Riemann Paper (1859) Is False

Transcription

1 Riemann Paper (859) I Fale Chun-Xuan Jiang P O Box94, Beijing 00854, China Jiangchunxuan@vipohucom Abtract In 859 Riemann defined the zeta function ζ () From Gamma function he derived the zeta function with Gamma function ζ () ζ () and ζ () are the two different function It i fale that ζ () replace ζ () Therefore Riemann hypothei (RH) i fale The Jiang function J ( ω ) can replace RH n AMS mathematic ubject claification: Primary M6

2 In 859 Riemann defined the Riemann zeta function (RZF) [] =Π P = P n= n, () ζ () ( ) where = σ + ti, i =,σ and t are real, P range over all prime RZF i the function of the complex variable with σ 0, t 0,which i abolutely convergent In 896 J Hadamard and de la Vallee Pouin proved independently [] In 998 Jiang proved [] ζ ( + ti) 0 () ζ () 0, () where 0 σ Riemann paper (859) i fale [] We define Gamma function [, ] For σ > 0 On etting t n π x Γ = t e t dt (4) 0 =, we oberve that π n x e dx (5) n π x Γ = 0 Hence, with ome care on exchanging ummation and integration, for σ >, π x e dx n π x Γ ς() = 0 n= ( x) ϑ = x dx 0, (6) where ζ () i called Riemann zeta function with gamma function n π x ϑ( x): = e, (7) n= i the Jacobi theta function The functional equation for ϑ ( x) i

3 ϑ ( ) ϑ ( and i valid for x > 0 Finally, uing the functional equation of ϑ ( x), we obtain x x = x (8) π ϑ( x) ζ () = + ( x + x )( ) dx ( ) (9) Γ From (9) we obtain the functional equation S π Γ ζ ( ) = π Γ ζ ( ) (0) The function ζ () atifie the following ζ () ha no zero for σ > ; The only pole of ζ () i at =, it ha reidue and i imple; ζ () ha trivial zero at =, 4, but ζ () ha no zero; 4 The nontrivial zero lie inide the region 0 σ and are ymmetric about both the vertical line σ = / The trip 0 σ i called the critical trip and the vertical line σ = / i called the critical line Conjecture (The Riemann Hypothei) All nontrivial zero of ζ () lie on the critical line σ = /, which i fale [] ζ () and ζ () are the two different function It i fale that ζ () replace ζ (), Pati proved that i not all complex zero of ζ () lie on the critical line: σ = / [4] Schadeck pointed out that the fality of RH implie the fality of RH for finite field [5, 6] RH i not directly related to prime theory Uing RH mathematician prove many prime theorem which i fale In 994 Jiang dicovered Jiang function Jn( ω ) which can replace RH, if Jn( ω) 0 then the prime equation ha infinitely many prime olution; and if Jn( ω ) = 0 then the prime equation ha finitely many prime olution By uing Jn( ω ) Jiang prove about 600 prime theorem including the Goldbach theorem, twin prime theorem and theorem on arithmetic progreion in prime [7, 8]

4 In the ame way we have a general formula involving ζ () = 0 0 n= n= n= x Fnxdx ( ) x Fnxdx ( ) = y F( y) dy = ζ ( ) y F( y) dy n 0 0, () where F( y ) i arbitrary From () we obtain many zeta function ζ () which are not directly related to the number theory The prime ditribution are order rather than random The arithmetic progreion in prime are not directly related to ergodic theory, harmonic analyi, dicrete geometry, and combinatorie Uing the ergodic theory Green and Tao prove that there exit infinitely many arithmetic progreion of length k coniting only of prime which i fale [9, 0, ] Fermat lat theorem (FLT) i not directly related to elliptic curve In 994 uing elliptic curve Wile proved FLT which i fale [] There are Pythagorean Theorem and FLT in the complex hyperbolic function and complex trigonometric function In 99 without uing any number theory Jiang proved FLT which i Fermat marvelou proof [7, ] n n Prime Repreented by P + mp [4] ()Let n = and m = We have P = P + P J( ω) = ( P P+ χ( P)) 0, P P Where χ ( P) = P if (mod P ); χ ( P) = P+ if / (mod P ); χ ( P) = otherwie Since Jn( ω) 0, there exit infinitely many prime P and P uch that P i a prime We have the bet aymptotic formula P 4

5 π (,) = { P, P : P, P, P + P prime} where ω J ( ωω ) P( P P+ χ( P)) P ~ = 6 ( ) log Φ ω P ( ) log Φ ( ) = ( P ) = P i called primorial, ω P P It i the implet theorem which i called the Heath-Brown problem [5] P0 ()Let n= P0 be an odd prime, m and m ± b we have P = P + mp P0 P0 We have J( ω) = ( P P+ χ( P)) 0, P P0 where χ ( P) = P+ if Pm; χ ( P) = ( P0 ) P P0 + if m P P (mod P0 P ); χ ( P) = P+ if m (mod P ); χ ( P) = otherwie Since Jn( ω) 0, there exit infinitely many prime P and P uch that P i a prime We have J ( ωω ) π (,)~ P0 Φ ( ω) log n The Polynomial P + ( P + ) Capture It Prime [4] ()Let n = 4, We have P = P + ( P + ), 4 J( ω) = ( P P+ χ( P)) 0, P Where χ ( P) = P if P (mod 4); χ ( P) = P 4 if P ( mod 8 ) ; χ ( P) = P+ otherwie 5

6 Since Jn( ω) 0, there exit infinitely many prime P and P uch that P i a prime We have the bet aymptotic formula 4 π (,) = { P, P : P, P, P + ( P + ) prime} J ( ωω ) ~ 8 ( ) log Φ ω It i the implet theorem which i called Friedlander-Iwaniec problem [6] ()Let n= 4m, We have P = P + ( P + ), 4m where m =,,, J( ω) = ( P P+ χ( P)) 0, P P i where χ ( P) = P 4m if 8 m ( P ); χ( P) = P 4 if 8( P ) ; χ ( P) = P if 4( P ) ; χ ( P) = P+ otherwie Since J ( ω) 0, there exit infinitely many prime P and P uch that P i a prime It i a generalization of Euler proof for the exitence of infinitely many prime We have the bet aymptotic formula J ( ωω ) 8 mφ ( ω) log π (,)~ ()Let n= b We have P = P + ( P + ), b where b i an odd J( ω) = ( P P+ χ( P)) 0, P where χ ( P) = P b if 4 b ( P ); χ( P) = P if 4( P ) ; χ ( P) = P+ otherwie We have the bet aymptotic formula 6

7 J ( ωω ) 4 bφ ( ω) log π (,)~ (4)Let n= P0, We have P = P + ( P + ), P0 where P 0 i an odd prime J( ω) = ( P P+ χ( P)) 0, P where χ ( P) = P0 + if P 0 ( P ); χ( P) = 0 otherwie Since J ( ω) 0, there exit infinitely many prime P and P uch that P i alo a prime We have the bet aymptotic formula J ( ωω ) π (,)~ P0 Φ ( ω) log The Jiang function Jn( ω ) i cloely related to the prime ditribution Uing J ( ) n ω we are able to tackle almot all prime problem in the prime ditribution Acknowledgement The Author would like to expre hi deepet appreciation to R M Santilli,G Wei, L Schadeck, A Conne, M Huxley and Chen I-wan for their help and upport Reference [] B Riemann, Uber die Anzahl der Primzahlen under einer gegebener Gröe, Monatber Akad Berlin, (859) [] PBormein,SChoi, B Rooney, The Riemann hypothei, pp8-0, Springer-Verlag, 007 [] Chun-Xuan Jiang, Diproof of Riemann hypothei, Algebra Group and Geometrie, -6(005) Riemann pdf [4] Tribikram Pati, the Riemann hypothei, arxiv: math/07067v, 9 Mar 007 [5] Laurent Schadeck, Private communication ov

8 [6] Laurent Schadeck, Remarque ur quelque tentative de demontration Originale de l Hypothèe de Riemann et ur la poiblilité De le prolonger ver une thé orie de nombre premier conitante, unpublihed, 007 [7] Chun-Xuan Jiang, Foundation of Santilli ionumber theory with application to new cryptogram, Fermat theorem and Goldbach conjecture, Inter Acad Pre, 00 MR004c: 00, pdf [8] Chun-xuan Jiang, The implet proof of both arbitrarily long arithmetic progreion of prime, Preprint (006) [9] B Kra, The Green-Tao theorem on arithmetic progreion in the prime: an ergodic point of view, Bull Am Math Soc 4, -(006) [0] B Green and T Tao, The prime contain arbitrarily long arithmetic progreion, Ann Math67, (008) [] TTao,The dichotomy between tructure and randomne, arithmetic progreion, and the prime In proceeding of the international congre of mathematician (Madrid 006) Europ Math, Soc Vol, , 007 [] A Wile, Modular elliptic curve and Fermat lat theorem, Ann Math 4, (995) [] Chun-Xuan Jiang, Fermat marvelou proof for Femart lat theorem, preprint (007), ubmit to Ann Math [4] Chun-Xuan Jiang, Prime theorem in Santilli ionumber theory (II), Algebra Group and Geometrie 0,49-70(00) [5] DRHeath-Brown, Prime repreented by x + y Acta Math 86, -84 (00) 4 [6] J Friedlander and H Iwaniec, The polynomial x + y capture it prime Ann Math48, (998) 8