A DIRECT APPROACH FOR THE LINDELÖF CONJECTURE RELATED TO THEORY OF THE RIEMANN ZETA FUNCTION

It is due to Littlewood that the truth of the Riemann theorem implies that of the Lindelöf conjecture. This paper aims to use the idea of Littlewood to prove the Lindelöf conjecture for the Riemann zeta function. The Lindelöf μ function at the critical line is zero, with use of the Riemann theorem for the entire Riemann zeta function, proved based on the work of Heath-Brown. Our result is given to show that the Lindelöf conjecture, connected with the proof of the moment conjecture, is true.


Introduction
The Lindelöf conjecture via Riemann zeta function, proposed in 1908 by Finnish mathematician Ernst Leonard Lindelöf, has been one of most important open problems in the history of mathematics [1]. More important, the Lindelöf conjecture is not only linked with the consequence of the Riemann conjecture [2] but also used to investigate the higher movement for the Riemann zeta function [3]. The Lindelöf conjecture has played the important role in the field of the analytic number theory [4].
Suppose that C, R and N are the sets of the complex numbers, real numbers and natural numbers, respectively. Let s = σ+it ∈ C such that Re (s) = σ ∈ R and Im (s) = t ∈ R are the real and imaginary parts of the complex variable s, where i = √ −1. Let As is well known, the Riemann zeta function ζ (s) of the complex variable s = σ+it is defined by the sum [5] (1) where k ∈ N and Re (s) > 1. As is stated in [5] that this allows Eq. (1) to be a meromorphic continuation to the entire complex plane s, with pole of residue 1 at s = 1. The trivial zeros for Eq. (1) reads s = −2ν with ν ∈ N . The nontrivial zeros for Eq. (1) are located on the critical line Re (s) = 1/2 and in the critical trip 0 < Re (s) < 1 [6,7]. The entire Riemann zeta function ξ (s) is expressed by the product of the Riemann zeta function ζ (s) or the series [5,8]: where Γ is the gamma function [9], (3) Π (s) = (s − 1) π −s/2 Γ (s/2 + 1) , Based on the above-mentioned results, it is stated in 2005 by Heath-Brown [7] that an equivalent statement for the Riemann theorem [5] is given as follows:

Theorem 1. Riemann theorem (Heath-Brown statement)
The Heath-Brown statem states the real part of all zeros of ξ (s) is 1/2.
It is equivalent to the Riemann statement [5] that the real part of the nontrivial zeros of ζ (s) is 1/2. It is known that Eq. (2) is the entire function of order 1 [10]. Although the Riemann statement has been achieved in [11], we would like to give the proof of the Heath-Brown statement for the Riemann theorem. For more details for the zeros, number of zeros and imaginary parts of zeros for ξ (s), see [8,10].
Based on the above, the Lindelöf conjecture [1] claims that for every positive ε > 0, where t → ∞. This easily yields that the equivalent statement for the Lindelöf conjecture as follows [8,10,12]: There exists There are a number of the equivalent statements for the Lindelöf conjecture. The equivalences of these various assertions were proposed in 1915 by Hardy and Riesz [12] and reported in 2015 by Conrey [13]. It is shown in 1912 by Littlewood that Conjecture 1 is the consequences of Theorem 1 implies [2]. In 1923, Hardy and Littlewood give two equivalences for Conjecture 1 states that [3] for ε > 0, m ∈ N and T → ∞, and for ε > 0, m ∈ N , σ ≥ 1 2 and T → ∞. It is stated in 2006 by Laurincikas and Steuding that the equivalence for the Lindelöf conjecture becomes [14] (9) for ε > 0, m ∈ N and T → ∞. Eq. (9) implies that the moment conjecture states [15] (10) 1 T for ε > 0, m ∈ N and T → ∞. There exist the advances for the Lindelöf conjecture, reported in 2006 by Conrey and Ghosh [16] and made in 2019 by Fokas [17] based on the estimation of relevant exponential sums.
Note that µ (σ) is the Lindelöf µ function, expressed in Eq. (5) [1]. It was proved that (see [2,18]; [12],p.18) Due to the idea of Littlewood [2], which is the only way of proving the Lindelöf conjecture, the target of the paper is to give the proof of the Lindelöf conjecture by the study of the Lindelöf µ function (see [12], p.18; [18], p.338), with the aid of Theorem 1, which is proved based on the work of Heath-Brown [7]. The structure of the paper is given as follows. In Section 2 we introduce the results for the Riemann zeta function. In Section 3 we present the proof of Conjecture 1. In Section 4 we prove the moment conjecture. Finally, we suggest the new results on the moment for the Riemann zeta function in Section 5.

Preliminaries
In this section we give the recent results on the Riemann Ξ and entire Riemann zeta functions.
Let s n , φ n and ϕ n run the nontrivial zeros of the Riemann zeta function ζ (s), the imaginary part of the Riemann zeta function ζ (s), and the positive imaginary part of the Riemann zeta function ζ (s), respectively.
Suppose that s n = Re (s n ) + ϕ n , then the following representations are equivalent: , , Proof. See the details for the proof of Lemma 1 [11].
Proof. For the details of the proof of Lemma 3, see [10,25]. Proof. For the proof of Lemma 4, see [11].
Remark. Lemma 4 can be derived from Lemmas 2 and 3.
where t ∈ C.
Suppose that s n = Re (s n ) + ϕ n , then the following representations are equivalent: Proof. For the details for the proof of Lemma 5, see [11].
Lemma 6. The Riemann Ξ function Ξ (t) with t ∈ C has infinitely many real zeros φ n ∈ R.

The Lindelöf conjecture is true
In this section we prove the Lindelöf conjecture by using the Heath-Brown statement of the Riemann theorem.

The new proof of the Riemann theorem. We now consider the new proof of Theorem 1.
From the Hadamard product [19] (42) and the Patterson product (see [22], p.34) we have [11] (44) where ξ (0) = 1/2, and for s = 1/2 + it we suggest [11] (45) where t ∈ C and By using Lemma 5 we have from Eq. (45) that where By Lemma 6 and the fact ϕ n run the Riemann-Siegel zeros, we have since by Lemma 2, the Turán inequalities where ξ ( 1 2 ) > 0 and t ∈ C. In view of Lemma 4 we show that there exist s n = Re ( s n ) + iϕ n such that Combining Eqs. (53) and (54) we have ) Thus, which is the same as Eq. (52). This implies that we prove the Heath-Brown statement [7] and that the zeros for ξ (s) are sample [7]. Hence, in other worlds that the Riemann theorem follows. All zeros for the entire Riemann zeta function ξ (s) in the whole complex plane s are demonstrated in Fig. 1. 3.2. The proof of the Lindelöf conjecture. In order to prove the Lindelöf conjecture, from Eq. (38) we have (see [18], p.338) Suppose that ℜ (T ) ≪ T ε for any positive ε > 0 and T → ∞ (see [18], p.338). By Lemma 8 we arrive at (62) µ (σ) = 0 for σ ≥ 1/2, and we give for T → ∞ and any positive ε > 0. Combining Eqs. (61) and (65) we show that for T → ∞ and any positive ε > 0. For T → ∞ and any positive ε > 0 we obtain Thus, we finish the proof of the Lindelöf conjecture.

The moment conjecture is true
In this section we give the proof of the moment conjecture based on the Lindelöf conjecture.
According to Montgomery and Vaughan [18] the alternative representation for the Lindelöf conjecture states that there is in which ℓ > 0 is an absolute constant, t → ∞ and any positive ε > 0 By Eq. (68) and m ∈ N we have where ℓ > 0 is an absolute constant, t → ∞ and any positive ε > 0 Making use of Eq. (69) there exist for every positive ε 1 = mε > 0 and t → ∞, which is in accordance with the result of Landau [34]. With Eq. (69) we present where ℓ > 0 is an absolute constant, T → ∞ and any positive ε > 0. From Eq. (71) we get (73) and from Eq. (72) we have if T → ∞ and ε 1 = mε > 0.
for every positive ε 3 = 2ε 1 > 0, m ∈ N and t → ∞. By Eq. (75) we give From Eq. (77) we have for any positive ε 4 = 2mε > 0 and T → ∞, which is in agreement with the result of Laurincikas and Steuding [14]. By virtue of Eq. (78) we present which is in accord with the result [15]. Thus, we prove the moment conjecture.

New results on the moment for the Riemann zeta function
In this section we propose the new results related to the moment conjecture for the Riemann zeta function.