A Proof of the Riemann Hypothesis Based on Hadamard Product

By unraveling a persistent misconception in the zeta Hadamard product expansion, and employing the zeta functional equation, a concise proof of the Riemann Hypothesis is presented, which conclusively demonstrate that the Riemann Hypothesis is true.


Introduction
The Riemann hypothesis is a conjecture that the Riemann zeta function ζ(s) has its zeros only at the negative even integers and complex numbers with real part 1 2 . It was proposed by Bernhard Riemann (1859) [1]. Jacques Hadamard in (1893) [2], based on Weierstrass's factorization theorem, showed that the Riemann zeta function ζ(s), can be expressed as an infinite product expansion over the non-trivial zeros of the zeta function. The Riemann Hypothesis is the eighth problem in David Hilbert's list of 23 unsolved problems published in (1900) [3], there has been tremendous work on the subject since then, which are illustrated in Titchmarsh (1930) [4], Edwards (1975) [5], Ivic (1985) [6] and Karatsuba (1992) [7]. However, it is still regarded as one of the most difficult unsolved problems and has been placed as the second most important problem in the list of the Clay Mathematics Institute Millennium Prize Problems (2000), as its proof would shed light on many of the mysteries surrounding the distribution of prime numbers [8] and [9].
The Riemann zeta function is the function of the complex variable s, defined in the half-plane (s) > 1 by the absolutely convergent series and in the whole complex plane by analytic continuation [8]. The Riemann hypothesis is concerned with the locations of the non-trivial zeros of ζ(s), and states that: The nontrivial zeros of ζ(s) have real part equal to 1 2 [8].
In this paper, I will prove the truth of the Riemann Hypothesis by employing the Hadamard product and the functional equation of the zeta function. The proof is given in a less abstract form and includes some 'simplistic steps' for it to be accessible to a wider audience. Proof: It was shown by Riemann [1], that the zeta function satisfies the following functional equation

Proof of the Riemann Hypothesis
in addition, we have the well known fact that where s = σ + it is a complex variable with σ and t being real numbers, and i is the imaginary unit. Now, if ζ(s) = 0, then equations (2) and (3) must be zero, thus Therefore, from equation (5) all the zeros of the zeta function are conjugates, and from equation (4) the zeros must satisfy or ζ(s) = 0.
Furthemore, we note from equations (4) and (5), that the set S of all the principle non-trivial zeros S and the set M of all the principle trivial zeros of ζ(s), these two sets S and M , must satisfy as well both ζ(1−s) = 0 and ζ(s) = 0, i.e. ζ(1 − s r ) = 0, ζ(s r ) = 0, ζ(1 + 2m) = 0 and ζ(−2m) = 0. Now using the Legendre duplication formula for Γ(z), we have and since thus, we have Therefore, the functional equation (2) becomes Now, consider the Hadamard product [2], which is an infinite product expansion over the principle non-trivial zeros s r of the zeta function, given by also we can obtain the Hadamard product expansion over both the set of the trivial zeros M and the set of the non-trivial zeros S, by noting that where γ is the Euler-Mascheroni constant. Thus, we have It is worh noting that there is some misconception in the literature about the zeros that should be included in the product expansions in eqaution (11).
Many assume that the product over the zeros includes the non-principle zeros of ζ(s), i.e. s = 1 − s r and s = 1 − s r , in the Hadamard product expansion (11). Such assunption is incorrect, for in this case we should also include the nonprinciple trivial zeros s = 2 + 2m in the product expansion (12) and s = 1 + 2m in the product expansion (13), as s = 2 + 2m are also zeros of 1/Γ 1 − s 2 , and s = 1 + 2m are also zeros of ζ(1 − s). Clearing this misconception is key to the understanding of the proof of the theorem. Now, from equation (11) i.e.
we note that which gives us Now, equating (10) and (19), we have which simplifies to Now, equation (21) is true only if 1 − 2σ r = 0, which gives σ r = 1 2 , for all the non-trivial zeros of ζ(s), concluding the proof of the Riemann Hypothesis, that the real part of every non-trivial zero of the Riemann zeta function is σ r = 1 2 .

Conclusion
The proof of the Riemann Hypothesis would unravel many of the mysteries surrounding the distribution of prime numbers, as the primes are at the heart of all encryption systems. Furthermore, the proof of the Riemann Hypothesis would as a consequence, prove many of the propositions known to be true under the Riemann Hypothesis. The proof shown in this paper was based on a basic insight into the relation of the Hadamard product expansion and the functional equation of the Riemann zeta function, which were available from Hadamard's publication in (1893) and Riemann's first publication in (1859). Sometimes the truth is hidden in plain sight.