ON THE DISTRIBUTION OF THE NONTRIVIAL ZEROS FOR THE DIRICHLET L-FUNCTIONS

This paper addresses a variant of the product for the Dirichlet L– functions. We propose a completely detailed proof for the truth of the generalized Riemann conjecture for the Dirichlet L–functions, which states that the real part of the nontrivial zeros is 1/2. The Wang and Hardy–Littlewood theorems are also discussed with removing the need for it. The results are applicable to show the truth of the Goldbach’s conjecture.


Introduction
The Dirichlet L-function L (s, χ) was formulated by German mathematician Johann Peter Gustav Lejeune Dirichlet in 1837 for the meromorphic continuation of the function defined by the series [1] (1) where χ (n) is a Dirichlet character (mod q > 1), Re (s) > 1, n ∈ N and s ∈ C. Here, q is the prime, N and C are the sets of the natural numbers and complex numbers, and Re (s) = σ ∈ R and Im (s) = t ∈ R are the real and imaginary parts of the complex variable s = σ+it ∈ C, where i = √ −1. For Re (s) > 1 the Euler product representation for Eq. (1) can be expressed in the form [2]: Let the primitive character χ * with the modulus q > 1 such that [3] (3) χ (n) = χ * (n) if gcd (n, q) = 1, 0 if gcd (n, q) = 1.
It is clearly seen that the nontrivial zeros of Eq. (13) exist and that Eq. (13) has infinitely many nontrivial zeros by using Eq. (14).
The generalized Riemann conjecture for Eq. (1) states the real part of the nontrivial zeros is 1/2.
According to Davenport (see [2], p.124), Conjecture 1 was formulated in 1884 by Adolf Piltz. There have been applicable to consider the Goldbach's conjecture by Hardy and Littlewood in 1923 [14, 15] and by Wang in 1962 [16]. Hardy and Littlewood [17,18,19] proved that if Conjecture 1 is true, almost all even numbers are sums of two primes and that every large odd number is the sum of three primes.
Recently, author have proposed two classes of the Riemann zeta function, which can be represented by (the first term was by Hadamard [9,11,20] and the second term was by author [9]) provided that there exists the entire Riemann zeta function, given as (the first term was by Hadamard [9,11,20] and the second term was by author [9]) with the Euler's constant a.
As is well known, Conjecture 1 is an unsolved important mathematical problem in analytic number theory up to now. By inspired by the above results to structure the variant of the product for the Riemann zeta function, the main targets of the paper are to propose a variant of the product for Eq. (13), to prove Conjecture 1 for Eq. (1) and to present a detailed account of applications of the Hardy-Littlewood and Wang theorems to the Goldbach's conjecture directly removing the need for Conjecture 1. The structure of the paper is given as follows. In Section 2, we introduce the results related to the Riemann zeta function and Dirichlet L-functions. In Section 3, we give the detailed proof of Conjecture 1. In Section 4 we apply to obtain the properties via Dirichlet L-functions, get the representations for Wang theorems, and apply the Hardy-Littlewood theorems to obtain the Goldbach's conjecture. Finally, we draw the conclusion in Section 5.

Fundamental results
We now consider the variants of the products for the Riemann zeta function and Dirichlet L-functions.

2.1.
A variant of the product for the Riemann zeta function. We now consider the variant of the product for the Riemann zeta function.  Proof. See the results of Landau [11].
where a is the Euler's constant. Then two classes of the entire Riemann zeta function are equivalent: where s k run the zeros for ξ (s), s ∈ C and k ∈ N. Proof. According to Hadamard [20] and Landau [11], we have which is derived by [6,11,20]  Thus, the proof of Lemma 3 is completed.
Remark. For the detailed proofs of Lemmas 3 and 4, see [8,9]. According to Patterson [23] we have which implies from Eq. (23) that [8] (42) , and [8] (44) where s ∈ C, s = 1, and Now, from Eq. (42), we see that s k , 1 − s k , s * k and 1 − s * k are the zeros for ξ (s) and the nontrivial zeros for ζ (s) [8]. Moreover, ξ (s) is a meromorphic continuation to the entire complex plane s with pole of residue 1 [6], and ξ (s) is an integral function of order 1 and simple in the entire complex plane s [8]. By Eqs. (31) and (44), it is proved that Re (s k ) = 1/2, in other words that the Riemann conjecture is true (for the detailed proof, see [8,9]).

2.2.
A variant of the product for the Dirichlet L-functions. In this part we present the variant of the product for the Dirichlet L-functions. We now denote the primitive character modulo q with q > 1 by χ * = χ * (n).

Lemma 6.
If χ * is a primitive character modulo q with q > 1, then we have that is an entire function of order 1, where s ∈ C and (χ * ) is defined in (9).
Proof. For the detailed proof of Lemma 6, see [3].
Lemma 8. Suppose that χ * is a primitive character modulo q with q > 1 and ρ k run the nontrivial zeros of L (s, χ * ). Then there is where s ∈ C and Proof. For the detailed proof of Lemma 8, see [3].
Lemma 9. Suppose that χ * is a primitive character modulo q with q > 1 and ρ k run the nontrivial zeros of L (s, χ * ). Then there is Proof. For the detailed proof of Lemma 9, see [3].
Lemma 10. Suppose that χ * is a primitive character modulo q with q > 1 and ρ k run the nontrivial zeros of L (s, χ * ). Then we have for s ∈ C, where (χ * ) is defined in (9), the Gauss sum τ (χ * ) of χ * is denoted by and χ * is the complex conjugate character to χ * .
Lemma 11. Suppose that χ * is a primitive character modulo q with q > 1. Then we have Proof. For the detailed proof of Lemma 11, see [3].
Thus, we see that if χ * is a primitive character modulo q with q > 1.
Theorem 1. Suppose that χ * is a primitive character modulo q with q > 1 and ρ k run the nontrivial zeros of L (s, χ * ) with k ∈ N. Then there exist the equivalent representations: where s ∈ C and Proof. From Eqs. (51) and (52), By Lemma 8, we get which is the required result.
Theorem 2. Suppose that χ * is a primitive character modulo q with q > 1 and ρ k run the nontrivial zeros of L (s, χ * ) with k ∈ N. Then there exist the equivalent representations: where s ∈ C and Proof. With use of Lemma 6 and Theorem 1, we present [11] (71) which are the desired results, where s ∈ C.
Proof. See the result of Davenport (see [2], p.82). If Conjecture 1 is true, then every odd number m > 5 is the sum of three primes.
Proof. See the work of Hardy and Littlewood [14] and the paper of Deshouillers and coauthors [15] under the condition of truth of Conjecture 1. Lemma 13 was proved in 2013 by Helfgott [26] without Conjecture 1 and a detailed account of the numerical verification for Lemma 13 was shown by Helfgott and Platt in 2013 [27].
For the sake of brevity, we denote the following proposition by (1, X) [16,28]: Every sufficiently large even integer is a sum of a prime and an almost prime of at most X prime divisors.
Proof. See the work of Wang [16]. Proof. See the work of Wang [16].
Proof. See the work of Hardy and Littlewood [17]. Based on it, the work of Granville [18] gives the detailed proof of Lemma 16 on the condition of truth of Conjecture 1.

A detailed proof for the generalized Riemann conjecture
We now apply the variant of the product for the Dirichlet L-functions to present the complete proof for Conjecture 1.  According to Lemma 3, we have the representation ∞ k=1 e s/s k where s ∈ C and s = 1, and so we have where ξ (0) = 0 and ξ (1/2) = 0 (for the details, see Lemma 2).
Taking β = β k , we have from Eq. (66) that By Eq. (67) and Lemma 7, we may arrive at where s ∈ C.
Since ξ (s, χ * ) is an integral function of order 1, by Theorem 1, we have from Eqs. (102) and (108) that Since Eq. (132) is an integral function of order 1, L (s, χ * ) is an integral function of order 1. Taking s = 1/2 + iβ into Eq. (132), we get We may find that Eq. (133) is an integral function of order 1. Substituting β = β k into Eq. (133), we have By the virtue of Eq. (134) and Lemma 6, we give and by Eq. (135), we arrive at Combining Eqs. (127) and (132) and (136) and using Theorem 2, we have In view of Eq. (137), we obtain which implies, by Lemma 6, that Thus, by Eq. (123) we have By Lemma 12 and using the above results, we clearly see that the real part of the zeros in the critical trip is 1/2 .
It is shown that Conjecture 1 is true. Hence, we finish the proof of Conjecture 1.
Remark. In short, we easily see the followings: • When χ = χ 0 and q = 1, L (s, χ) = ζ (s) is extended to be a meromorphic continuation to the entire complex plane s, and has a pole at s = 1 with residue 1, the trivial zeros s = −2h with h ∈ N, and the nontrivial zeros s k = 1/2 + i k with k ∈ N, which lie on the critical line s = 1/2 and in the critical trip 0 < Re(s) < 1.
As shown in Section 1, Case 1 has been proved by authors in [8,9]. • When χ = χ 0 and q > 1, L (s, χ) = L (s, χ 0 ) is extended to be a meromorphic continuation to the entire complex plane s, and has a pole at s = 1 with residue υ (q) /q = p|q (1 − p −1 ), the pure imaginary zeros s = 2πij/ (log p) with p |q and j ∈ Z, the trivial zeros s = −2h with h ∈ N, and the nontrivial zeros s k = 1/2 + i k for k ∈ N. It is seen that Cases 2 and 1 have the same as the nontrivial zeros s k = 1/2 + i k , a pole at s = 1 with different residues, and trivial zeros s = −2h with h ∈ N. Its nontrivial zeros for L (s, χ) = L (s, χ 0 ) lie on the critical line s = 1/2 and in the critical trip 0 < Re(s) < 1. As shown in Case 3 of Section 1, we have followings: • When χ = χ * and χ * (−1) = 1, is an integral function of order 1, and has the zeros (all zeros are the nontrivial zeros) ρ k = 1/2 + iβ k with k ∈ N.
• When χ = χ * and χ * (−1) = −1, is an integral function of order 1, and has the zeros (all zeros are the nontrivial zeros) ρ k = 1/2 + iβ k with k ∈ N. It is observed that in Case 3 of Section 1, they have the nontrivial zeros ρ k = 1/2 + iβ k with k ∈ N, which lie on the critical line s = 1/2 and in the critical trip 0 < Re(s) < 1.

New results and applications
In this section we report the new formulas associated with the Dirichlet L-functions. Then we also give the new representations for the Wang theorems. Main target of the part is to present the applications of Conjecture 1 and Hardy-Littlewood theorems to obtain the Goldbach's conjecture.

4.1.
New formulas for the Dirichlet L-functions. We now give the properties for the Dirichlet L-functions.
Theorem 3. Suppose that χ * is a primitive character modulo q with q > 1. Let β k be the Siegel zeros for the Dirichlet L-functions with the primitive character χ * with k ∈ N. Then there exist the equivalent representations: where s ∈ C, and Proof. By Conjecture 1 and Theorem 1, we give the desired results.
Theorem 4. Suppose that χ * is a primitive character modulo q with q > 1 and β k run the Siegel zeros for the Dirichlet L-functions with the primitive character χ * with k ∈ N. Then there exist the equivalent representations: where s ∈ C, (χ * ) is defined in Eq. (9), and Proof. By Conjecture 1 and Theorem 2, we give the desired results.
Theorem 5. Suppose that χ * is a primitive character modulo q with q > 1 and β k are the Siegel zeros. Then there exist the equivalent representations: where β ∈ C, (χ * ) is defined in Eq. (9), and Proof. By Theorem 3 and Conjecture 1, we obtain the desired results.
We now define the function by where s ∈ C and s = 1.
Remark. Suppose that χ 0 is the principal character modulo q with q > 1. Then we have that where s ∈ C and s = 1.
Because Conjecture 1 is true, 1 − ρ k , ρ k and 1 − ρ k are the nontrivial zeros for the Dirichlet L-functions, which is in agreement with the results of Montgomery and Vaughan [3].

4.2.
The truth of the Goldbach's conjecture. We now give new representations for the Wang theorems and present the applications of the Hardy-Littlewood theorems to the Goldbach's conjecture because Conjecture 1 is proved and true. Theorem 9. (The strong Goldbach's conjecture) Every even number m > 2 is sums of two primes.
Proof. By Lemma 16, Theorem 9 is true because of true of Conjecture 1.
Remark. By the work of Deshouillers and coauthors [15], we have that that the weak Goldbach's conjecture is true because Conjecture 1 is valid. By the work of Granville [18], we also see that the strong Goldbach's conjecture is true because Conjecture 1 is valid. Thus, the Goldbach's conjecture is true.

Conclusion
The present paper has proved that Conjecture 1 is true with the variant of the product for the entire function related to the Dirichlet L-functions. We have presented the applications of it to the Wang theorems for the Goldbach's problems. By using the Hardy-Littlewood theorems, we have shown that every odd number m > 5 is the sum of three primes and that every even number m > 2 is sums of two primes. The obtained result is proposed to solve the mathematical problems under the assumption of the truth of Conjecture 1.