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.

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. □

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. □

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. □

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 .

In a similar way as Section 5, we will show Theorem 1.2.

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.

*The second author was partially supported by the INOUE ENRYO Memorial Grant 2023, TOYO University.