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A Concise Proof of the Riemann Hypothesis Based on Hadamard Product Fayez Alhargan

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A Concise Proof of the Riemann Hypothesis Based on Hadamard Product Fayez Alhargan A Concise Proof of the Riemann Hypothesis Based on Hadamard Product Fayez Alhargan To cite this version: Fayez Alhargan. A Concise Proof of the Riemann Hypothesis Based on Hadamard Product. 2021. hal-01819208v5 HAL Id: hal-01819208 https://hal.archives-ouvertes.fr/hal-01819208v5 Preprint submitted on 7 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A CONCISE PROOF OF THE RIEMANN HYPOTHESIS BASED ON HADAMARD PRODUCT∗ FAYEZ A. ALHARGANy Abstract. A concise proof of the Riemann Hypothesis is presented by clarifying the Hadamard product expansion over the zeta zeros, demonstrating conclusively that the Riemann Hypothesis is true. Then, an accurate zero-counting function exhibiting the expected step function behaviour is developed. Also, based on the Heaviside function a unifying analysis of the prime-counting function is presented and its relation to the zeta function is developed via the Laplace transform and the residue theorem. Furthermore, a new s-domain definitions of the prime-counting function and the Chebyshev function are developed, revealing a profound underlining relationship to the zeta function. The paper encompass a new paradigm shift in the prime analysis with a fresh perspective from the s-domain. Key words. the Riemann Hypothesis, the functional equation, the Riemann zeta function, Hadamard Product AMS subject classifications. 11M26 1. Introduction. In his landmark paper in 1859, Bernhard Riemann [1] hypoth- esized that the non-trivial zeros of the Riemann zeta function ζ(s) all have a real part 1 equal to 2 . Major progress towards proving the Riemann hypothesis was made by Jacques Hadamard in 1893 [2], when he 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. In 1896 [3], he also proved that there are no zeros on the line <(s) = 1. The Riemann Hypothesis is the eighth problem in David Hilbert's list of 23 un- solved problems published in 1900 [4]. There has been tremendous work on the subject since then, which has been illustrated by Titchmarsh (1930) [5], Edwards (1975) [6], Ivic (1985) [7], and Karatsuba (1992) [8]. It is still regarded as one of the most diffi- cult unsolved problems and has been named 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 [9, 10]. The Riemann zeta function is a function of the complex variable s, defined in the half-plane <(s) > 1 by the absolutely convergent series 1 X 1 (1.1) ζ(s) := ns n=1 and in the whole complex plane by analytic continuation [9]. The Riemann hypothesis is concerned with the locations of the non-trivial zeros 1 of ζ(s), and states that: the non-trivial zeros of ζ(s) have a real part equal to 2 [9]. In this paper, 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. Furthermore, new concise and accurate results have been obtained for the prime- counting function and the zero-counting function. These were achieved by utiliz- ing new techniques; that included the Heaviside function, Dirac delta function, the ∗Revised Version Submitted to the editors 5 March 2021. yPSDSARC, Riyadh, Saudi Arabia ([email protected]). 1 2 F.A. ALHARGAN Laplace transform, Mittag-Leffler’s theorem, and the residue theorem. Such tech- niques provided concise and elegant solutions both in the x-domain and s-domain, revealing profound underlining connections between the Heaviside prime-counting function, the Dirac delta function, and the zeta function. Which I hope will provide another angle to address prime computations, primality testing, and prime factoriza- tion. 2. The Riemann Hypothesis. 2.1. 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 πs (2.1) ζ(s) = 2sπs−1 sin Γ(1 − s)ζ(1 − s); 2 where the symmetrical form of the functional equation is given as s 1−s − 2 s 2 1−s (2.2) π Γ( 2 )ζ(s) = π Γ( 2 )ζ(1 − s): We note that ζ(s) has zeros at s = sm = σm + itm, s =s ¯m = σm − itm, and s = −2m with m = 1; 2; 3;::: . Many assume, from the functional equation (2.2) for ζ(1 − s), that s = 1−sm 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 = sm, s =s ¯m, and s = −2m. Therefore, the sums and products of ζ(s) should only be over the zeros s = sm, 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 clarity, I have rephrased the statement to "the ζ(s) infinite product is understood to be taken in an order which pairs each root sm 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 f2; 3; 5; : : : ; pm;::: g, given by Y 1 (2.3) ζ(s) = ; 1 − 1 p ps 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 πs to the sin( 2 )Γ(1 − s) term. Jacques Hadamard (1896) [3] and Charles Jean de la Vall´ee-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 1 shown that there are infinitely many non-trivial zeros sm on the critical line s = 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 1 located in the strip 0 < <(s) ≤ 2 . Although this is a minor definition clarification, it is critical in proving the Riemann Hypothesis. This has been overlooked, as 1−sm =s ¯m for all the known zeros; thus, the product or sum over the zeros (1 − sm) is the same as the product or sum overs ¯m for the first ten trillion known zeros [13]. PROOF OF RIEMANN HYPOTHESIS 3 t The Critical Strip ζ(s) 1 − s¯m sm σ = 0 1 σ = 1 σ σ = 2 1 − sm s¯m ζ(1 − s) Fig. 1. The Critical Strip. 2.2. 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 s s − 2 (2.4) f(s) = 2ξ(s) = ζ(s)(s − 1)sΓ( 2 )π ; which is an entire function with f(s) = f(1 − s), f(1) = f(0) = 1, and has principle zeros only at s = sm and s =s ¯m. Thus, the ζ(s) infinite product is understood to be taken in an order which pairs each root sm with the corresponding conjugate root s¯m. Now, taking the log, we have s s (2.5) ln f(s) = ln 2 + ln ξ(s) = ln ζ(s) + ln(s − 1) + ln s + ln Γ( 2 ) − 2 ln π: Differentiating, we have 0 0 0 0 s f (s) ξ (s) ζ (s) 1 1 Γ ( 2 ) 1 (2.6) = = + + + s − 2 ln π; f(s) ξ(s) ζ(s) (s − 1) s Γ( 2 ) which gives 0 f (0) (2.7) = ln 2π − 1 − 1 γ − 1 ln π: f(0) 2 2 0 f (s) Note that f(s) has simple poles at the same zeros of ξ(s) (i.e., the poles are at s = sm and s =s ¯m). Now, using Mittag-Leffler’s theorem for the sum over the poles of the function 0 f (s) f(s) , we obtain 0 0 s ζ (s) 1 1 Γ ( 2 ) 1 + + + s =[ln 2π − 1 − 2 γ] ζ(s) (s − 1) s Γ( 2 ) (2.8) 1 X 1 1 1 1 + + + + : (s − s ) s (s − s¯ ) s¯ m=1 m m m m 4 F.A.
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