ON ALL REAL ZEROS FOR A CLASS OF EVEN ENTIRE FUNCTIONS

. The present paper deals with a class of even entire functions of order ρ = 1 and genus ϑ = 0 of the polynomials form, where Φ (0) ̸ = 0, real numbers x , nonnegative integers m , and ℓ k ̸ = 0 are all of the nonzero roots with 1 ∑ k =1 1 / | ℓ k | < ∞ and natural numbers k . We provide an eﬃcient criterion for the polynomials with only real zeros. We also prove that the conjecture of Jensen is our

uniformly to them since they are expressed by the Weierstrass primary factors (see [3], p.18; [4], p.25). Laguerre in 1882 [5] and Pólya in 1913 [6] proposed the class of the Laguerre-Pólya class [7]. The theory of entire functions in Laguerre-Pólya class has been a increasing interest for finding their real zeros of entire functions [8,9]. As an example of the progress made, the integral transforms of the entire functions in the Laguerre-Pólya class were reported in [10]. A fundamental paper makes a nice progress in study of an analog of the linear finite difference operators [11]. Moreover, another work on an entire function with the increasing Taylor coefficients was discussed in [12]. To discover the zeros of them, the sign regularity of Maclaurin coefficients of entire functions was also considered in [13].
1.1. The statement of the problem. Let R, N and C denote the sets of real, natural and complex numbers, respectively.
We now consider the theory of the product of the cosine and hyperbolic cosine, which were proposed by Euler ([14], p.127-128).
Euler [14] suggested that the cosine can be expressed by the Taylor series and the product: Euler [14] also suggested that the hyperbolic cosine can be expressed by the Taylor series and the product: , 4.5.63, 4.5.68, p.85). It is know that the cosine and hyperbolic cosine are the even entire functions of order ρ = 1 and genus ϑ = 0 (for the definitions of the order and genus of the even entire functions, see in Section 1).

1.2.
A class of even entire functions. By the observation of the above works of Euler, we now suggest a class of even entire functions of order ρ = 1 and genus ϑ = 0, which is structured as follows: Definition 1. A real even entire function of order ρ = 1 and genus ϑ = 0, With (4) we know that ℓ k ̸ = 0 are all zeros of Φ (x) ∈ Y by applying the theory of entire functions. The behavior of Φ (x) ∈ Y as one of subclasses of entire functions of real and complex variables in the Laguerre-Pólya class is considered in the present paper.
1.3. The conjecture of Jensen. Riemann in 1859 [16] proposed the Riemann Ξ function Ξ (x) by (5) log which leads to the product which was discovered by Cahen [17], Landau [18] and Titchmarsh [19], where run all of the positive real roots of Ξ (x) = 0 for k ∈ N. It is easy to verity that (6) can be also derived from the product of Hadamard [20] (7) where ρ k run all of the real roots of Ξ (t) = 0 with k ∈ N. Based on the work of Jensen [21], Pólya in 1927 [22] considered that Ξ (x) is represented as the polynomials associated to its Taylor expansion, i.e., (8) Ξ By connection with the product of Hadamard for the entire Riemann zeta-function ξ (x) [20], Eq. (7) was rewritten by Edwards [23] as Jensen in 1913 [21] proposed the following assert: The conjecture of Jensen is that the roots ρ k of the polynomials associated to its Taylor expansion of Ξ (t) are all real, where k ∈ N.
The conjecture of Jensen for the zeta-function was studied, completed, and expanded by Pólya in 1927 [22] and further discussed by Titchmarsh [24]. Recently, a breakthrough for new progress on conjecture of Jensen for the zeta-function was have made by Griffin et al. [25]. Up to now, a interesting paper by Bombieri [26] makes a progress report in the conjecture of Jensen that remains a unsolved problem in analytic number theory and mathematical physics.
1.4. The main targets of this paper. In this paper we mainly plan to prove the following theorems: has the critical line Im (x) = 0, then all of its zeros are real.

Theorem 2. The conjecture of Jensen is true.
The structure of this paper is designed as follows. In Section 2 we introduce the theory of the entire functions. In Section 3 we give the proof of Theorem 1. In Section 4 we present the proof of Theorem 2.

Preliminaries
In this section we introduce some results in the theory of the entire functions. Let s ∈ C. We now start with the definition of the Weierstrass primary factors. where ϑ = 0, and where ϑ > 1 and ϑ ∈ N.
be a sequence of complex numbers and k ∈ N such that then the product converges uniformly on every compact set ∆.

Lemma 2.
Let Γ = {µ k } ∞ k=1 be a sequence of complex numbers and k ∈ N. If then an even entire function

The proof of Theorem 1
We now present the proof of Theorem 1. In order to prove it, we give two hypothesis tests and we prove that if they are false, our result is true. Since Here, all of its zeros are ℓ k for k ∈ N. Now, we consider two cases as follows: Case 1. Now, we assume that With (25) we obtain where k ∈ N. From (26) we get (27) From (27) it follows that Φ (x) has the critical line Im (ℓ k ) = k , where k ∈ R\ {0}. This is contradicted against the fact Φ (x) has the critical line Im (x) = 0. Case 2. Now, we assume that and we find that Φ (x) has the critical line Im This implies that (31) is contradicted against the fact Φ (x) has the critical line Im (x) = 0.
To sum up, two cases are contradicted against the fact Φ (x) has the critical line Im (x) = 0.
Hence, we complete the proof of Theorem 1.
We now introduce an alternative method to study the of the real zeros of Φ (x) ∈ Y as follows: has a real zero, then all of its zeros are real.
Let us consider that Φ (x) has a real zero η ∈ R.
Then we obtain such that From (34) we give Hence, which implies that Φ (x) has the critical line Im (x) = 0. By Theorem 1, we deduce that all of the zeros of Φ (x) are real.
It is easy to give the following result: Then, under the condition of the truth of Theorem 1 (or Corollary 1), we have where ℓ k > 0 run all of the positive roots of Φ (t) = 0 with k ∈ N.
Proof. From Theorem 1 we deduce that all of the zeros of Φ (x) are real. Hence, where ℓ k = |ℓ k | > 0 run all of the positive roots of Φ (t) = 0 with k ∈ N.
In a similar way for Corollary 1, we obtain the same result. Hence, the desired result follows.

The proof of Theorem 2
In order to prove the conjecture of Jensen, we at first prove Ξ (x) ∈ Y and by Theorem 1, we give the proof of Theorem 2, in other words that all of its zeros are real. Now, we suggest the following result for the behavior of Ξ (x): Proof. From the work of Hadamard [20], we have Let us recall the Maclaurin formula of Ξ (x), where x ∈ R and m ∈ N. By (41) and (43), we arrive at If by Lemma 2, we write the Weierstrass primary factor By (17) and (48), Hence, it is easy to see that Ξ (x) is a real even entire function of order ρ = 1, exponent of convergence λ = 1 and genus ϑ = 0. By above results and (44), we obtain Ξ (x) ∈ Y, which is the desired result.
We now give the proof of Theorem 2. First proof.
By Theorem 1, we obtain the desired result. Second proof.
By Corollary 1, all of the zeros for Ξ (x) are real. To sum up, by Corollary 2 and (44), we have where ρ k = |ρ k | run all of the positive roots of Ξ (t) = 0 with k ∈ N.
Hence, the desired result follows. By the above proofs, the conjecture of Jensen holds.