Riemann Paper (1859) Is False

Transcription

1 Riema Paper (859) I Fale Chu-Xua Jiag P O Box94, Beijig 00854, Chia Jiagchuxua@vipohucom Abtract I 859 Riema defied the zeta fuctio ζ () From Gamma fuctio he derived the zeta fuctio with Gamma fuctio ζ () ζ () ad ζ () are the two differet fuctio It i fale that ζ () replace ζ () Therefore Riema hypothei (RH) i fale The Jiag fuctio J ( ω ) ca replace RH AMS mathematic ubject claificatio: Primary M6

2 I 859 Riema defied the Riema zeta fuctio (RZF) [] =Π P = P =, () ζ () ( ) where = σ + ti, i =,σ ad t are real, P rage over all prime RZF i the fuctio of the complex variable with σ 0, t 0,which i abolutely coverget I 896 J Hadamard ad de la Vallee Poui proved idepedetly [] I 998 Jiag proved [] ζ ( + ti) 0 () ζ () 0, () where 0 σ Riema paper (859) i fale [] We defie Gamma fuctio [, ] For σ > 0 O ettig t π x Γ = t e t dt (4) 0 =, we oberve that π x e π x dx (5) Γ = 0 Hece, with ome care o exchagig ummatio ad itegratio, for σ >, π π x Γ ς() = x e 0 = dx ( x) ϑ = x dx 0, (6) where ζ () i called Riema zeta fuctio with gamma fuctio π x ϑ( x): = e, (7) = i the Jacobi theta fuctio The fuctioal equatio for ϑ ( x) i

3 ad i valid for x > 0 ϑ ( ) ϑ ( Fially, uig the fuctioal equatio of ϑ ( x), we obtai x x = x (8) π ϑ( x) ζ () = + ( x + x )( ) ( ) dx (9) Γ From (9) we obtai the fuctioal equatio S π Γ ζ ( ) = π Γ ζ ( ) (0) The fuctio ζ () atifie the followig: ζ () ha o zero for σ > ; The oly pole of ζ () i at =, it ha reidue ad i imple; ζ () ha trivial zero at =, 4, but ζ () ha o zero; 4 The otrivial zero lie iide the regio 0 σ ad are ymmetric about both the vertical lie σ = / The trip 0 σ i called the critical trip ad the vertical lie σ = / i called the critical lie Cojecture (The Riema Hypothei) All otrivial zero of ζ () lie o the critical lie σ = /, which i fale [] ζ () ad ζ () are the two differet fuctio It i fale that ζ () replace ζ (), Pati proved that i ot all complex zero of ζ () lie o the critical lie: σ = / [4] Schadeck poited out that the fality of RH implie the fality of RH for fiite field [5, 6] RH i ot directly related to prime theory Uig RH mathematicia prove may prime theorem which i fale I 994 Jiag dicovered Jiag fuctio J ( ω ) which ca replace RH, if J ( ω) 0 the the prime equatio ha ifiitely may prime olutio; ad if J ( ω ) = 0 the the prime equatio ha fiitely may prime olutio By uig J ( ω ) Jiag prove about 600 prime theorem icludig the Goldbach theorem, twi prime theorem ad theorem o arithmetic progreio i prime [7, 8]

4 I the ame way we have a geeral formula ivolvig ζ () x Fxdx ( ) = x Fxdx ( ) 0 0 = = = = = y F( y) dy ζ ( ) y F( y) dy 0 0, () where F( y) i arbitrary From () we obtai may zeta fuctio ζ () which are ot directly related to the umber theoryuig Jiag fuctio we prove the followig theorem Prime Repreeted by P + mp [9] ()Let = ad m = We have P = P + P We have Jiag fuctio Where χ ( P) = P if (mod P ); χ ( P) = otherwie Sice ( ) 0 prime We have the bet aymptotic formula ( ω) = ( + χ( )) 0 P P (mod P ); χ ( P) = P+ if P / J ω, there exit ifiitely may prime ad uch that P i a P P π (,) = { P, P : P, P, P + P = P prime} where J P P P P ~ 6 ( ) log ( ) log ω = P ( ωω ) ( + χ( )) = Φ ω P P P i called primorial, Φ ( ω) = ( P ) P It i the implet theorem which i called the Heath-Brow problem [0] P0 ()Let = P 0 be a odd prime, m ad m ± b we have 4

5 P = P + mp P0 P0 We have ( ω) = ( + χ( )) 0 P P0 where χ ( P) = P+ if Pm; χ ( P) = ( P0 ) P P0 + if m P P0 P ); χ ( P) = P+ if m (mod P ); χ ( P) = otherwie P (mod Sice J ( ω) 0, there exit ifiitely may prime P ad P uch that P i a prime We have J ( ωω ) π (,)~ P0 Φ ( ω) log The Polyomial P + ( P + ) Capture It Prime [9] ()Let = 4, We have P = P + ( P + ) 4 We have Jiag fuctio ( ω) = ( + χ( )) 0 P Where χ ( P) = P if P (mod 4); χ ( P) = P 4 if P ( mod 8 ) ; χ ( P) = P+ otherwie Sice J ( ω) 0, there exit ifiitely may prime P ad P uch that P i a prime We have the bet aymptotic formula π (,) = { P, P : P, P, P + ( P + ) = P prime} 4, J ( ωω ) ~ 8 ( ) log Φ ω It i the implet theorem which i called Friedlader-Iwaiec problem [] ()Let = 4m, We have 5

6 4m P = P + ( P + ), where m =,,, L We have Jiag fuctio ( ω) = ( + χ( )) 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 Sice J ( ) 0 ω, there exit ifiitely may prime ad uch that P i a P P prime It i a geeralizatio of Euler proof for the exitece of ifiitely may prime We have the bet aymptotic formula ()Let J ( ωω ) 8 mφ ( ω) log π (,)~ = b We have P = P + ( P + ) b where b i a odd We have Jiag fuctio ( ω) = ( + χ( )) 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, J ( ωω ) 4 bφ ( ω) log π (,)~ (4)Let = P 0, We have P = P + ( P + ) P0, P 0 where i a odd prime We have Jiag fuctio ( ω) = ( + χ( )) 0 P 6

7 χ J ω, there exit ifiitely may prime ad uch that P i where χ ( P) = P0 + if P 0 ( P ); ( P) = 0 otherwie Sice ( ) 0 a prime We have the bet aymptotic formula P P J ( ωω ) π (,)~ P0 Φ ( ω) log The Jiag fuctio J ( ω ) i cloely related to the prime ditributio Uig J ( ) ω we are able to tackle almot all the prime problem i the prime ditributio Ackowledgemet The Author would like to expre hi deepet appreciatio to R M Satilli,G Wei, L Schadeck, A Coe, M Huxley ad Che I-wa for their help ad upport Referece [] B Riema, Uber die Azahl der Primzahle uder eier gegebeer Gröe, Moatber Akad Berli, (859) [] PBormei,SChoi, B Rooey, The Riema hypothei, pp8-0, Spriger-Verlag, 007 [] Chu-Xua Jiag, Diproof of Riema hypothei, Algebra Group ad Geometrie, -6(005) Riema pdf [4] Tribikram Pati, the Riema hypothei, arxiv: math/07067v, 9 Mar 007 [5] Lauret Schadeck, Private commuicatio ov [6] Lauret Schadeck, Remarque ur quelque tetative de demotratio Origiale de l Hypothèe de Riema et ur la poiblilité De le prologer ver ue thé orie de ombre premier coitate, upublihed, 007 [7] Chu-Xua Jiag, Foudatio of Satilli ioumber theory with applicatio to ew cryptogram, Fermat theorem ad Goldbach cojecture, Iter Acad Pre, 00 MR004c: 00, pdf [8] Chu-xua Jiag, The implet proof of both arbitrarily log arithmetic progreio of prime, Preprit (006) 7

8 [9] Chu-Xua Jiag, Prime theorem i Satilli ioumber theory (II), Algebra Group ad Geometrie 0,49-70(00) [0] DRHeath-Brow, Prime repreeted by x + y Acta Math 86, -84 (00) 4 [] J Friedlader ad H Iwaiec, The polyomial x + y capture it prime A Math48, (998) 8