Riemann Hypothesis Proof

Transcription

1 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Riea Hypothesis Proof H. Vic Dao March, 009 Revised Deceber, 009 Abstract We show that i 0 < x < ay factor of Riea s fuctio ξ () ust have its ero o the lie / x =. Sice () ξ has i 0 < x < the sae eros as ζ (), this proves Riea s Hypothesis that all the eros of ζ () i 0 < x <, are o / x =. Keywords: Riea Hypothesis, Zeta fuctio, Gaa fuctio, Aalytic fuctios. 000 Matheatics Subject Classificatio:M6, 33E0, 30A0, 6D5, 30D0. Itroductio I his 859 Zeta paper, Riea obtaied a forula for the cout of the pries up to a give uber. Riea s forula has four ters. But oly the first ad the third ters have o-egligible values. The first is the doiat ter, ad ca be coputed precisely. The third is saller ad depeds o the provisio that all the eros of the Zeta fuctio i 0 < x < are o the lie x = /. This provisio becae kow as the Riea Hypothesis, but it was ever hypothesied by Riea, or was it used by hi. Not seeig a easy proof for it, Riea used oly the first ter of his forula ad obtaied a approxiatio far superior to Gauss for the cout of the pries.

2 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. I 935, ad 936, Titcharsh ad Corie [Hasel, p. xii] cofired by coputatios that the first 04 eros of Zeta i 0 < x < with 0 < I < 468 lie exactly o the lie x = /. Followig that, Tourig (953) exteded the upper liit to I = 540, ad Leher (956) cofired that the first 5,000 eros of Zeta i 0 < x < with 0 < I <, 944 lie o the lie x = /. By 00, the first 50 billio Zeta eros have bee located o the lie x = /, ad i a 008 eetig a far greater uber was etioed. The fact that the first 50 billio eros are o x = /, does ot costitute a proof for the ifiitely ay eros i the ifiite area of the strip 0 < x <. But expectig assive edless coputatios to disprove the Riea Hypothesis by fidig a ero off the lie x = /, is statistically iplausible. I fact, statistical tests idicate that the Riea Hypothesis holds with extreely high statistical certaity. I 008, we applied a Chi-Squared Goodess-of-Fit-Test to the Riea forula for the cout of the pries, ad cofired that the Riea Hypothesis holds with certaity that is liited oly by the software [Da]. I sectios to 4, we preset Riea s fuctio ξ (), ad kow facts about its eros. I sectio 5, we recall that ξ( x iy) = ξ( x + iy). This fact serves as a key result for the proof. I sectios 6-8, we recall the d key result that is due to Hadaard. The proof of this well-kow result, is i the Appedix. The 3 rd key result due to Titcharsh, is preseted i sectio 9.

3 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. There, we prove that i 0 < x <, ay factor of () the lie x = /. ξ ust have its ero o Sice ξ () has i 0 < x < the sae eros as ζ (), this proves that all the eros of ζ () i 0 < x <, are o x = /.. ξ (), ad () ζ have the sae eros i 0 < x < For x + iy =, Riea gave the Defiitio () Γ( )() ζ π. ξ ( ) Sice Γ () has o eros i 0 < x <, [Saks], we have the Propositio. ξ () has i 0 < x < the sae eros as ζ (). () ξ has i 0 < x <, ifiitely ay eros that are sequeced by sie ad icrease to. Sice ζ () has ifiitely ay eros i 0 < x < [Titch], we have the Propositio. ξ () has ifiitely ay eros,, 3,... i 0 < x <. Also, ξ () is a etire fuctio, ot idetically ero. Thus, by [Saks, p. 96] we have the Propositio. The eros,, 3,..., are sequeced by sie, ad icrease to

4 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/ the ultiplicity of each ero of ξ () is fiite Sice a aalytic fuctio that vaishes o a covergig sequece is idetically ero, there is o ifiite sequece of idetical ξ () eros. That is, we have the Propositio 3. The ultiplicity of each ero of ξ () is fiite. 4. Diagoally syetric eros of ξ () For ζ (), Riea derived the fuctioal equatio ξ() = ξ( ). Therefore, we have the Propositio 4. If = x + iy is a ero of ξ (), the so is = x iy. These two eros are diagoally syetric with respect to =, because deotig x = + α, we have x = α. 5. Observig ξ( x iy) = ξ( x + iy), for 0 < x <, Proof: Riea obtaied a itegral forula for ξ (), t = t = ( ( + ) ) ξ( ) = + ( ) ψ( t) t + t dt, where the ifiite series ψ( t) , πt e πt 3 πt e e coverges uiforly for t. Therefore, t = t = ( ( + ) ) ξ( ) = + ( ) ψ( t) t + t dt 4

5 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. t = t = ( ( + ) ) = + ( ) ψ( t) t + t dt That is, we have = ξ() = ξ(). ξ( x iy) = ξ( x + iy). 6. A coverget ifiite product is ero if ad oly if at least oe of its factors is ero Aalytic fuctios are effectively represeted as ifiite products [Saks, VII]. By the defiitio i [Saks, p. 86], The Ifiite Product 0, so that all the ters after it a0+ a0+... a 0 0+ q. Cosequetly, p = aa... a q. 0 aaa 3... coverges to p if ad oly if there is a idex a, a, a,... Therefore, [Saks, p. 8], has the Propositio are o-ero, ad The value of a coverget ifiite product is equal to ero if ad oly if at least oe of its factors is ero. The product of a = that has o such q, diverges to ero, although it satisfies the covergece ecessary coditio a [Saks, p. 8]. If a coverget ifiite product equals ero, this is due to oly fiitely ay vaishig factors, without which, the reaiig ifiite product is o-ero.. Absolutely Coverget Ifiite Product 5

6 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. ( + u )( + u )... coverges if ad oly if u + u +... coverges. Here, we ll eed Absolute Covergece, so that the ifiite products will coverge idepedetly of the order of their factors. By the defiitio i [Saks, p. 88], ( + u )( + u )... coverges absolutely if ad oly if( + u )( + u )... coverges. Therefore, [Saks, p. 89] has the Propositio. ( + u)( + u)... coverges absolutely u + u +... coverges. Cosequetly, sice the value of a absolutely covergece series does ot deped o the order of the suatio, we have Propositio. The value of a absolutely coverget ifiite product does ot deped o the order of its factors. 8. The Hadaard factoriatio of ξ () Hadaard factoriatio for ξ () is the d key result for the Hypothesis proof. Due to Hadaard, we have Propositio 8. ξ() =..., where the product coverges absolutely, ad uiforly with respect to the s, ' its value does ot deped o the order of its factors, ad it vaishes oly at the eros of ξ ().,,,,,,... do ot appear i the The cojugate roots, 3 3 factoriatio of ξ (). A proof of this well-kow result, is i the Appedix. We ote here that the do ot appear i the factoriatio, because that will require the iclusio 6

7 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. of the factors. These factors are ot differetiable with respect to, ad icludig the will ake ξ () ot differetiable with respect to. We tur to prove the Hypothesis 9. x =, for ay ero Proof: To keep it readable, we ll take =, ad assue that has ultiplicity. Say, = 8, ad 6 < = 8 < 9 Sice = x + iy is a ero of ξ, By 5, Thus, ξ ( x + iy ) = 0. ξ( x iy ) = 0. ξ( x iy ) = 0. Applyig 8. to ξ( x iy), x iy x iy x iy x iy 0 =... Sice we assued = 8, we have x iy x iy x iy x iy 0... = 6 6 x iy x iy x iy x iy Clearly, for, 8, the factors with are all o-ero. Ideed, x iy = 0 =, which cotradicts 8..

8 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Ad, x iy = 0 = x iy =, which cotradicts 8.. Sice all the factors with are o-ero, by 6., we ust have Thus, Hece, xiy xiy ( ) ( ) = 0. x iy x iy 0 =. 4 y x + x iy x iy = 0. That is, 4y x + ( ) + x y x y = 0 Now, the three factors that iclude To see that, ote the 3 rd Titcharsh, [Titch, p.39-33], y are o-vaishig. key result for the Hypothesis Proof, due to i 0 < x <, ad 0 y y, ζ () has oly oe ero o the lie x = /, at y = , Thus, y ust be out of the Titcharsh rectagle. That is, y y, ad ay of the factors with Therefore, we have That is, x = 0. x =. y is greater tha y > 96. 8

9 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Replacig with, ad with k, we coclude that if is a ero of ultiplicity k, the we ll apply the arguets above to obtai that x =. APPENDIX: Proof of Hadaard Factoriatio 8. A. The Weierstrass Factoriatio of ξ () Due to Weierstrass, ξ () has a ifiite product represetatio A. h ( ) Q ( ) ( )( ), = ξ() = e e where the polyoials Q () guaratee the uifor covergece of the product i the ope plae, where h () is a etire fuctio. ad where,, 3,... are the eros of ξ () i 0 < x <, that are sequeced by sie, icrease to, ad do ot iclude,,.... This represetatio is well kow. We detail the proof i [Ed]. Proof: By., the eros of ξ () i 0 < x <,,, 3,..., are sequeced by sie, ad icrease to Sice ξ () is a etire fuctio so that ξ(0) 0, ad sice the ' s are sequeced by their sie ad icrease to, by Weierstrass [Saks,VII,.3], we have h ( ) Q ( ) ( ) e, = ξ() = e where the polyoials Q () guaratee the uifor covergece of the 9

10 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. product i the ope plae, ad where h () is a etire fuctio. The eros of 4. appear i the factoriatio, ad we have h ( ) Q ( ) ( )( ). = ξ() = e e The appearace of the eros of 5, will require the iclusio of the factors. These factors are ot differetiable with respect to, ad icludig the i the product will ake ξ () ot differetiable. Hadaard showed that for ξ () ad Q ( ) e =, h ( ) e =. Thus, siplifyig the Weierstrass product represetatio of ξ (). I B we derive Hadaard s first siplificatio, ad i D we derive the secod. B. Q ( ) e = To establish Q ( ) e =, we ai to show that for all i 0 < x <, ( ) = ( ) coverges absolutely. = By., we eed to show that ( ) coverges. ( ) For istace, cos( π) = ( ) coverges absolutely, because = 0

11 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. coverges. ( ) = Note that, ( ) ( ) 4 = iy 4 = ( x + ) 4 = ( x ) y + i( x ) y ( ) 4 = ( x ) y + 4( x ) y = ( x ) + y + ( x ) + y + ( x ) y ( ) 6 = ( x ) + y + ( x ) + y 4 ( 6 ) = + x + y Now, x 6 y ( ) + > 0, because x, ad by a result of Titcharsh (that we state i sectio 3), y y > 4. Therefore, Hece, ( ) >. < ( ), ad it is sufficiet to show that <. > N

12 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Note that the ecessary coditio, To show the covergece, for large eough, 0 = N, N +, N +,...,, holds, sice. defie positive ubers R > so that The, ad >, 4 R log R =. log >. Hadaard showed [Da3, 0.3], that the uber of eros of ξ () i R is bouded by R log R. Sice we took 4 R log R Therefore, =, we have > R. > N N R > = 4 (log ) R > N 4 > N ( log ) ( log ) = 4. 3/ / > N

13 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. log Sice ( ) ad Therefore, ad / 0, as, we have ( log ) / > N >, <, for N <. = coverges absolutely, Q ( ) e =. Sice the Weierstrass product coverges uiforly, we have h ( ) ξ() = e ( )( ), = where the ifiite product coverges absolutely, ad uiforly with respect to the s. ' C. The order of a etire fuctio By [Holl, p. 68], the Hadaard Factoriatio Theore applies to a etire fuctio f () for which li sup log log ax ( ) f R = R <. is called the order of f (). For istace, if f () cos( π) =, 3

14 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. iπ iπ cos( π) = e + e ( iπ iπ e e ) + iπ( x+ iy) iπ( x+ iy) ( e e ) = + ( πy πy e e ) = + e πy Thus, e π. log log ax cos( π) log log ax log R = R = R e π = = log log log( πr) R e π Hece, f () = cos( π) has order = = (log π + log R ) log π = +, as R.. h ( ) D. e =. Hadaard showed that for () h ( ) ξ, e =. Hadaard replaced Weierstrass etire fuctio 4

15 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. with a polyoial so that h (), Q (), deg Q ( ). It is well-kow ([Ed] or [Da3, 0.]), that if R is large eough, log ξ( ) R i R. Hece, Sice by L'Hospital we coclude log log ax ξ( ) log R = R li R log ( ) log = +. log RR li R R = = 0, li sup log log ax ξ( ) =. = R R That is, Hece, That is ξ () is of order = deg Q ( ).. Q () = A+ B ( ), for soe costats A, ad B. Sice the factoriatio factors are we have, 4 4 ( ) ( ) = =, ( ) ( ) ( )( ) 5

16 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. ξ() = e, where the product is a eve fuctio of But Riea showed [Da3, 9.] that is a eve fuctio of ( ). Cosequetly, ad Settig = 0, 0 A+ B( ) ( ) 4 ( ) 4. 4 ( ) A ( ) ξ( ) = A + A B = 0, ( ) A ξ() = e ( ). A e = ξ(0). By Riea s Itegral forula for ξ (), that we used i 5, Hece, we have Therefore, t = t = 4 4 ( ( + ) ) ξ( ) = + ( ) ψ( t) t + t dt. ξ (0) =. ( ) ξ() = ( ) 4 4. = where the ifiite product coverges absolutely, ad uiforly with respect to the s. ' 6

17 Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Refereces [Da] Dao, Vic, Riea s Zeta paper: The Product Forula Error, Gauge Istitute Joural of Math ad Physics, May 006. [Da] Dao, Vic, Chi-Squared Goodess-of-Fit-Test of the Riea Hypothesis, Gauge Istitute Joural of Math ad Physics, August 008. [Da3] Dao, Vic, Riea s Zeta Fuctio: the Riea Hypothesis Origi, the Factoriatio Error, ad the Cout of the Pries, Gauge Istitute Joural of Math ad Physics, Noveber 009. [Ed]. Edwards, H. M. Riea s Zeta Fuctio, Acadeic Press, 94. [Hada] Hadaard, J., Etude sur les Proprietes des foctios Etieres et e particulier d'ue foctio cosideree par Riea J. Math. Pures Appl. [4] 9, 893, pp. -5. Also, i [Ed, chapter ] [Hasel] Haselgrove C. B. ad Miller J.C.P., Tables of the Riea Zeta Fuctio, Cabridge Uiversity Press 960. [Holl] Hollad, A. S. B., Itroductio to the Theory of Etire Fuctios, Acadeic Press, 93. [Rie]. Riea, B., O the Nuber of Prie Nubers less tha a give quatity, 859, i God Created the Itegers: The Matheatical Breakthroughs that chaged History, edited by Stephe Hawkig, Ruig Press, 005. Page 86. Riea s paper was traslated also by H. M. Edwards, i Appedix to [Ed]. Ad by Roger Baker, Charles Christeso, ad Hery Orde, I Collected Papers, Berhard Riea, Kedrick Press, 004. [Saks]. S. Saks, ad A. Zygud, Aalytic Fuctios. Secod Editio, Polish Scietific Publishers, 965. [Titch]. Titcharsh, E. C., The Theory of the Riea Zeta Fuctio. Oxford U. Press, 95. Soe web sources,