ON THE RIEMANN-HARDY CONJECTURE FOR THE RAMANUJAN ZETA-FUNCTION

In this article we propose the integral, series and product representations for the Ramanujan zeta-function. We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function. We present some variants for the product for the Ramanujan Ξ-function. We prove that all of its zeros are real. Along the way we obtain the truth of the Riemann-Hardy conjecture.

Hardy in 1940 (see [3], p.174) proposed the Riemann-Hardy conjecture for the Ramanujan zeta-function associated with the Ramanujan's tau-function, that in his honor is as an analogue of Riemann conjecture for the Riemann zeta-function. Up to now, it remains an important unsolved problem in the analytic number theory. Moreover, a great many of the interesting problems similar to the interesting topics for the Riemann zeta-function have been reported by Hardy [3]. For example, an analogue of the theorem of Hardy for the entire Ramanujan zeta-function was proposed by Hardy (see [3], p.174) and proved by Lekkerkerker [16]. An analogue of the von Mangoldt-like formula for the entire Ramanujan zeta-function was conjectured by Hardy (see [3], p.174) and proved by Ki [17]. Ferguson and coauthors [18] reported 18 nontrivial zeros for the Ramanujan zeta-function. An analogue of the Rieman-Siegel-like formula for the Ramanujan zeta-function (1) was conjectured by Hardy (see [3], p.174) and proved by Keiper [19].
Let φ ϑ run the positive zeros for Ξ τ (t). In 1983, Hafner [13] proposed the equivalent form of Conjecture 1, which states that Moreover, other statements equivalent to Conjecture 1 were reported in Moreno [7]. The main targets of our article are to proceed to prove Conjecture 1, and to propose the series and product formulas to give the structure of the product formulas to obtain this conjecture. The structure of this article is given as follows. In Section 2 we propose the integral and series representations for (1) and the Lekkerkerker theorem. In Section 3 we suggest the product formulas for (1) and (4). In Section 4 we prove that all zeros of the Ramanujan Ξ-function (9) are real. In Section 5 we also prove the truth of Conjecture 1. Finally, we propose some equivalent theorems in Section 6.

The integral and series representations
In this section we consider the integral and series representations for the Ramanujan zeta-function. Now we consider the remark on the work of Wilton to consider the integral representations of them.

Remark.
In 1929, Wilton (see [10], formula (5.2)) proved that which leads to ξ τ (s) Thus, we have . (4), then we have for Proof. By the definition of the entire Ramanujan zeta-function and (7), we have By (2) and (11), we arrive at the identity which leads to It follows that Then, the desired result follows. Hence, we finish the proof of Theorem 1.
Proof. By Theorem 1, we now show that Thus, the result follows.
Proof. Making use of (9) and Theorem 1, we give and the result follows.
Proof. By Theorem 2 and (9), one obtains Thus, we finish the proof. Remark. It is clearly seen that Theorem 3 also reduces to the desired result. By (27) and (29), we find that which is in agreement with the result of Conrey and Ghosh [12]. Moreover, there exist In view of (34) and (30), we arrive at Similarly, we also present By using the above remark, one has the following corollaries: In 1955, Lekkerkerker [16] showed that Ξ τ (t) has infinitely many zeros by Lekkerkerker theorem. (9) and Proof. See [16].

The product formulas
In this section we propose a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function and some product for Ramanujan zeta-function. Now, we introduce the Conrey-Ghosh product by applying the result of the Weierstrass and Hadamard [11].
Theorem 5. Let ρ ϑ ∈ C, s ∈ C and ϑ ∈ N. Suppose that ρ ϑ run the nontrivial zeros of ξ τ (s) and γ is a constant defined in (6). If ξ τ (s) is defined in (4)  Proof. By (5) and Lemma 2, one has Thus, we complete the proof of Theorem 5.
Theorem 7. Let ρ ϑ ∈ C and ϑ ∈ N. Suppose that ρ ϑ run the nontrivial zeros of ξ τ (s) and γ is a constant. Then we have for s ∈ C, Proof. By (4) and Lemma 2, one obtains and by (4) and Theorem 6, one gives Thus, we finish the proof of Theorem 7.

Real zeros
In this section we will prove that all of the zeros for the Ramanujan Ξ-function Ξ τ (t) are real by the Lekkerkerker theorem and Theorem 6. (9), then all of the zeros of Ξ τ (t) are real.
Proof. Applying Theorem 6, we have where t ∈ R, ϑ ∈ N, and the product runs over zeroes ρ ϑ of the entire Ramanujan zeta-function ξ τ (s). In order to investigate the zeros of Ξ τ (t), we now rewrite (56) as where t ∈ R, ϑ ∈ N, and the product runs over the zeroes ψ ϑ of Ξ τ (t) and the zeroes ρ ϑ of the entire Ramanujan zeta-function ξ τ (s). Now, let us consider the first hypothesis that all zeros of Ξ τ (t) are given by where σ ϑ ∈ R\ {0} and τ ϑ ∈ R\ {0}, and the second hypothesis that all zeros of Ξ τ (t) are where ℓ ϑ ∈ R\ {0}. Our idea is that if two cases are false, then we have that all zeros of Ξ τ (t) are real.
We now start to prove that they are false. Hypothesis 1. Now, we consider the first hypothesis that the complex zeros for Ξ τ (t) can be expressed in the form By (57) and (60), we now consider Putting t = (s − 6) /i into (6), we now give By (64), we present where σ ϑ ∈ R\ {0} and τ ϑ ∈ R\ {0}. By σ ϑ ∈ R\ {0} and (65), that the critical line is expressed as which is in contradiction with Lemma 1 (Lekkerkerker theorem), which states infinitely many zeros for ξ τ (s) lie on the critical line Re (s) = 6. Hypothesis 2. Now, we give the hypothesis that Ξ τ (t) has zeros By inserting (67) into (57), we suggest that Substituting t = (s − 6) /i and (68), we give From (69), we show that From (70) we have that which is in contradiction with the well-known fact where Re (ρ ϑ ) ∈ R\ {0} and ψ ϑ ∈ R\ {0}. Moreover, (72) and Lemma 1 (Lekkerkerker theorem) conflict with each other.
In sum, all zeros of Ξ τ (t) are real since two cases are false. Hence, we finish the proof.

Remark.
We have to point out that Corollary 3 is a sufficient condition for (9) to have all infinitely many real zeros.

The nontrivial zeros
In this section we present the detailed account of the proof of Conjecture 1.
To begin with, we consider the product of the Ramanujan Ξ-function with the aid of Corollary 3.
By Theorems 4 and 6, we now structure the entire Ramanujan Ξ-function by where β m are the coefficients and γ is the constant. In view of Corollary 3, the second term of (74) gives us to show that all of the zeros of Ξ τ (t) are real.

The equivalent theorems
In this section we propose some equivalent theorems for the Ramanujan zetafunction based on the true of Conjecture 1.
In view of (92), we have Thus, we finish the proof of Corollary 4.
Thus, the result follows.
Remark. By the above results, we show the following comments.
• By theory of the entire functions [20], it is observed that Ξ τ (t) is an even entire function of order α = 1 with the exponent of convergence λ = 1 and genus β = 0, and of growth (1, 0). ξ τ (s) is an entire function of order α = 1 with the exponent of convergence λ = 1 and genus β = 0, and is of growth (1, 0). • It was proposed in 1940 by Hardy [3] that the nontrivial zeros ρ ϑ for L τ (s) lie on the critical line Re (s) = 6 and in the critical strip 11/2 < Re (s) < 13/2, and that the trivial zeros for L τ (s) are s = −w with w ∈ N ∪ {0}. The trivial and nontrivial zeros, critical line and critical trip for the Ramanujan zeta-function in the entire complex plane are shown in Fig. 1. Thus, it is clear to see that Conjecture 1 is true.