Riemann Hypothesis and Primorial Number. Choe Ryong Gil

Transcription

1 Rieann Hyohesis Priorial Nuber Choe Ryong Gil Dearen of Maheaics Universiy of Sciences Gwahak- dong Unjong Disric Pyongyang DPRKorea Eail; Augus 8 5 Absrac; In his aer we consider he Rieann hyohesis by he riorial nubers Keywords; Rieann hyohesis Priorial nuber Inroducion ain resul of aer Le N be he se of he naural nubers The funcion ϕ ( n = n ( n is called he Euler s funcion of n N([3] Here n noe is he rie divisor of n Robin showed in his aer [5] (also see [4] [Robin Theore] If he Rieann hyohesis (RH is false hen here exis consans < β < / γ c > such ha σ ( n e n n+ c n n/ ( n β holds for infiniely any n N where σ n = d is he divisor funcion of n N([5] γ = 577L is Euler s consan ([3] dn Fro his we have [Theore ] If here exiss a consan c such ha ( γ n/ ϕ n e c n ex n n (* holds for any n hen he RH is rue For n N ( n we define Φ n = ex ex e n/ ϕ n / n ex n n Then we give [Theore ] For any n we have Φ ( n 4 [Corollary] For any n 5 we have ( ( ( n n γ e n+ 483 ϕ ( n n Proof of Theore I is clear haσ n ϕ n n for any n If (* holds bu he RH is false hen ( n ( n ( n ( ( γ c n σ n γ e n+ e c n ex n n β n ϕ holds for infiniely any n N On he oher h since ( + ( > we have ( c n ( n ( n n c n ( n ex = + + = ( n c ( n = n n n c n n Therefore for infiniely any n N we have e ( n n ( n c n c + Fro his we β n n c n have < e c + / ( n bu i is a conradicion β β n n n

2 3 Reducion o he riorial nuber Le = = 3 3 = 5 L be firs consecuive ries Then is h rie nuber The L is called he riorial nuber ([8] Assue n= q λ L q λ is he rie nuber ( facorizaion of n N Here q L q are disinc ries λ L λ are nonnegaive inegers Pu I = L hen i is clear ha n I n I = ( qi ( i= i= i = ϕ n ϕ I ( Φ ( n Φ ( I so This shows ha he boundedness of he funcion Φ for n N is reduced o one for he riorial nubers 4 Soe sybols b E by [6] where E ( =Ο ex ( a ( a > b= γ + ( / / 6 is a real nuber Pu hen we + = L F =I / ϕ ( I have ( F = ( / i = / i / i / i= i= + + i = i= = ( / / i i i b E = = = ( / / ( / / i i i γ E = = = + γ + E + ε I is known = + + n where ε ( = ( / + / Fro his we have > ( e γ γ F = e ex( e F = e where e = ex( E( + ε ( e = ex( ( e Siilarly we have ( e F = e ex( e F = e where e = ex( E( + ε ( e = ex( ( e ϑ = ([3] Then by he rie nuber heore We recall he Chebyshev s funcion ([3] i is known ha ϑ ( = ( + θ ( where ( ( ( a ( a I = α I = α where α = + θ( α θ( Now we u N = I ( I ( i = C = Φ ( I ( i i i θ = Ο ex > Then we see = + 5 Soe nuerical esiaes 5 An esiae of e e 4 We u = = below For he heoreical calculaion we assue e The discussion 4 for e is suored by MATLAB Since( e F = e < + / ( by (33 of [6] we resecively have ( 4 ( 4 e < + / < 5 e e < ex(/ < 75 e 4 e e < 8( e 5 An esiae of ( e e Since if e hen e we have e e On he oher h i is known ha by (37 (3 of [6] / E = b / > Hence since ε < if e > a: E ε / 5( e hen we have < = + < 4 so

3 We have ( n /! / 5 = a n= e a a n a a a a ( b= ( + a+ 5 a 3 ( 4 e e = ex a+ e + b+ b / b where Therefore we have ε 53 An esiae of K : = ( e α ( e α e 4 ( ε e e + + E E + e > e I is clear ha α α = I I = = ( = ( / i ( / i= i= i E E b b = / + = ε = / / Fro his we have ε e e e = + = ex e e Thus we have e e K = e = e ex = ( μ e e e where μ = ex Hence we ge e e e e μ ex + / e e e+ / e+ 53 e > e 4 μ e e e + 55 e > e + E 4 54 An esiae of ( Pu f ( = ( E( θ ( g( ( α ( ( real nuber α = + θ is a osiive consan such ha α ( f ( g( are coninuously differeniable funcions on he inerval( + funcions ( / b ϑ ( = are consans on( + we have ( ( f = d( = + where is a g E = b = θ ( /4 + /4 Then boh In fac since he ϑ( ϑ( θ( = = = f so on hence f ( = ( + E( where f ( is he derivaive of he funcion Thus he funcion d( is also coninuously differeniable on he inerval( + since Now we will arbirary ake x x such ha < x < x < + fix i Then we have d( x d( x = ( f ( x f ( x d ( g( x g( x g 3 g >

4 hence x F( d = where g g( x x ( = d = d( x F( = d ( f ( d g ( ( α g = + 4 ( α for ( x x g ( By he ean value heore for inegrals of [7] here exiss a oin ξ such ha x < ξ < x x F( d F( ξ = ( x x = On he oher h since d ( = ( f ( d( g ( x g = + for any ( x x we have d( d ( ( ( ( g( g g( g g f d g d g ( ( ( ( ( ( F = f g = g g d d d g F ξ = ( g g ξ d ( ξ ( d( ξ d g ( = g ξ hence 54 Proof of d ( < ( ( α g ( As above enioned since( / E( ( / 4 > e we easily see E( 8 f ( = + < = < 4 ( α f < g for any x x where f is he second-order derivaive funcion of hence ( f ( Fro his we have d ( < for any ( x x In fac i is clear ha d ( ( where d ( = f ( d( g ( g ( d ( g 4 < is equivalen o A < + / where β = + α f ( 8 f ( A= E( + f ( β + β 4 ( α On he oher h i is known ha ( / θ ( ( / ( 4 4 since α ( e we have ( α α /4 by (35 (36 of [6] And = > hence 4 A < e This shows ha d ( ( α < for any x x 54 Proof of F ( < under d ( g( We here assue d ( g( for any ( x x Then we have F ( In fac since 4 we will call i he condiion (d below < for any ( x x under he condiion (d where F ( = ( g g( d ( ( d( d g ( d ( g ( g ( g > for any x x i is clear ha F ( ( < is equivalen o ( g g d ( ( d( d ( g ( d ( g ( + < 4

5 Since g( is he increasing funcion on he inerval ( sufficien o show exiss a oin ( d d ( g ( d ( g ( x x so g g( d < i is ( < By he ean value heore of [7] here such ha x < < d( d( x d ( ( x ( > hence d ( ( x d ( ( x x d ( = Fro he condiion (d we have d because x x + = Also by he ean value heore of [7] here exiss a oin such ha < < d d = d On he oher h for any x x we have ( ( ( ( 4 ( d ( g( A ( e = Fro his since d ( > we have ( d ( ( d ( ( g( 5 5 g = + + g( g( g ( ( ( ( 5 g g( g g g 4 g By he condiion (d we have ( d( d ( g ( d ( ( g ( d ( ( d ( + ( g ( hence ( ( ( ( (( ( ( ( d + / g g 3 d g < d g F < 543 An esiae for he oin ξ of Here we will obain ( x x F ξ = under he condiion (d ξ of ξ = + / for he oin F ξ = under he condiion (d By he ean value heore for inegrals of [7] by x ξ x F ( d = F( d+ F( d = x x here exis oins λ λ such ha x < λ < ξ < λ < x F λ ξ x + F λ x ξ = (F ( ( ( ( Since F ( < for ( x x under he condiion (d we here have F( λ F( ξ F( λ On he oher h we denoe by y( x he line assing hrough he oins ( ( ( x F ( λ hen we have F( λ F( λ y( x = ( x x + F ( λ x x ξ If he line y( x inersecs he line y = a he oin x hen F( λ ( x x F( λ ( x x F ( λ ( x ξ ( x x Fro (F (F we have = = F ( λ ( ξ x ( x x (F we also obain > = > y x = hence we have x F λ + = (F ( λ + ( λ ( = ( λ ( λ ( F F x x F F x ξ x ξ x F + F =F = F ( λ ( λ ( λ ( λ ξ x x x 5 so x + x = x + ξ Fro (F ξ

6 Since F where δ ( λ ( ξ ( x + x+ δ ξ + ( x+ δ x = x ( ξ + x+ δ x + ( x+ δ ξ = F ( λ F ( λ = x x δ F λ F λ = x x F λ F λ F λ > we have wo quadraic equaions wih resec o ξ x ; Here since δ δ = x ξ we firs have x + x + δ = ξ + x + δ Nex fro (F (F x + x = x + ξ we also have ( x+ δ x = ( x+ δ ξ This shows ha above wo equaions have coon roos Thus we have x = ξ = ( x+ x / 544 An esiae of ( + ( E By he ean value heore of [7] here exis oins η η such ha x < η < ξ < η < x d( ξ d( x = d ( η ( ξ x g( x g( ξ = g ( η ( x ξ Then we have g ( ξ g ( η since g ( is he decreasing funcion on ( x x If he condiion (d holds hen d ( > for any x x hence by F ( ξ = we have ( d( ξ d( x ξ x g ( ξ d ( ξ = g ( ξ = d ( η d ( η g( x g( ξ x ξ g ( η bu his is a conradicion o ha d ( is he decreasing funcion because d ( < for any x x Thus he condiion (d is iossible Fro his we have ha here exiss a oin such ( ha x < < x d g so f ( d( g ( arbirary aken i is clear ha Therefore we have as x E d + Since x x were we have f d g + / as ( α ( 55 An esiae of G R ( N N Here R : = I / N : I 4 < ( e I 4 ( α I 4 I = + I I is known ha k+ L k for by 46 of [] hence we have N Since ( ( / ( / k + < < firs ( ( (( ( N = I I I + I I I ( ( I + I I I I = I = ( I + I I + I I I + I I I I 7 6

7 ( I ( ( 3/ ( I I ( I R I N N I + I I I I + I I I I + I + 3/ ( I 4 I ( On he oher h i is known k k by (36 of [6] So ( + / G ( I + for 7 by 47 of[] / < ϑ 4 I + 3/ ( ( I 4 I N + I 4 I k 4 if e hen we have α ( 3 /4 4 ( e + + α 4 + α S : = / 56 An esiae of Pu + s ( = / = + b+ E ( Then we have d S ds de + d E( + + E( = + + d + + ( = ( = + ( ( ( E d 4 E + + d = E( S( + d 4 3 If is a firs rie e 3 hen = 69 i is 938-h rie And we have 6 S( 8 Now we are ready for he roof of he following lea Lea For any 4 we have C < roof Le D = ( e α /( α ( α ( 4 ThenC < is equivalen o D < 4 4 And we here have D < for 7 e D a : = S( for any e In fac i 4 is easy o see ha for e by MATLAB (see he able he able 7 7 =

8 Nex ( e F R : = I + I I < = 69 hen we have D938 = 38L S ( 4 < Now assue 4 e D a Le us see D a We have ( e α N K D = = ( ( e α + K = D + N N N N N a + e a + b ( μ N N = ( ( ( where b μ e a N N / N We have o obain b / By he assuion D a we have ( α α α e α+ a = α + a α e = e + a α by aking arih of boh sides we have θ We also have e + θ + a α ( α a ε θ ( α ( α E + + a α Thus we see θ ( α ( E ε α d θ ε f E a = = g α α α By above 54 we have ( + E d ( α + + ( α ( α 4 ( α ε 4 4 = a α ( α ( α 4 a + ( + ε + α ( α 4 4 since ε < + ( + ( e α /4 Thus we see ( α ( α 4 ( + ( E + ε a + + α ( α If e > hen since < we also have a E ε ( α 4 + ( α ( α 4 4 ( α α ( α α ( α ( α α 8

9 Finally we have ( μ e a N N ( + ( E + ε a ( N N + 55 ε ( α E + G N α ( α b G α N N 4 ( + ( + α G / α α α α ( e 4 Nex if e hen we have b 55 ( N e 6 Proof of Theore Le n= q λ Lq λ be he rie facorizaion of any naural nuber n Then i is clear 4 7 e hen we have since we have by he Lea Therefore we have 4 C < R < (see he able he able if C < ( n C ax{ C } 7 Proof of Corollary Fro he heore heore for 8 Noe Algorih The able shows he values + Φ Φ I = 4 n 5 we have n γ γ 4 ( n ( n e n+ e + ϕ ( n n n ( n γ e n+ 483 n ( C =Φ I R o ( n q If 4 e hen ω = of n N There are only values of 4 C R for here Bu i is no difficul o verify he for3 e Noe if ore 4 inforaions hen i should be aken R < noc < for 63 e by reason of he liied values of MATLAB 65 The able shows he values R for Of course all he values in he able he able are aroxiae R The algorih for o ω n = by MATLAB is as follows: Funcion Phi-Index clc gaa= ; fora long P= [ ]; M=lengh(P; for =:M; =P(:; q=-/; F=-gaa+(rod(/q; N=su((^; N=(N^(/; N3=((N^; N4=N*N3; N5=N+N4; P R=F-((N5 end 9

10 Table C R e e e e e Table R Acknowledgens We would like o hank all of he whose have an ineresing in his aricle References [] P Sole M Plana Robin inequaliy for 7-free inegers arxiv: 67v [ahnt] 3 Dec [] J Sor D S Mirinovic B Crsici Hbook of Nuber heory Sringer 6 [3] H L Mongoery R C Vaugnan Mulilicaive Nuber Theory Cabridge 6 [4] J C Lagarias An eleenary roble quivalen o he Rieann hyohesis Aer Mah Monhly 9 ( [5] G Robin Gres valeurs de la foncion soe des diviseurs e hyohese de Riann Journal of Mah Pures e al 63 ( [6] J B Rosser L Schoenfeld Aroxiae forulars for soe funcions of rie nubers IIlinois J Mah 6 ( [7] Sudhir R Ghorade Balohan V Liaye A Course in Calculus Real Analysis Sringer 6 [8] J L Nicolas Sall values of he Euler funcion he Rieann hyohesis arxiv: 79v [ahnt] 5 Ar Maheaics Subjec Classificaion; M6 N5