A CONCISE PROOF OF THE RIEMANN HYPOTHESIS BASED ON HADAMARD PRODUCT∗

A concise proof of the Riemann Hypothesis is presented by clarifying the Hadamard product expansion over the zeta zeros, demonstrating that the Riemann Hypothesis is true.

1 n s and in the whole complex plane by analytic continuation [9].
The Riemann hypothesis is concerned with the locations of the non-trivial zeros of ζ(s), and states that: the non-trivial zeros of ζ(s) have a real part equal to 1 2 [9]. In this article, the truth of the Riemann Hypothesis is demonstrated by employing the Hadamard product of the zeta function and clarifying the principle zeros for the product expansion. The process is outlined in a less abstract form, to be accessible for a wider audience.
The paper is organized as follows. The principle zeros and poles are defined in section 2, the relations between sums and products are shown in section 3, the Proof of the Riemann Hypothesis is demonstrated in section 4 , and the conclusions follow in section 5.
2. Principle Zeros of the Zeta Function. For the case of the Riemann zeta function ζ(s), it has been shown, by Riemann [1], that the zeta function satisfies the following functional equation where the symmetrical form of the functional equation is given as We note that ζ(s) has zeros at s = s m = σ m + it m , s =s m = σ m − it m , and s = −2m with m = 1, 2, 3, . . . . Many assume, from the functional equation (2.2) for ζ(1 − s), that s = 1 − s m and s = 1 −s m are also zeros of zeta. Nevertheless, the principle zeros of ζ(s) are determined only by using the pure argument s in ζ(s); hence, the principle zeros are only at s = s m , s =s m , and s = −2m. Therefore, the sums and products of ζ(s) should only be over the zeros s = s m , s =s m , and s = −2m, whenever appropriate, contrary to the usual statement that "the infinite product is understood to be taken in an order which pairs each root ρ with the corresponding root 1 − ρ" [6] p.39. For correctness, I have rephrased the statement to "the ζ(s) infinite product is understood to be taken in an order which pairs each root s m with the corresponding conjugate roots m "; the difference is minor though the impact is tremendous. Now, the locations of the non-trivial zeros are determined by considering the Euler product of ζ(s) over the set of the prime numbers {2, 3, 5, . . . , p m , . . . }, given by which shows that ζ(s) does not have any zeros for (s) > 1, and by the functional Equation (2.1), no zeros for (s) < 0; save for the trivial zeros at s = −2m, due to the sin( πs 2 )Γ(1 − s) term. Jacques Hadamard (1896) [3] and Charles Jean de la Vallée-Poussin [11] independently proved that there are no zeros on the line (s) = 1. In addition, considering the functional equation and the fact that there are no zeros with a real part greater than 1, it follows that all non-trivial zeros must lie in the interior of the critical strip 0 < (s) < 1. Hardy and Littlewood (1921) [12] have shown that there are infinitely many non-trivial zeros s m on the critical line s = 1 2 +it. We note that the non-trivial principle zeros of ζ(s) are located only in the strip 1 2 ≤ (s) < 1, as shown in Figure (1), whereas the non-trivial zeros of ζ(1 − s) are located in the strip 0 < (s) ≤ 1 2 . Although this is a minor definition clarification, it is critical in proving the Riemann Hypothesis. This has been overlooked, as 1−s m =s m for all the known zeros; thus, the product or sum over the zeros (1 − s m ) is the same as the product or sum overs m for the first ten trillion known zeros [13].

Sums and Products for Zeta Function.
In this section, the sum over the principle poles of a reciprocal function of zeta is developed based on Mittag-Leffler's theorem, in order to showcase the linkage to the Hadamard product over the principle zeros of zeta, by considering a normalized function of ξ(s) given by  Differentiating, we have Note that f (s) f (s) has simple poles at the same zeros of ξ(s) (i.e., the poles are at s = s m and s =s m ). Now, using Mittag-Leffler's theorem for the sum over the poles of the function which was proved by Hadamard [2]. Note the 1 2 ln π term canceled out, as it appears on both sides of the equation. Also, using Mittag-Leffler's theorem for the following function Integrating and taking the antilog, we have that is, which was given by Riemann [1], in a logarithmic form with minor difference from the modern definition of ξ(s). He set s = 1 2 + ti to obtain that is, Proof. It has been shown, by Riemann [1], that the zeta function satisfies the following functional equation:

A Proof of the Riemann Hypothesis.
and, considering the case of the limit when s → 1, we have It is well-known that lim s→1 ζ(s)(s − 1) = 1 and Γ( 1 2 ) = π 1 2 .
Therefore, Equation (4.5) becomes and since (4.7) for all the principle non-trivial zeros (s m = σ m ± it m ) of ζ(s), it implies that Therefore, Equation (4.6) is true only when (2σ m − 1) = 0, which requires that σ m = 1 2 for all the non-trivial zeros of ζ(s). This concludes the proof of the Riemann Hypothesis that: the real part of every non-trivial zero of the Riemann zeta function is σ m = 1 2 .
Also, the proof can be stated in a concise form as (4.9) To validate the result, with σ m = 1 2 , Equation (3.10) can be restated as from which we see that the right hand side of Equation (4.10) is unchanged when s is replaced by (1 − s), obtaining the expressions for ζ(1 − s) and ξ(1 − s) as Therefore, Equations (4.10) and (4.11) are equal, as validated by the well-known ξ(s) functional equation, given by If any zero of ζ(s m ) has σ m = 1 2 in Equation (4.4), it implies that ξ(s) = ξ(1−s), which would contradict Equation (4.12). Therefore, all σ m must be equal to 1 2 . From this, we can hypothesize that the product form of the ξ(s) in Equation (3.11) developed by Riemann [1] was very likely to have been the source of inspiration for the Riemann Hypothesis. Now, as a consequence of the proof of Riemann Hypothesis, combining Equation (2.3) and Equation (4.10), to obtain a relation between all non-trivial zeros of the zeta function and all prime numbers as in particular for s = 2, we have (4.14)

Conclusions.
Proof of the Riemann Hypothesis would unravel many of the mysteries surrounding the distribution of prime numbers, which are at the heart of all encryption systems. In addition, proof of the Riemann Hypothesis would, as a consequence, prove many of the propositions known to be true under the Riemann Hypothesis.
The proof demonstrated here was based on a basic insight into the product expansion of the Riemann zeta function, as available from Hadamard's publication in 1893 and Riemann's publication in 1859, as well as clarifying that the product expansion is only over the principle non-trivial zeros of zeta. Sometimes, the truth is hidden in plain sight.