On the Non-Real Roots of the Riemann Zeta Function ϛ(s)

The Riemann zeta function ϛ(s) plays a very important role in analytic number theory; its importance comes essentially from the very close connection it has with prime numbers; this connection that the great German mathematician Georg Friedrich Bernhard Riemann (1826-1866) had shown in his famous manuscript published in 1859(it is the same date when Charles Darwin (1809-1882) published his work "On the Origin of Species") when he gave an explicit formula linking the function counting the prime numbers with the roots of zeta function ϛ(s), namely the solutions s ∈ of the equation ϛ(s)=0, this explicit formula given by Riemann is ( ) ( ) ( ) ( ) 2 log 2 1 log


Introduction
The Riemann zeta function ϛ(s) plays a very important role in analytic number theory; its importance comes essentially from the very close connection it has with prime numbers; this connection that the great German mathematician Georg Friedrich Bernhard Riemann represents all the non-real roots of ϛ(s) and in his manuscript, Riemann claimed that it is very likely that all these non-real roots have real part equal exactly to 1/2. The connection between the prime numbers and zeta function ϛ(s) which is defined on the half-plane ( ) ( ) comparing the margins of error, we note that The Method of trapezes and half-ellipses THE is more accurate than the following three known methods: The method of rectangles on the left R (l) n , The method of rectangles on the right R (γ) n and The method of trapezes T n .
validity of the Riemann hypothesis and it should be noted here that the key result of the concise proof below lies in the expression of ŋ(s).

The proof of the Riemann hypothesis
We know that the Riemann zeta function ϛ(s) is the analytic function of the complex variable s, defined on the half-plane R is being over the prime numbers p ∈ P={2, 3, 5, 7, 11,...} and the function ϛ(s) is defined in the complex plane C\{l} by analytic continuation. As shown by Riemann, ϛ(s) can be continued analytically to C\{l} as a meromorphic function and has a first order pole at s=1 with residue 1. On the other hand, we know that the Riemann zeta function ϛ(s) is defined for any complex number s different from 1 and with real part strictly greater than 0 by ϛ(s)= η (s)/(1-2 1-s ) [2] where η is the Dirichl et eta function which is defined on the half-plane R (s)>0 by η (s)= ( ) 1 1 n s n − − − ∑ [3].
We also know that there is an important relationship between ϛ(s) and ϛ(1-s). This relationship is defined for any s in the complex plane  by the following Riemann functional equation: ξ(s)=ξ (1-s) [4] where ξ (s)=π -s/2T Ґ (s/2)ϛ(s), Ґ being the Euler gamma function which is defined for any complex number s such that R (s)>0 by: Ґ(s)= . We also know that in 1896, the two mathematicians Jacques Salomon Hadamard (1865-1963) and Charles-Jean de LaVallee Poussin (1866-1962) independently proved that no root of ϛ(s) could be on the line R (s)=1, and that all the nonreal roots have to be in the interior of the critical strip 0 < R (s)<1, and it is known that this demonstration was a key result in the first complete proof of the prime number theorem And the Russian mathematician Ivan Vinogradov (1891-1983), in the thirties, had obtained a region without root of form R (s)>1-k (logτ) -γ where τ=|t|+3, t= J (s), 3/4<γ< 1 and k is a certain positive constant; The largest known region that contains no roots of the function ϛ(s) is given asymptotically by R (s)>1-k(logτ) -2∕3 (1oglogτ) -1∕3 where τ=|t| +3, t=J (s) and k is a certain positive constant; In 1942, the Norwegian mathematician AtleSelberg  demonstrates that at least some fraction of roots of ϛ(s) is on the critical line R (s)=1/2, but its fraction is less than 1%; In 1974, the American mathematician Norman Levinson  increases this fraction to 1/3; In 2004, the two mathematicians Xavier Gourdon and Patrick Demichel use the Odlyzko-Schonhage algorithm and find that the first 10,000,000,000,000 roots of ϛ(s) are on the critical line R (s)=1/2;.
In 2011, an other American mathematician John Brian Conrey (1955) proves that at least 41.05% of the non-real roots of ϛ(s) are aligned on the critical line R (s)=1/2 and are simple; According to the Vinogradov-Korobov method, we have the following property: there are two constants c and C strictly positive such that for all 1/2≤σ≤1and |t| ≥ 2, we have: |ϛ(σ+it)|≤ ( ) ( ) In the current state of knowledge, according to the American mathematic cian Kevin Ford, we can take c=4.45 and C=76.2; Littlewood also has proved the following theorem: "Either the function ϛ(s) or the function ϛ'(s) has an infinity of roots in the strip 1-δ<σ<1, σ= R (s) and o is an arbitrarily small positive quantity"; In 1934, the Swiss mathematician Andreas Speiser (1885-1970) has proved the following theorem: '∀s ∈ C with 0< R (s)<1/2: ϛ'(s) ≠ O"; The Turkish mathematician Gem Yalpn Yildirzm (1961) has proved that: "The Riemann hypothesis implies that ϛ"(s) and ϛ"' (s) do not vanish in the strip 0< R (s)<1/2"; There are still more proven results about the non-real roots of ((s) apart from these results just mentioned, but all these results are not enough to say that the Riemann hypothesis is true. Fortunately, after several attempts we were able to solve this great problem and we believe that if there are many proofs to the Riemann hypothesis, our proof would probably be the simplest and the most beautiful. has to be finite or infinite? it could be equal to 0? why not?". It is from this intelligent and relevant question and based on the explicit expression of η(s) which is described as an infinite complex series for R (s)>0 and based on the special form of its general term (-1) n-1 n -s and referring to universally known formulas and proven theorems that we started our reflection and our reasoning and carefully following the logical and rational implications we were able to find the right answer to the previous question and then to prove the absolute , applying this condition we keep the same infinite number of ηterms in order to converge exactly to the same function in numerator and denominator, To prove the Riemann hypothesis, it seemed to us that it is more convincing to prove the conjecture: If η (s)=0 and 0< R (s)<1, then: In the conjecture above, we denote: 0={0, i0, 0+i0} and ∞  ={ ∞, i∞,  To prove the conjecture [C], we have to prove the following proposition: If η (s)=0, and η (1-s)=0 and 0<R(s)<1, then If m*=m (m *= m is a necessary condition in the following calculation of limits), then we have: exists, then we can write: exists for R (s)>0 and is equal to 0, but we have to prove that L also exists, where: And we know that x ∀ <1: That is to say if z,m → s,∞ with R (s)>0, then: So, if ŋ(z) → 0when z,m → s,∞, then we have: So, according to (*), it follows that if ŋ (s)=0 and ŋ(l-s)=0, then we must have: So, the proposition [P] implies the following result: Thus it has been shown that: that is equivalent to say that However, for any complex s with number s with 0 < R (s) <1, we have shown that if ŋ(s)=0 and 0 < R (s) <1, then R (s)=1/2, and according to [P], we also note that: if ŋ(s)=0 and 0 < R (s) <1, then J(s)=πx/logy where x ∈ Z* and y ∈ ] 0, l[∪]1+∞[.

The proposition [P]
can be generalized by the following statement:  For example, for t being the imaginary part of a non-real root of ϛ(s) and belonging to the critical line R (s)=σ=1/2 and k E *, we have: If we look for more precision and rigor, we have to take into consideration in our calculation of limits the following obvious condition (note: this condition is respected in the previous calculation of limits) [6]: By this condition, we want to say that if someone wants to replace z by z' with z'≠ z in the expression of ŋ, he has to keep the same number of ŋ-terms beginning from n=1 to n=m=∞ to keep exactly the same function and to change only the complex variable z in the expression of ŋ and this condition is valid in the general case.

Numerical verification:
We have made the numerical verification for (*) and (**) using the first 5 roots of the Riemann zeta function ϛ(s) whose real parts are equal to 1/2 and the imaginary parts are positive and we should note that this numerical verification was done using

Conclusion
Based on the following condition: and on the special property that when ŋ (s)=0 for R (s)>0 and z, m → s,∞: and refering to this result, it has been deduced that:

The Method of Trapezes and Half-Ellipses T.H.E Purpose
This method is original and its main purpose is the approximate calculation of the integral of any continuous real function f on a closed interval [a, b] with a<b replacing each arc of f -curve (M i M i+1 ) by an elliptical half-arc (H E) and we remind that this method can be used