2})).

This completes the proof of Theorem 1.1.

3 Application

In this section we compute the absolute tensor product of the Hasse zeta functions of finite fields F_{p^{N_{1}}} and F_{p^{N_{2}}}. We first recall the definition of the absolute tensor product of meromorphic functions. Let Z_{j} (j = 1, 2) be meromorphic functions of order μ_{j}. We put the Hadamard product as

Z_{j}(s) = s^{k_{j}}e^{Q_{j}(s)} \begin{matrix}∏ & ′ \\ ρ∈ℂ \end{matrix} P_{μ_{j}} (\frac{s}{ρ})^{m_{j}(ρ)} ,  (3.1)

where P_{r}(u) := (1 − u) exp(u + \frac{u^{2}}{2} + ⋯ + \frac{u^{r}}{r}), m_{j} denotes the multiplicity function with k_{j} := m_{j}(0), and Q_{j} is a polynomial with deg Q_{j} ≤ μ_{j}. Here the product over ρ ∈ ℂ means

lim_{R→∞} ∏_{0<|ρ|<R} P_{μ_{j}} (\frac{s}{ρ})^{m_{j}(ρ)}. The absolute tensor product is defined by

(Z_{1} ⊗ Z_{2})(s) := s^{k_{1}k_{2}}e^{Q(s)}\begin{matrix}∏ & ′ \\ ρ_{1},ρ_{2}∈ℂ \end{matrix}P_{μ_{1}+μ_{2}} (\frac{s}{ρ_{1} + ρ_{2}} )^{m(ρ_{1},ρ_{2})} ,  (3.2)

where Q(s) is a polynomial with deg Q ≤ μ_{1} + μ_{2} and

m(ρ_{1}, ρ_{2}) := m_{1}(ρ_{1})m_{2}(ρ_{2}) × \{ \begin{matrix}1 & if Im(ρ_{1}), Im(ρ_{2}) ≥ 0, \\ −1 & if Im(ρ_{1}), Im(ρ_{2}) < 0, \\ 0 & otherwise. \end{matrix}

Here we do not give the precise definition of the polynomial Q(s), since it is not necessary for our purpose.

In this section we will compute this absolute tensor product for the Hasse zeta functions for finite fields:

Z_{1}(s) = ζ(s, F_{p^{N_{1}}}) = (1 − p^{−N_{1}s})^{−1},
Z_{2}(s) = ζ(s, F_{p^{N_{2}}}) = (1 − p^{−N_{2}s})^{−1},

with p a prime number and N_{1}, N_{2} positive integers.

The following theorem extends our previous results on ζ(s, F_{p})⊗ζ(s, F_{q}) with p, q primes in [KoKu2].

Proposition 3.1 The absolute tensor product of the Hasse zeta functions for finite fields F_{p^{N_{1}}} and F_{p^{N_{2}}} is given as follows:

ζ(s, F_{p^{N_{1}}}) ⊗ ζ(s, F_{p^{N_{2}}}) = e^{Q(s)}S_{2} (\frac{isN_{1}N_{2} log p}{2π}, (N_{1}, N_{2}))

= e^{Q(s)} ∏_{k_{1}=0}^{(N_{2}/N_{0})−1} ∏_{k_{2}=0}^{(N_{1}/N_{0})−1} S_{2} (N_{0} (\frac{is log p}{2π} + \frac{k_{1}}{N_{2}} + \frac{k_{2}}{N_{1}} )) ,

where N_{0} = (N_{1}, N_{2}) and Q(s) is a polynomial of degree at most two, which depends on p.

Proof. The second equality is seen from Theorem 1.1. In what follows we prove the first one. The Hadamard product (3.1) for the Hasse zeta function is given for j = 1, 2 by

ζ(s, F_{p^{N_{j}}}) = s^{−1}e^{Q̃_{N_{j}} (s)}\begin{matrix}∞ \\ ∏ & ′ \\ n=−∞ \end{matrix} P_{1} ( \frac{s}{\frac{2πi}{N_{j} log p}n})^{−1}

with Q̃_{p,j}(s) a linear polynomial depending on N_{j}. Thus by the definition (3.2) of the absolute tensor product,

ζ(s, F_{p^{N_{1}}}) ⊗ ζ(s, F_{p^{N_{2}}}) = se^{Q̃_{N_{1},N_{2}}(s)}\begin{matrix}∏ & ′ \\ k,n∈ℤ \end{matrix}P_{2} ( \frac{s}{\frac{2πi}{N_{1} log p}k + \frac{2πi}{N_{2} log p}n})^{m_{k,n}} ,

where Q̃_{N_{1},N_{2}}(s) is a polynomial of degree at most two and

m_{k,n} := m (\frac{2πi}{N_{1} log p}k, \frac{2πi}{N_{2} log p}n) = \{ \begin{matrix}1 & if k, n ≥ 0 \\ −1 & if k, n < 0 \\ 0 & otherwise. \end{matrix}

Hence

ζ(s, F_{p^{N_{1}}}) ⊗ ζ(s, F_{p^{N_{2}}}) = e^{Q̃_{N_{1},N_{2}}(s)}s\frac{\begin{matrix}∞ \\ ∏ & ′ \\ k,n=0 \end{matrix} P_{2} ( \frac{s}{\frac{2πi}{N_{1} log p}k + \frac{2πi}{N_{2} log p}n})}{∏_{k,n=1}^{∞} P_{2} (− \frac{s}{\frac{2πi}{N_{1} log p}k + \frac{2πi}{N_{2} log p}n})}.

We appeal to the r = 2 case of the formula [KuKo, Proposition 2.4]:

S_{2}(z, (ω_{1}, ω_{2})) = e^{Q\underline{ω}(z)}z\frac{\begin{matrix}∞ \\ ∏ & ′ \\ k,n=0 \end{matrix} P_{2} (−\frac{z}{ω_{1}k + ω_{2}n})}{∏_{k,n=1}^{∞} P_{2} (\frac{z}{ω_{1}k + ω_{2}n})}

where Q\underline{ω}(z) a polynomial with deg Q\underline{ω} ≤ 2. The proof is complete.

Taking the following exponential expression of the normalized double sine functions into account, we see that the absolute tensor product in Theorem 1.2 has an “Euler product” expression for Re(s) > 0.

Proposition 3.2 The following expression holds for Imz > 0.

S_{2}(z) = exp (−\frac{1}{2πi}Li_{2}(e^{2πiz}) + (1 − z) log(1 − e^{2πiz}) + Q(z)) ,

where Q(z) = \frac{πi}{2}z^{2} − πiz + \frac{5πi}{12}.

Proof. We recall the formulas of the double sine functions:

S_{2}(z) = S_{2}(z)^{−1}S_{1}(z),  ([KuKo, Example 3.6])  (3.3)

where S_{r}(z) (r = 1, 2) are the primitive multiple sine functions [KuKo]. We have by definition

S_{1}(z) = 2 sin πz = exp (−πiz + \frac{πi}{2} + log(1 − e^{2πiz}))

and the expression [KuKo, Theorem 2.8 (2.12)]:

S_{2}(z) = exp (\frac{1}{2πi}Li_{2}(e^{2πiz}) + z log(1 − e^{2πiz}) − \frac{πi}{2}z^{2} − \frac{ζ(2)}{2πi})

for Im(z) > 0. Thus (3.3) equals

exp (−\frac{1}{2πi}Li_{2}(e^{2πiz}) + (1 − z) log(1 − e^{2πiz}) + \frac{πi}{2}z^{2} − πiz + \frac{5πi}{12}) .

Lemma 3.3 Assume r ∈ ℂ satisfies that r^{N} = 1. Then we have

\frac{1}{N} ∑_{n=0}^{N−1} nr^{n} = \{ \begin{matrix}(r − 1)^{−1} & (r ≠ 1) \\ \frac{N−1}{2} & (r = 1). \end{matrix}

Proof. The r = 1 case is well-known. Differentiating the formula ∑_{n=0}^{N−1} r^{n} = (1−r^{N})/(1−r) in case r ≠ 1 leads to the result.

Proof of Theorem 1.2: By the above propositions all we should compute is the following product:

∏_{k_{1}=0}^{(N_{2}/N_{0})−1} ∏_{k_{2}=0}^{(N_{1}/N_{0})−1} S_{2} (N_{0} (\frac{is log p}{2π} + \frac{k_{1}}{N_{2}} + \frac{k_{2}}{N_{1}} ))

= exp ( ∑_{k_{1}=0}^{(N_{2}/N_{0})−1} ∑_{k_{2}=0}^{(N_{1}/N_{0})−1} (−\frac{1}{2πi}Li_{2} (e^{−N_{0}s log p+2πi( \frac{N_{0}k_{1}}{N_{2}} + \frac{N_{0}k_{2}}{N_{1}} )})  (3.4)

+ (1 − \frac{isN_{0} log p}{2π} − \frac{N_{0}k_{1}}{N_{2}} − \frac{N_{0}k_{2}}{N_{1}} ) log (1 − e^{−sN_{0} log p+2πi( \frac{N_{0}k_{1}}{N_{2}} + \frac{N_{0}k_{2}}{N_{1}} )}) (3.5)
+Q (\frac{isN_{0} log p}{2π} + \frac{N_{0}k_{1}}{N_{2}} + \frac{N_{0}k_{2}}{N_{1}} ))) .  (3.6)

Put N_{1} = N_{0}N′_{1} and N_{2} = N_{0}N′_{2}. First the double sum in (3.4) is computed as follows:

− ∑_{k_{1}=0}^{N′_{2}−1} ∑_{k_{2}=0}^{N′_{1}−1} \frac{1}{2πi}Li_{2} (e^{−N_{0}s log p+2πi(_{\frac{k_{1}}{N′_{2}}} +_{\frac{k_{2}}{N′_{1}}} )})

= −\frac{1}{2πi} ∑_{k_{1}=0}^{N′_{2}−1} ∑_{k_{2}=0}^{N′_{1}−1} ∑_{n=1}^{∞} \frac{(p^{−sN_{0}}e^{2πi(_{\frac{k_{1}}{N′_{2}}} +_{\frac{k_{2}}{N′_{1}}} )})^{n}}{n^{2}}

= −\frac{1}{2πi} ∑_{n=1}^{∞} \frac{p^{−sN_{0}n}}{n^{2}} ( ∑_{k_{1}=0}^{N′_{2}−1} e^{2πi_{\frac{nk_{1}}{N′_{2}}}} ) ( ∑_{k_{2}=0}^{N′_{1}−1} e^{2πi_{\frac{nk_{2}}{N′_{1}}}} )

= \frac{1}{2πi} − ∑_{N′_{1}N′_{2}|n} \frac{p^{−sN_{0}n}}{n^{2}} N′_{1}N′_{2}

= \frac{1}{2πi} − \frac{1}{N′_{1}N′_{2}} ∑_{n=1}^{∞} \frac{p^{−sN_{0}N′_{1}N′_{2}n}}{n^{2}}

= \frac{1}{2πi} − \frac{N_{0}^{2}}{N_{1}N_{2}} ∑_{n=1}^{∞} \frac{p^{−snN_{1}N_{2}/N_{0}}}{n^{2}} .

Next we compute the double sum of (3.5) over k_{1}, k_{2}:

∑_{k_{1}=0}^{N′_{2}−1} ∑_{k_{2}=0}^{N′_{1}−1} (1 − \frac{isN_{0} log p}{2π} − \frac{k_{1}}{N′_{2}} − \frac{k_{2}}{N′_{1}} ) log (1 − e^{−sN_{0} log p+2πi(_{\frac{k_{1}}{N′_{2}}} +_{\frac{k_{2}}{N′_{1}}} )})

= − (1 − \frac{isN_{0} log p}{2π}) ∑_{n=1}^{∞} \frac{p^{−sN_{0}n}}{n} ∑_{k_{1}=0}^{N′_{2}−1} e^{2πi_{\frac{nk_{1}}{N′_{2}}}} ∑_{k_{2}=0}^{N′_{1}−1} e^{2πi_{\frac{nk_{2}}{N′_{1}}}}

+ ∑_{n=1}^{∞} \frac{p^{−sN_{0}n}}{n} ( ∑_{k_{1}=0}^{N′_{2}−1} ∑_{k_{2}=0}^{N′_{1}−1} (\frac{k_{1}}{N′_{2}} + \frac{k_{2}}{N′_{1}} ) e^{2πi(_{\frac{k_{1}}{N′_{2}}} +_{\frac{k_{2}}{N′_{1}}} )n})

= (\frac{isN_{0} log p}{2π} − 1) ∑_{n=1}^{∞} \frac{p^{−sN_{0}nN′_{1}N′_{2}}}{n}

+ ∑_{n=1}^{∞} \frac{p^{−sN_{0}n}}{n} ( ∑_{k_{1}=0}^{N′_{2}−1} \frac{k_{1}}{N′_{2}} e^{2πi_{\frac{k_{1}n}{N′_{2}}}} ∑_{k_{2}=0}^{N′_{1}−1} e^{2πi_{\frac{k_{2}n}{N′_{1}}}} + ∑_{k_{2}=0}^{N′_{1}−1} \frac{k_{2}}{N′_{1}} e^{2πi_{\frac{k_{2}n}{N′_{1}}}} ∑_{k_{1}=0}^{N′_{2}−1} e^{2πi_{\frac{k_{1}n}{N′_{2}}}} )

= (\frac{isN_{0} log p}{2π} − 1) ∑_{n=1}^{∞} \frac{p^{−sN_{0}nN′_{1}N′_{2}}}{n}

+ ∑_{n=1}^{∞} \frac{p^{−snN_{0}N′_{1}}}{nN′_{1}} ∑_{k_{1}=0}^{N′_{2}−1} \frac{k_{1}}{N′_{2}} e^{2πi \frac{k_{1}nN′_{1}}{N′_{2}}} N′_{1} + ∑_{n=1}^{∞} \frac{p^{−snN_{0}N′_{2}}}{nN′_{2}} ∑_{k_{2}=0}^{N′_{1}−1} \frac{k_{2}}{N′_{1}} e^{2πi \frac{k_{2}nN′_{2}}{N′_{1}}} N′_{2}

= (\frac{isN_{0} log p}{2π} − 1) ∑_{n=1}^{∞} \frac{p^{−sN_{0}nN′_{1}N′_{2}}}{n}

+ ∑_{n=1}^{∞} \frac{p^{−snN_{0}N′_{1}}}{nN′_{2}} ∑_{k_{1}=0}^{N′_{2}−1} k_{1}e^{2πi \frac{k_{1}nN′_{1}}{N′_{2}}} + ∑_{n=1}^{∞} \frac{p^{−snN_{0}N′_{2}}}{nN′_{1}} ∑_{k_{2}=0}^{N′_{1}−1} k_{2}e^{2πi \frac{k_{2}nN′_{2}}{N′_{1}}}

= (\frac{isN_{0} log p}{2π} − 1) ∑_{n=1}^{∞} \frac{p^{−snN_{1}N_{2}/N_{0}}}{n} + ∑_{n=1}^{∞} \frac{p^{−snN_{1}}}{n} f_{1}(n) + ∑_{n=1}^{∞} \frac{p^{−snN_{2}}}{n} f_{2}(n)

by Lemma 3.3.

References

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[KoKu1] S. Koyama and N. Kurokawa: Zeta functions and normalized multiple sine functions. (preprint)

[KoKu2] S. Koyama and N. Kurokawa: Multiple zeta functions (I). (preprint)

[KuKo] N. Kurokawa and S. Koyama: Multiple sine functions. Forum Math. (in press, 2002)

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