RIEMANN HYPOTHESIS PROOF

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1 RIEMANN HYPOTHESIS PROOF IDRISS OLIVIER BADO October, 08 Abstract The mai cotributio of this paper is to achieve the proof of Riema hypothesis. The key idea is based o ew formulatio of the problem ζ(s) = ζ( s) re(s) =. This proof is cosidered as a great discovery i mathematic. Itroductio The Riema Hypothesis is a cojecture cocerig the zeros of the Riema zeta fuctio. First proposed by Berhard Riema i 859 [], the hypothesis has yet to be tested despite great efforts for over 00 years. The hypothesis remais oe of Hilbert s usolved problems [], as well as beig a Milleium Prize Problem [3].To fully uderstad the Riema Hypothesis we must first itroduce the Riema zeta fuctio ζ(s). zeta fucti is defie as ζ(s) = + s C such as re(s) ], + [ Aother form of the zeta s fuctio which liks prime to zeta is give by Euler re(s) ], + [ ζ(s) = p P. By simple itegratio we ca see that : p s s, re(s) ], + [, l(ζ(s)) = s + π(u)du u(u s ) I order to use zeta fuctio to explai the Riema Hypothesis, we must first exted the domai of the fuctio to all complex values of s. through the olivier.bado@esea.edu.ci,ewdiscoveryresearch.blogspot.com

2 method of aalytic cotiuatio. First, we defie the Dirichlet eta fuctio, which coverges for ay complex umber s with re(s) ]0, + [, ad is give by the followig Dirichlet series: re(s) ], + [ ζ(s) = p P p s Zeta fuctio has bee exteded to all of complex by usig Dirichlet eta ζ(s) = η(s) with η(s) = ( ) s fuctio through the method of aalytic cotiuatio Riema s hypothesis attracts attetio from emiet s mathematicias because may mathematical theorems rely o its validity: a proof of this cojecture would be a real achievemet i mathematics For example, it would improve the Miller-Rabi primality test - a algorithm to determie whether a give umber is prime - is based o the geeralized Riema hypothesis]. If it turs out to be wrog it would be a upheaval. The Riema hypothesis ca also be vital to improve the error term i the prime umber theorem [3]. This theorem attempts to estimate the rate at which prime umbers appear, or more precisely, the rate at which they become less commo. The prime umber theorem states that the fuctio of coutig prime umbers π(x) x.this approximatio was doe by Gauss ad Legedre. By usig Riema zeta fuctio Hadamard ad de la Vallee-Poussi l x i 896 prove the estimatio π(x) = x dt l t + l x (xe a ) The Riema Hypothesis will brushig up the error term ad we would have : x dt π(x) = l t + ( x l x) May have tried to prove Riema s hypothesis, but without success. Several statemets implyig the hypothesis of riema have emerged. Here we will show some examples statemets equivalet to the Riema hypothesis all offerig a differet way of approach the proof of the hypothesis. the first equivalet statemet doe by Robi asserts that d d < eγ l, > 5040 [4] where γ = lim [ l ] + I cotiuatio with Robi works Jeff Lagarias shows that Riema hypothesis will follow if d d + e l( )

3 [5]. Priciple of the proof Our proof is based o the followig assertio, ζ(s) = ζ( s) re(s) = From the Fuctioal relatioship ζ(s) = η(s) s where ζ(s) ζ( s) = B (s) = The the formulatio becomes : We deduce ( ) B (s) s s s s ( ) B (s) = 0 re(s) =. For the proof we must first establish the followig theorem Theorem It exists a C- module E cotaiig B ad a moomorphism of C- module O E which zero is B. Proof As far as the proof is cocered, let us cosider a complex umber such as re(s) ]0, [ ad let s desigate by F(C C)the set of complex fuctios ad A set defied as follows A = {f F(C C) : s C, re(s) ]0, [, f(s) + f( s) = 0 f = 0} Let E be a C- module formed by providig stability property by additio ad subtractio to A Let Θ a complex fuctio Θ(s) = s, s C Let T be a applicatio o E defie as follow : f E, T f = f + f Θ It is clear that T is moomorphism of C- module Furthermore T B (s) = (B + B Θ)(s) = 0, s the T B = 0 3

4 . The proof of Riema hypothesis Let cosider ad suppose that so As so F (s) = T (s + s + ζ(s) ζ Θ(s)) ζ(s) = ζ( s) F (s) = T (s + s ) T (s + s ) = T (id + id )(s) T (s + s ) = (id + id )(s) + (id + id ) Θ(s) = 0 T (id + id )(s) = 0 (id + id )(s) = 0 hece s + s = for the followig re(s) = re(s) = the s + s = so Reciprocally Let s such as F (s) = T (ζ ζ Θ)(s) = T (ζ(s) ζ Θ(s)) = T ( ) B (s) As T is moomorphism of C-module. Providig a topology o E which make T cotiuous the T ( ) B (s) = ( ) T B (s) = 0 F (s) = T (ζ ζ Θ)(s) = 0 (ζ ζ Θ)(s) = 0 the ζ(s). = ζ( s).3 corollary Let s be a complex umber such as re(s) ]0, [ the ζ(s) = 0 re(s) = 4

5 .4 Proof Let s be a complex umber such as re(s) ]0, [ ad suppose that ζ(s) = 0, accordig to the fuctioal relatio ζ(s) = s π s si( πs )Γ( s)ζ( s) the ζ( s) = 0. We have i this case ζ(s) = ζ( s) the re(s) = Reciprocally,let suppose that re(s) = the ζ(s) = ζ( s) fially ζ(s) = 0 3 Coclusio I this article we have proved Riema s hypothesis which is such a importat problem. our fudametal result derives its essece from the ew formulatio of the problem. This opes a ew trajectory for sciece. 4 Ackowledgemet I would like to express my deepest appreciatio to all those who provided me the possibility to complete this article. A special gratitude I give to Professor Taoe Fraçois for his ecouragemet. 5 Refereces [] Berhard Riema. Ueber die Azahl der Primzahle uter eier gegebee Grosse. Ges. Math. Werke ud Wisseschaftlicher Nachlaß, 45 55, 859. [] Wikipedia, Hilbert s problems, accessed 6//05, s problems. [3] Erico Bombieri. Problems of the milleium: The Riema hypothesis 000 [4]YougJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Sole, O Robi s criterio for the Riema hypothesis, J. Theor. Nombres Bordeaux 9 (007), o., [5] J. C. Lagarias, A elemetary problem equivalet to the Riema hypothesis, Amer. Math. Mothly 09 (00), o. 6,