Riemann s Hypothesis and Conjecture of Birch and Swinnerton-Dyer are False

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1 Riemann s Hypothesis and Conjecture of Birch and Swinnerton-yer are False Chun-Xuan Jiang. O. Box 3924, Beijing 854 China jcxuan@sina.com Abstract All eyes are on the Riemann s hypothesis, zeta and L-functions, which are false, read this paper. The Euler product converges absolutely over the whole complex plane. Using factorization method we can prove that Riemamn s hypothesis and conjecture of Birch and Swinnerton-yer are false. All zero computations are false, accurate to six decimal places. Riemann s zeta functions and L functions are useless and false mathematical tools. Using it one cannot prove any problems in number theory. Euler totient function ( n) and Jiang s function J ( ) replace zeta and L functions.. Introduction n will The function () s defined by the absolute convergent series () s () s n in complex half-plane Re () s is called the Riemann s zeta function. The Riemann s zeta function has a simple pole with the residue at s n and the function () s is analytically continued to whole complex plane. We then define the () s by the Euler product s () s ( ), (2) where the product is taken all primes, s it, i, and t are real. The Rieman s zeta function () s has no zeros in Re () s. The zeros of () s Re () s are called the nontrivial zeros. In 859 G. Riemann conjectured that every zero of () s would lie on the line Re () s /2. It is called the Riemenn s hypothesis. [] We have We define the elliptic curve [2] where is the congruent number. ( s it, ) (3) E : y x x, (4) in Assume that is square-free. Let be a prime number which does not divide 2. Let N

2 denote the numbers of pairs ( x, y) where x and y run over the integers modulo,which satisfy the congruence ut y x x mod. (5) a N (6) We then define the L function of E by the Euler product s 2s (7) (,2 ), s) ( a ) where the product is taken over all primes which do not divide 2. The Euler product converges absolutely over the half plane Re () s 3/2, but it can be analytically continued over the whole complex plane. For this function, it is the vertical line Re () s which plays the analogue of the line Re () s /2 for the Riemann zeta function and the irichlet L functions. Of course, we believe that every zero of, s) in Re () s should lie on the line Re () s. It is called a conjecture of Birch and Swinnerton-yer (BS). We have 2. Riemann s Hypothesis is false, s it, 3/2) (8) Theorem. Euler product converges absolutely in Re () s. Let s /2 it, using factorization method we have ( s /2 it) (9) roof. Let s 2 s,2.2 s,2.8 s,3 s,4 s,5 s, s We have the following Euler product equations s (2 s) ( s) ( ), ().2s.2s (2.2 s) ( s) ( ), () s.8s.8s (2.8 s) ( s) ( ), (2) s, (3) 2 (3 ) ( ) ( s s s s ) 2

3 , (4) 2 (4 ) ( ) ( s s s s ) ( ) s 2s 3s 4s s s (5 ) ( ) ( ), (5) ( s ) ( s) ( ) s ( ), (6) Since the Euler product converges absolutely in Re () s, the equation ()-(6) are true. From ()-(6) we obtain ( s ) (9) All zero computations are false and approximate, accurate to six decimal places. Using three methods we proved the RH is false [3]. Using the same Method we are able to prove that all Riemann s hypotheses also are false. All L functions are false and useless for number theory. 3. The Conjecture of Birch and Swinnerton-yer is false. Theorem 2. Euler product converges absolutely in Re () s 3/2. Let s it. Using factorization method we have, s it) (7) roof. Let s 2 s,3 s,4 s, we have the following Euler product equations. 2s 3s 2s ( a ) a,2 s), s) s 2s (,2 ) a (8) 3s 4s 5s 4s a a,3 s), s) s 2s (,2 ) a (9) 4s 6s 7s 6s a a,4 s), s) s 2s (,2 ) a (2) Since the Euler product converges absolutely in Re () s 3/2, equations (8)-(2) are true. From (8)-(2) we obtain, s) (7) All zero computations are false and approximate. Using the same method we are able to prove all LEs (, ) in whole complex plane. The elliptic curves are not related with the iophantine equations and number theory [4]. Frey and Ribet did not prove the link between the elliptic curve and Fermat s equation [4,5]. Wiles proved Taniyama-Shimura conjecture based on the works of Frey, Serre, Ribet, Mazuer and Taylor, which 3

4 have nothing to do with Fermat s last theorem [6]. Taniyama-Shimura conjecture was in obscurity for about 2 years till people seriously started thinking about elliptic curves. Mathematical proof does not proceed by personal abuse, but by show careful logical argument. Wiles proof of Fermat s last theorem is false [7-9]. In 99 Jiang proved directly Fermat s last heorem [,]. 4. Conclusion. The zero computations of zeta functions and L functions are false. Riemann s zeta functions and L functions are useless and false mathematical tools. Using it one cannot prove any problems in number theory [2]. The heart of Langlands program(l) is the L functions [3]. Therefore L is false. Wiles proof of Fermat last theorem is the first step in L. Using L one cannot prove any problems in number theory, for example Fermat s last theorem [6]. Euler totient function ( n) and Jiang s function J ( ) n will replace Riemann s zeta functions and L functions [3-5]. Acknowledgments The author would like to thank professor R.M.Santilli for help and support. References [] B. Riemann, Uber die Anzahl der rimzahlen under enier gegebener GrÖsse, Monatsber. Akad. Berlin (859) [2] John Coates, Number theory, Ancient and Modern, In: ICCM 27. vol. I, 3-2. [3] Chun-Xuan Jiang, isproofs of Riemann s hypothesis. Algebras, Groups and Geometries, 22, 23-36(25) [4] G. Frey, Links between stable elliptic curves and certain diophantine equations, Annales Universitatis Saraviensis (986), -4. [5] K. A. Ribet, On modular representations of Gal ( Q/ Q) arising from modular forms, Invent. Math. (99), [6] A. Wiles, Modular elliptic curves and Fermat s last theorem, Ann. of Math. 4(995), [7] G. erelman, erelman disproves Wiles proof of Fermat s theorem. [8]Y.G.Zhivotov,Fermat slasttheoremandkennethribet smistakes, html. [9] Y. G. Zhivotov, Fermat s last theorem and mistakes of Andrew Wiles, [] Chun-Xuan Jiang, Automorphic function and Fermat s last theorem () 4

5 [] Chun-Xuan Jiang, Automorphic function and Fermat s last theorem (3)(Fermat s proof of FLT), [2] Arthur, et al., editors, On certain L functions. AMS, CMI. 2. volume3. [3] S. Gelbart, An elementary introduction to the Langlands program, Bull, of AMS. (984) [4] Chun-Xuan Jiang, The Hardy-Littlewood prime k tuple conjecture is false. ( ( [5] Chun-Xuan Jiang, Jiang s function J ( ) n in prime distribution, ( ( [6] Chun-Xuan Jiang, Foundations of Santilli s isonumber theory with applications to new cryptograms, Fermat s theorem and Goldbach s conjecture, Inter. Acad. ress, 22. MR24c:/. ( ( 5