Euler Product Expressions of Absolute Tensor Products of Dirichlet L-Functions ()
1. Introduction
In 1992 Kurokawa [1] defined the absolute tensor products (Kurokawa tensor products). The definition is given by
for some zeta functions 
, where the symbol 
, which was introduced by Deninger [2], represents the zeta regularized product (see below) and the integer 
is defined by
where 
denotes the order of 
which is a zero of 
; now, we regard the poles of as the zeros with negative orders in this paper. Here the zeta regularized products are defined by
where 
and 
are complex sequences such that
converges locally, uniformly and absolutely in some s-region included in 
for 
with some constant 
and is a meromorphic function of w at 
. If 
then 
is a meromorphic function of s in the whole 
and has zeros only at 
. The integer 
contributes to the order of 
. See [3] for more details concerning the zeta regularized products. The factors of the zeta regularized products are derived from the summands of 
, so we call the “factors series” in this paper.
Kurokawa [1] also predicted that the absolute tensor product of r arithmetic zeta functions which have the expression by the Euler product over primes would have the Euler product over r-tuples 
of primes. The validity of Kurokawa’s prediction has been confirmed in some cases, for example, the cases of the Hasse zeta functions of finite fields by Koyama and Kurokawa [4] for 
, by Akatsuka [5] for 
and by Kurokawa and Wakayama [6] for general r. Also, the case of the Riemann zeta function for 
was first proved by Koyama and Kurokawa [4], and then by Akatsuka [7] in a different way.
In [7], Akatsuka successfully eliminated the parameter α in the absolute tensor square of Koyama and Kurokawa. In that sense he obtained the true form of the Euler product expression of the absolute tensor square of the Riemann zeta function. He did so by establishing an equation which links the zeros of the Riemann zeta function to prime numbers. In this paper, according to Akatsuka’s method in [7], we will reach the Euler product expression of the absolute tensor product 
, where 
denotes the Dirichlet L-function corresponding to a primitive Dirichlet character 
to the modulus 
. The key item which leads to our goal is an equation which links the factors series of 
to r-tuples of prime numbers (see Theorem 4.1 below). We name such equation the “key equation”. In the following, let 
denote the non-trivial zeros of 
corresponding to a non-principal primitive Dirichlet character 
to the modulus 
and let 
. We shall count the zeros with multiplicity. Then, letting 
, where r is a parameter in the key equation, we obtain the zeta regularized product expression of 
:
Theorem 1.1 We have the following expression for 
:
(1.1)
From (1.1) and the definition of the absolute tensor products, we find that has the following expression:
Now, let 
be primes and 
be positive integers, and let α be any fixed number with 
. For the complex numbers 
with 
, we define 
; we fix 
arbitrarily with
. Also, we define 
. Define that
where 
and 
, 
and 
denote the Euler constant, the gamma function and the Gauss sum respectively, that is,
Then, letting 
in the key equation, we can deduce the Euler product expression of 
as follows:
Theorem 1.2 In 
we have
where 
, that is,
The proofs of Theorem 1.1 and Theorem 1.2 are given in Section 5 and Section 6 respectively. The contents of the other sections are as follows. In Section 2 some lemmas are proved which are made use of in Section 3 or later. In Section 3 a series is introduced which includes information on the zeros of the Dirichlet L-functions and some properties of the series is shown. In Section 4 the key equation is deduced.
2. Lemmas
In this section, we prove some lemmas which are used later.
Define that
(2.1)
This function will appear in Section 3 in the properties of a series involving zeros of Dirichlet L-functions in Theorem 3.3.
Remark 2.1 In the following, it is found that 
has an analytic continuation, and let the same symbol denote its continuation.
We show the properties of 
in the following lemma:
Lemma 2.2 (i) 
has the following asymptotic behavior at 
:
(ii) 
is a single-valued meromorphic function on the whole 
.
(iii) has the simple poles at 
with residue 
.
Remark 2.3 Let 
and the argument lie in 
. It follows from Lemma 2.2 (ii) that 
is a meromorphic function because
is such one.
Proof of Lemma 2.2.
(i) It was proved by Cramér [8, p.116, (19); p.117, (20)] that for 
(2.2)
where 
was a power series of z which converged for 
. By replacing z for 
in (2.2), we obtain
(2.3)
for 
, where 
is a power series of t which converges for . We can derive the desired result from this.
(ii) Note that the integral (2.1) also converges if 
and that the integrand has pole at 
with residue 
. Now, if t moves counterclockwise around the origin from the quadrant 
into the half-plane 
across the negative imaginary axis, then the pole at 
moves from the forth quadrant into the upper half-plane across the positive real axis. Since the positive real axis is the integral path in (2.1), the analytic continuation of 
into is given by subtracting 
times the residue of the integrand at from the integral (2.1), that is, for
(2.4)
The right-hand side of (5) is meromorphic unless it is on the non-negative real axis, so we find that changes by 
when it moves counter-clockwise around the origin, making one complete circuit. Therefore 
is unchanged by the analytic continuation around the origin, so it is a single-valued function on 
. Furthermore, we find that
is meromorphic for from (4). The proof of (ii) is complete.
(iii) By (2.1) it is easily found that is holomorphic if 
. From this and Lemma 2.2 (ii), we can obtain the desired result. □
Next, we will show that the Euler product of 
converges locally and uniformly on 
. This fact will be used to justify the change of the order of limit and integration in the proof of Theorem 3.3.
Lemma 2.4 (i) Let 
be the von Mangoldt function, that is
Then, we have
(2.5)
(ii) The Euler product of 
,
converges locally and uniformly on 
.
Proof of Lemma 2.4. (i) By the Abel’s summation formula, we have
It was proved in ([9], Theorem 4.4.2) that
so there exists some positive constant M such that
We find from this that
because
where 
is the logarithmic integral, that is
Therefore, we have
We obtain the desired result.
(ii) It suffice to prove the local and uniform convergence of
on 
. We will first show that 
tends to 0 uniformly on 
as 
, where
and then show that 
converges locally and uniformly as 
.
Let s be on 
. In 
, since the sum over m converges absolutely, we can exchange the order of the sums:
so we have
The series 
converges absolutely as 
if 
, so 
tends to 0 uniformly on as 
.
Since (2.5) holds and 
is regular at 
, we can derive the local and uniform convergency of 
on from M. Riesz’s statement ([10], Satz I): if the coefficients of a Dirichlet series 
meet the condition
(2.6)
and if the function 
, which is regular due to the condition (2.6) for 
, is also regular in certain points of the line 
then the series converges at these points. The convergence is uniform in any finite interval, which consists only of regularity points.
This completes the proof. □
Lemma 2.5 was proved by Akatsuka [7].
Lemma 2.5 (i) ([7], Lemma 2.5) For any 
satisfying 
(ii) ([7], Remark 2.1) 
.
(iii) ([7], p.639, (4.4)) For any fixed 
and any 
Also, we shall prove a formula for the gamma function in the following lemma.
Lemma 2.6 Let any fixed 
satisfy 
and let 
and 
. Then, we have
Proof of Lemma 2.6. For any fixed 
satisfying , let 
. Then, we have
When w is fixed in 
, the both sides are holomorphic in
This completes the proof. □
3. Properties of a Series Concerning the Zeros of the Dirichlet L-Functions
For a series 
where 
with 
for the imaginary zeros 
of the Riemann zeta function, Cramér [8] and Guinand [11] deduced the properties: the explicit formula, the meromorphic continuation, the poles, the functional equation and the approximate behavior. Akatsuka [7] introduced 
and proved the properties on the basis of the results of Cramér and Guinand. Kaczorowski [12] introduced
and deduced the properties according to Cramér and Guinand.
In this section, we define
(3.1)
rewrite the results of Kaczorowski into the ones for 
and derive the further properties with reference to the methods of Cramér, Guinand and Akatsuka.
Kaczorowski deduced the following assertions concerning 
:
Lemma 3.1 Let 
be a Riemann surface of logarithmic type.
(i) ([12], Theorem 3.1) The function 
can be continued analytically to the meromorphic function on 
and
is a single-valued meromorphic function on 
for 
.
(ii) ([12], Theorem 3.2, (3.4)] The meromorphic function on 
satisfies the following functional equation:
(3.2)
where 
noting that can be uniquely written as 
.
Any single-valued function 
on 
can be considered as a function on 
due to the natural projection 
and then we have 
. From this and Lemma 3.1 (i), it follows that
so, adding 
to the both sides of (3.2), we obtain
Noting that 
, we have
(3.3)
If 
and the argument lies in 
then we can derive
because 
, so under the same assumption, by replacing z by it and multiplying by 
the both sides in (3.3), we have
Replacing 
by 
respectively in this formula, we have
Note that
because 
is a zero of 
with the same order as 
. From the above and the meromorphy of 
if 
, we obtain the following theorem:
Theorem 3.2 
has a meromorphic continuation to 
for which
(3.4)
where the argument lies in 
.
Next, we deduce the explicit formula, the approximate behavior and the poles of 
. Kaczorowski also deduced the explicit formula of 
, in the proof of which he used an integral path contained in the absolute convergence domain of the Euler product of 
. The path selection influences the convergence domain of the Euler product of 
, so we use a different path.
In preparation for the proof of Theorem 3.3, we need to choose a branch of 
. First, we cut the s-plane from 
to 
straight and also remove the area, determined by the inequalities
In the remaining part of the cut plane, each branch of 
is unique. We choose the one represented for 
by the series
Theorem 3.3 Define 
.
(i) 
has the following expression for 
:
(ii) has the following expression for 
:
(3.5)
(iii) has the following approximate behavior at 
:
(iv) has simple poles in 
only at the following points:
where 
.
In (ii)-(iv), the argument lies in 
.
Remark 3.4 It follows from Lemma 2.5 (ii) that the sums over p and m in Theorem 3.3 (i)(ii) converge absolutely and uniformly on any compact subset of 
.
Proof of Theorem 3.3. (i) If then we have by Cauchy’s theorem
(3.6)
where, choosing 
that satisfies the condition that 
has no zeros on the interval 
,
and we go around the integral path in the counterclockwise direction. By the integration by parts, (3.6) becomes
where we choose the branch of 
satisfying the following condition:
and s moves in the cut s-plane 
. Now, by the result of Montgomery and Vaughan ([13], Theorem 10.16], i.e., there exists a constant 
such that
where we use the representation
and 
denotes the completed Dirichlet L-function, that is
so
Therefore, we have
so (3.6) can be rewritten into
(3.7)
where
From the functional equation
(3.8)
we have
where 
satisfies the following relation: arg(LHS of (3.8)) = arg(RHS of (3.8)) + 
. Then, the integral of the path 
becomes
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
The integrals concerning (3.9) and (3.10) become
(3.15)
and
(3.16)
respectively. By the result of Cramér [8, p. 114, (12)]:
the integral concerning (3.11) is equal to
(3.17)
The integral concerning (3.12) is equal to
(3.18)
Since the third term of (3.18) becomes
we have
(3.18)
(3.19)
The integral concerning (3.13) becomes
(3.20)
The integral concerning (3.14) becomes
(3.21)
The integral of the path 
of (3.7) becomes
(3.22)
The above changes of the orders of the sums and integrations are justified by Lemma 2.4(ii), Remark 3.4 and the Dini’s statement ([8], p.112, footprint) that if 
converges uniformly for 
with every 
and 
also converges uniformly for all 
then we have
Applying (3.15), (3.16), (3.17), (3.19), (3.20), (3.21) and (3.22) to (3.7), we obtain the desired result.
(ii) Let the argument lie in 
. By Theorem 3.3 (i), we find that for 
By using the equation for 
deduced in Theorem 3.2, we obtain (3.5) for 
. Since the right-hand side of (3.5) is meromorphic for 
if the argument lies in 
, the proof of (ii) is completed.
In the following, let 
.
(iii) By Theorem 3.3 (ii) and Lemma 2.2 (i), we find that
(iv) By (3.1), we find trivially that 
is holomorphic for 
. From this and the expression obtained in Theorem 3.3 (ii), the desired result follows. □
We consider the bounds of 
which is needed later. Let 
denote 
.
Lemma 3.5 (i) For 
(ii) For 
(iii) If 
with 
and 
, then
In (i)-(iii), the argument lies in 
.
Proof of Lemma 3.5. Let the argument lie in 
.
(i) If 
then we have
from (3.1). Since
we obtain the desired result.
(ii) For 
, we have
(3.23)
by Theorem 3.2. Concerning the first and third term of the right-hand side of (3.23), we have
and
respectively. Hence, we obtain the desired result.
(iii) When 
with 
and 
, we have
(3.24)
by estimating trivially each term of the right-hand side of (3.5) in Theorem 3.3 except the first term.
If 
, then 
, so 
. Therefore, we have
In the last equation, we use Lemma 2.5 (ii). This completes the proof. □
Now, we fix 
arbitrarily with 
and 
.
Corollary 3.6 (i) For 
(3.25)
(3.26)
(ii) If 
and 
then
(iii) If 
, 
and 
then
Proof of Corollary 3.6. (i) First, by Lemma 3.5 (i) we find
In the last equation we use the fact that 
because 
. Hence, (3.25) has been proved.
Next, by Lemma 3.5 (ii) we have
(3.27)
Now,
|(the first term of RHS of (3.27))| 
Hence, we can deduce
where in the last equation we use the fact that 
because 
. Hence, (3.26) holds.
(ii) From Lemma 3.5 (i) and 
, we can easily deduce the desired result.
(iii) If 
(respectively 
) then we can trivially deduce the desired result from Lemma 3.5 (i) (respectively Lemma 3.5 (ii)).
If 
then we can derive
from Lemma 3.5 (iii). Concerning the first term of the right-hand side, we find that
(3.28)
and that by Lemma 2.5 (i)
(the first term of (3.28)) 
(the second term of (3.28)) 
Hence, we can obtain
This completes the proof. □
4. The “Key Equation”
In this section, we prove an equation we name the “key equation” which links the “factors series” of 
to r-tuples of prime numbers 
.
Define that
(4.1)
(4.2)
Then, we will show the following theorem:
Theorem 4.1 (“key equation”) Let 
satisfy
and 
. Then,
(4.3)
Proof of Theorem 4.1. Let 
be any fixed real number with 
and we define
where 
is the union of 
, 
and 
when
By Corollary 3.6 (i), for large enough u
Therefore, 
converges absolutely and uniformly on any compact subset of 
.
Now, when and 
, we have
by Theorem 3.3 (iv) and Cauchy’s theorem, where
and we go around the integral path in the counterclockwise direction. If 
then by Theorem 3.3 (iii) we have
(4.4)
By replacing t with -t, using Theorem 3.2 and taking note of
we find that the first term of (4.4) is equal to
Hence, we have
Next, we define that 
for 
and let with 
and 
with 
. By Theorem 3.3 (iv) and the residue theorem, we have
(4.5)
where
and we go around the integral path in the counterclockwise direction. First, we consider the limit of (4.5) as 
. Concerning the integral of the path 
, we have, by Corollary 3.6 (ii),
(4.6)
where in the last inequality we use the fact that 
. From 
because , it follows that (4.6) vanishes as . Hence, we have
(4.7)
where
and we go around the integral path in the counterclockwise direction. Next, we consider the limit of (4.7) as 
. Concerning the integral of the path 
, we have
(4.8)
About the first term of (4.8), by using Corollary 3.6 (iii) we can deduce
(4.9)
Since
we have
where in the last limit we use the fact that
(4.10)
and
because and 
. About the second term of (4.8), by using Corollary 3.6 (iii) we have
where in the last limit we use 
. About the third term of (4.8), by Corollary 3.6 (iii) we have
(4.11)
(4.12)
where in transforming (4.11) into (4.12) we use 
because , and in the last limit we use (4.10). Hence, we obtain
This completes the proof. □
In the following sections, it is necessary that the left-hand side of (4.3) be a meromorphic function of w at 
. To obtain the property we show a lemma. It is the generalization of the lemma proved by Hirano, Kurokawa and Wakayama ([14], Lemma 1].
Let 
be any fixed real number and 
be a locally integrable function on 
. We define
Now, assume that satisfies
for 
with 
and any 
; 
converges absolutely, so is an analytic function, in 
. Then, the following lemma holds.
Lemma 4.2 Suppose that has the following approximate behaviors as 
and 
:
(4.13)
where 
are non-negative and finite integers for each k and 
and 
are complex sequences with 
and 
monotonically increasing. Then has a meromorphic continuation into 
with poles at 
and 
for each k. Especially the poles at 
are simple if 
.
Proof of Lemma 4.2. First we define 
as
Then, in 
, we have
(4.14)
The first and third terms of the right-hand side of (4.14) are analytic function of w in 
and in 
respectively. The second term becomes
and then by partial integration we can transform it into
Hence, we see that is a meromorphic function of w with having poles at 
in 
, especially the orders of which at 
are simple if 
. Since
, it is shown that the meromorphy of in the left half plane 
.
In a similar way, we can obtain a meromorphic continuation into the right half plane 
. □
The meromophy of the left-hand side of (4.3) follows from Lemma 4.2.
Corollary 4.3 If 
and
, then 
and
are meromorphic functions of w on the whole 
.
Proof of Corollary 4.3. By the consideration about 
in the proof of Theorem 4.1, 
and 
are holomorphic functions of w under the assumption that
We can remove 
because it follows from Theorem 3.3 (iii) that
and
which appear in 
satisfy the condition concerning 
in (48). By putting 
we obtain the desired results. □
5. The Zeta Regularized Product Expression of 
Our goal in this section is to prove Theorem 1.1. We will first obtain an equation which links the factors series of 
to prime numbers by calculating the both sides of (4.3) with 
and then prove Theorem 1.1 .
5.1. The Key Equation for r = 1
Lemma 5.1 Let 
satisfy 
and 
. Then,
Proof of Lemma 5.1. Since and 
, we have
and from this we find 
. Therefore, by using Lemma 2.6 as 
we obtain
In a similar way as 
we can reach the desired result concerning 
. □
Lemma 5.2 If 
,
and 
then we have
(5.1)
(5.2)
where the argument lies in 
. The series in (5.1) and (5.2) converge absolutely, locally and uniformly in the given 
-region above.
Proof of Lemma 5.2. Putting 
in Lemma 5.1, we obtain the conditions concerning 
and have
Now, since 
is derived from 
and 
, we find
In the same way, we obtain (5.2).
The absolute and locally uniform convergences of the series in (50) and (51) in 
and 
are easily derived from
The desired convergency follows immediately from and including the given 
-region. □
Lemma 5.3 If 
,
and then we have
(5.3)
The series converges absolutely and uniformly on any compact subset of 
.
Proof of Lemma 5.3. By Theorem 3.3 (ii) and (iv), we find that the residue in 
is equal to
From this (5.3) follows.
It follows from Lemma 2.5 (iii) that the series in (5.3) converges absolutely and uniformly on any compact subset of 
. □
By using the above three lemmas we derive the desired equation.
Theorem 5.4 If 
,
and 
, we have
(5.4)
Proof of Theorem 5.4. We put 
and 
in Theorem 4.1 and then by applying Lemma 5.2 and 5.3 we have
(5.5)
under the conditions that
Then, replacing 
with 
in (5.5), we obtain (5.4). □
5.2. Proof of Theorem 1.1
Proof. The left-hand side of (5.4) is a meromorphic function of w on the whole 
by Corollary 4.3. Hence, by using the definition of the zeta regularized product we have
(5.6)
On the other hand, since 
, we have
By the property of the zeta regularized products, (5.6) is a meromorphic function on the whole 
. Hence (1.1) holds. □
6. The Euler Product Expression of 
In a similar way as Section 5, we will show Theorem 1.2.
6.1. The Key Equation for r = 2
Lemma 6.1 If 
, 
and 
then we have
The series which appear here converge absolutely, locally and uniformly in the given 
-region above.
Proof of Lemma 6.1. In a similar way as Lemma 5.1 and 5.2 we can prove them. □
Lemma 6.2 If 
, 
and 
then we have
(6.1)
Proof of Lemma 6.2. Let p and m be any fixed prime number and positive integer respectively. By Theorem 3.3 (ii) and Remark 2.3 we have
Applying this to
leads to (6.1). □
In the following lemma we will show the convergencies of 
which can be proved in almost the same way as Akatsuka’s method used in ([7], Theorem 1.2).
Lemma 6.3 For 
, 
converges absolutely and uniformly on any compact subset of 
, where
Proof of Lemma 6.3. The desired results follow from Lemma 2.5 (iii) immediately except for 
, 
and 
.
Concerning , we can easily prove its absolute and locally uniform convergence by Lemma 2.5 (iii).
We consider . Let 
satisfy 
and 
for any fixed real numbers δ, A and B with 
and 
. Then, for any prime numbers 
and any 
we have
where 
. From Lemma 2.5 (ii), we have
From Lemma 2.5 (ii)(iii), we have
Hence, we find that 
converges absolutely and uniformly on any compact subset of 
.
We consider 
. Let 
satisfy 
and 
for any fixed real numbers δ, A and B with 
and 
. Then, for any prime numbers 
and any 
we have
where 
. In the case of 
, from
for any 
and Lemma 2.5 (ii), it follows that
In the case of 
, we have
(6.2)
Concerning the third term of (6.2), we have 
because 
. Therefore, from Lemma 2.5 (ii)(iii), we have
(the third term of (6.2)) 
(6.3)
Concerning the second term of (6.2), from Lemma 2.5 (i), we have
Hence, from Lemma 2.5 (iii), we find
(the second term of (6.2))
(6.4)
Concerning the first term of (6.2), from Lemma 2.5 (i), we have
Hence, from Lemma 2.5 (iii), we find
(the first term of 6.2)
(6.5)
From (6.3), (6.4) and (6.5), it follows that (6.2) converges. This completes the proof. □
From Lemma 6.1, Lemma 6.2 and Lemma 6.3 we derive the “key equation” for 
.
Theorem 6.4 If 
, 
and 
then the following equation holds:
Proof of Theorem 6.4. We put 
and 
in Theorem 4.1 and then by applying Lemma 6.1 and Lemma 6.2 and replacing 
with 
we obtain the desired result. □
6.2. Proof of Theorem 1.2
Proof. The left-hand side of the formula in Theorem 6.4 is a meromorphic function of w on the whole 
by Corollary 4.3. Hence, by using the definition of zeta regularized products we have
On the other hand, by Theorem 6.4 and noting that
, we have
for 
. This completes the proof. □
Acknowledgements
The first author really thanks Ki-ichiro Hashimoto for his special support. The first author also thanks Hirotaka Akatsuka for his showing me the beneficial information for this study.
NOTES
*The second author was partially supported by the INOUE ENRYO Memorial Grant 2023, TOYO University.