Closed-Form Solution for the Nontrivial Zeros of the Riemann Zeta Function

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1 Closed-Form Soluio for he Norivial Zeros of he Riema Zea Fucio Frederick Ira Moxley III (Daed: April, 27) I he year 27 i was formally cojecured ha if he Beder-Brody-Müller (BBM) Hamiloia ca be show o be self-adjoi, he he Riema hypohesis holds rue. Herei we discuss he domai ad eigevalues of he Beder-Brody-Müller cojecure. Moreover, a secod quaizaio of he BBM Schrödiger equaio is performed, ad a closed-form soluio for he orivial zeros of he Riema zea fucio is obaied. I. INTRODUCTION I was recely show i [] ha he eigevalues of a Beder-Brody-Müller (BBM) Hamiloia operaor correspod o he orivial zeroes of he Riema zea fucio [2]. Alhough he BBM Hamiloia is pseudo-hermiia, i is cosise wih he Berry-Keaig cojecure [3, 4]. The eigevalues of he BBM Hamiloia are ake o be he imagiary pars of he orivial zeroes of he zea fucio ζ(z) = k z k= = Γ(z) z d. () exp() The idea ha he imagiary pars of he zeroes of Eq. () are give by a self-adjoi operaor was cojecured by Hilber ad Pólya [5]. Formally, Hilber ad Pólya deermied ha if he eigefucios of a self-adjoi operaor saisfy he boudary codiios ψ () =, he he eigevalues are he orivial zeroes of Eq. (). The BBM Hamiloia also saisfies he Berry-Keaig cojecure, which saes ha whe ˆx ad ˆp commue, he Hamiloia reduces o he classical H = 2xp. Remark. If here are orivial roos of Eq. () for which R(z) /2, he correspodig eigevalues ad eigesaes are degeerae []. II. STATEMENT OF PROBLEM Theorem. The eigevalues of he Hamiloia A. Beder-Brody-Müller Hamiloia Ĥ = are real, where ˆp = i x, =, ad ˆx = x. e iˆp (ˆxˆp ˆpˆx)( e iˆp ) (2) Corollary.. [] Soluios o he equaio Ĥψ = Eψ are give by he Hurwiz zea fucio ψ z (x) = ζ(z, x ) = (x ) z (3) = o he posiive half lie x R wih eigevalues i(2z ), ad z C, for he boudary codiio ψ z () =. Moreover, R(z) >, ad R(x ) >. As ψ z () is he Riema zea fucio, i.e., Eq. (), his implies ha z belogs o he discree se of zeros of he Riema zea fucio. Proof. Le ψ z (x) be a eigefucio of Eq. (2) wih a eigevalue λ = i(2z ): Ĥψ z (x) = λψ z (x). (4)

2 2 The we have he relaio Leig e iˆp (ˆxˆp ˆpˆx)( e iˆp )ψ z (x) = λψ z (x). (5) ϕ z (x) = [ exp( x )]ψ z (x), = ˆ ψ z (x), (6) where ˆ ψ z (x) = ψ z (x) ψ z (x ), ad ˆ exp( x ), (7) is a shif operaor. Upo iserig Eq. (6) io Eq. (5) wih ˆp = i x, =, ad ˆx = x, we obai The we have [ ix x i x x]ϕ z (x) = λϕ z (x). (8) (x x ϕ z (x)) ϕ z (x)dx R ( x xϕ z (x)) ϕ z (x)dx = iλ R ϕ z(x)ϕ z (x)dx. R (9) As ϕ z (x ), ex we iegrae he firs erm o he LHS of Eq. (9) by pars o obai xϕ z (x) x ϕ z(x)dx = ϕ z(x)ϕ z (x)dx ϕ z(x)x d R R R dx (ϕ z(x))dx, () ad he secod erm o he LHS of Eq. (9) by pars o obai xϕ z (x) x ϕ z (x)dx = ϕ z (x)ϕ z(x)dx ϕ z (x)x d R R R dx (ϕ z(x))dx. () Upo subsiuig Eqs. () ad () io Eq. (9), we obai ϕ z(x)x d R dx (ϕ z(x))dx ϕ z (x)x d R dx (ϕ z(x))dx = (iλ 2)N, (2) where Nex, we spli ϕ z (x) io real ad imagiary compoes, such ha N = ϕ z(x)ϕ z (x)dx. R (3) ϕ z (x) = R(ϕ z (x)) ii(ϕ z (x)), (4) ad subsiue Eq. (4) io Eq. (2) such ha R(ϕ z (x))x d R dx R(ϕ z(x))dx I(ϕ z (x))x d R dx I(ϕ z(x))dx N = iλ N. (5) 2 Upo seig λ = i(2z ), Eq. (5) ca be wrie R(ϕ z (x))x d R dx R(ϕ z(x))dx I(ϕ z (x))x d R dx I(ϕ z(x))dx N = 2z N. (6) 2 I ca be see ha all erms o he LHS of Eq. (5) are real, hereby verifyig Theorem. Q.E.D. Remark. If he Riema hypohesis is correc [2], he he eigevalues of Eq. (2) are degeerae [].

3 3 Give ha ψ z (x) = ˆ ψ z (x) = ψ z (x) ψ (x ) = = (x ) z = (x ) z, (7) he secod erm o he LHS of Eq. (6) goes o zero, as I(ϕ z (x)) =. Hece, we are lef wih z = ϕ z (x)x d N R dx ϕ z(x)dx 3 2. (8) Moreover, i ca be see ha Muliplyig Eq. (9) by ϕ (x), we obai x d dx (ϕ z(x)) = x d dx ψ z(x) x d dx ψ z(x ) = x d dx = (x ) z x d dx = (x ) z = xzζ(z, x ) xzζ(z, x). (9) ϕ z (x)xzζ(z, x ) ϕ z (x)xzζ(z, x) = ϕ z (x)[xzζ(z, x ) xzζ(z, x)] = ζ(z, x )xzζ(z, x ) ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x ) ζ(z, x)xzζ(z, x). (2) From he RHS of Eq. (2), i ca be see ha ad R ζ(z, x )xzζ(z, x )dx = (( ) 2z (( )( () () )2z ( 4z) )), (2) (2z(2z )) ζ(z, x)xzζ(z, x)dx = (( ) 2z ( ( )2z 2z)), (22) R (2(2z )) R ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx = ( ) z ( ) z[ (z ) (z) (z ) where he hypergeomeric fucio is 2F (, 2z, z, x ) = ] (() z ( ) z (() 2 F (, 2z, z, x ) )), (23) (2z ) () ( 2z) ( x ). (24) ( z)! =

4 Sice N = ϕ z(x)ϕ z (x)dx R = [ψ z (x) ψ z (x )] 2 dx R = [ψz(x) 2 2ψ z (x )ψ z (x) ψz(x 2 )]dx R = [( x ) 2z 2( x) z ( x ) z ( x) 2z ]dx = R ζ(2z, x) 2( x) z ( x) z ( x ) z 2F ( z, z, 2 z, x ) = ( 2z) z = ( x ) 2z, R(z) >, (25) ( 2z) = wih he hypergeomeric fucio 2F ( z, z, 2 z, x ) = ( z) (z) ( x ), (26) (2 z)! Eq. (8) ca be rewrie [ ( 2z) z = ζ(2z, x) z 2( x) z ( x) z ( x ) z 2F ( z, z, 2 z, x ) = ( 2z) ][ ( x ) 2z (( ) 2z (( )( () )2z ( 4z) )), (2z(2z )) (( ) 2z ( ( () )2z 2z)) ( ) z ( ) z[ (2(2z )) (z ) (z) ] (z ) = (() z ( ) z (() 2 F (, 2z, z, ) )) (2z ) for R(z) >, ad R(x ) >. Upo imposig he boudary codiio ψ () = z = = Γ(z) ] 3 2, (27) z exp(), (28) Eq. (27) are he orivial zeros of Eq. (), i.e., [ ( 2z) z = ζ(2z ) z 2( ) z () z ( ) z 2F ( z, z, 2 z, ) = ( 2z) ][ ( ) 2z (() 2z (( )( () )2z ( 4z) )), (2z(2z )) (() z ( ) z (() 2 F (, 2z, z, ) )) (2z ) for he boudary codiio x =, ad he covergece crieria =. 4 ] 3 ± cos, (29) 2 B. Covergece For breviy, le us examie he covergece of he iegral represeaio of he discree orivial zeros of he Riema zea fucio o he posiive half lie x R, z C, R(z) >, ad R(x ) >. From Eq. (8), he

5 iegral represeaio of he discree orivial zeros of he Riema zea fucio are give by z = ζ(z, x )xzζ(z, x )dx N R ζ(z, x)xzζ(z, x)dx N R ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx 3 N R 2, (3) where N = R Lemma.. From he firs erm o he RHS of Eq. (3), if [ ( x ) 2z 2( x) z ( x ) z ( x) 2z] dx. (3) 5 exiss for every umber, he ζ(z, x )xzζ(z, x )dx (32) ζ(z, x )xzζ(z, x )dx = provided his i exiss as a fiie umber. Proof. From L Hospial s Rule, we have ζ(z, x )xzζ(z, x )dx, (33) ζ(z, x )xzζ(z, x )dx = (( ) 2z (( )( () )2z 2z )). (34) (2z(2z )) (( ) 2z (( )( () )2z 2z )) (2z(2z )) (( ) 2z (( )( () = )2z 2z )) ( ) 2z (2z(2z )) ( ) 2z 2z( ) 4z (( () = )2z ( () )2z 4z ) 4( 2z)z 2 ( ) 2z. (35) Upo evaluaig Eq. (35) wih a series expasio a =, we obai ( ( )( () )2z 4z) (2( ) 2z z( 2z)) = (( ) 2z (( )( () )2z ( 4z) )). (36) (2z(2z )) Hece, i ca be see ha he firs erm o he RHS of Eq. (3) is coverge, give ha he i see i Eq. (35) exiss as a fiie umber as see i Eq. (36). Lemma.2. From he secod erm o he RHS of Eq. (3), if exiss for every umber, he ζ(z, x)xzζ(z, x)dx (37) provided his i exiss as a fiie umber. ζ(z, x)xzζ(z, x)dx = ζ(z, x)xzζ(z, x)dx, (38)

6 6 Proof. From L Hospial s Rule, we have ζ(z, x)xzζ(z, x)dx = (( ) 2z ( ( () 2z 2z)). (39) (2(2z )) (( ) 2z ( ( () 2z 2z)) (2(2z )) (( ) 2z ( ( () 2z 2z)) ( ) 2z = (2(2z )) ( ) 2z ( ) 4z ( ( () = )2z 2z) 2(2z )( ) 2z (4) Upo evaluaig Eq. (4) wih a series expasio a =, we obai (( ) 2z ( ( () )2z 2z)) (( ) 2z ( ( () )2z 2z)) = (( ) 2z ( ( () )2z 2z)). (4) (2(2z )) Hece, i ca be see ha he secod erm o he RHS of Eq. (3) is coverge, give ha he i see i Eq. (4) exiss as a fiie umber as see i Eq. (4). Lemma.3. From he hird erm o he RHS of Eq. (3), if exiss for every umber, he = provided his i exiss as a fiie umber. Proof. From he RHS of Eq. (23) i ca be see ha ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx (42) ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx, (43) ζ(z, x )xzζ(z, x) ζ(z, x)xzζ(z, x )dx = (( ) z ( ) z (( ) 2 F (, 2z, z, ) 2z)) (2z ) (() z ( ) z (() 2 F (, 2z, z, ) )). (44) (2z ) Sice he secod erm o he RHS of Eq. (44) is idepede of, we are oly cocered wih he i of he firs erm o he RHS of Eq. (44). As such, we cosider he i Here, i is useful o employ Gauss heorem, i.e., (( ) 2 F (, 2z, z, ) 2z) ( ) z ( ) z. (45) (2z ) 2F (, 2z, z, ) = Γ( z)γ(z ) Γ( z)γ(z) (46)

7 where R(z) >, =, ad Γ(z) = is he gamma fucio. Therefore, Eq. (45) ca be wrie (( ) Γ( z)γ(z ) Γ( z)γ(z) 2z) ( ) z ( ) z (2z ) x z e x dx (47) ( ) z ( ) z ( z) =. (48) (z ) Upo evaluaig Eq. (48) wih a series expasio a =, we obai (( ) 2 F (, 2z, z, ) 2z) ( ) z ( ) z = ( ) z ( ) z[ (2z ) (z ) (z) ]. (49) (z ) Hece, i ca be see ha he hird erm o he RHS of Eq. (3) is coverge, give ha he i see i Eq. (45) exiss as a fiie umber as see i Eq. (49). Fially, we mus cosider he covergece of he ormalizaio facor N. Lemma.4. From he firs hree erms o he RHS of Eq. (3), if [ ( x ) 2z 2( x) z ( x ) z ( x) 2z] dx (5) exiss for every umber, he [ ( x ) 2z 2( x) z ( x ) z ( x) 2z] dx = provided his i exiss as a fiie umber. Proof. [ ( x ) 2z 2( x) z ( x ) z ( x) 2z] dx (5) [ ( x ) 2z 2( x) z ( x ) z ( x) 2z] dx (( ) 2z (( )( () = )2z )) (2z ) (( ) 2z ((( () )2z ) )) (2z ) (2( ) z ( ) z ( ) z 2 F ( z, z, 2 z, )) (z ) (2( )z () z ( ) z 2F ( z, z, 2 z, )), (52) (z ) where he las erm o he RHS of Eq. (52) omis he i, as i is idepede of. The is see o he RHS of Eq. (52) ca be evaluaed i a similar maer o hose see i Eqs. (36), (4), ad (45), respecively. 7 C. Domai of he Beder-Brody-Müller Hamiloia For he BBM Hamiloia operaor as give by Eq. (2), he Hilber space is H = L 2 (R, dx). Moreover, ˆp ad ˆx are self-adjoi operaors ha ac i H. I order o sudy he domai of he BBM Hamiloia operaor, we firs iroduce a auxiliary operaor Ô, such ha Ô = ˆpˆp ˆxˆx, (53)

8 8 where ˆpˆp = 2, ad ˆxˆx = x 2. The se of fiie liear combiaios of Hermie fucios is a core of Ô, ad herefore he Schwarz space S is also a core of Ô. Lemma.5. [6] If ϕ is i D(Ô), he ˆpˆpϕ 2 ˆxˆxϕ 2 Ôϕ 2 c ϕ 2. (54) Proof. [6] We esimae ϕ for a core of Ô via a double commuaor o make he esimae [7], Therefore, i Eq. (54) c = 2. Afer rewriig Eq. (8) as ÔÔ = ˆpˆpˆpˆp ˆxˆxˆxˆx ˆpˆpˆxˆx ˆxˆxˆpˆp [ ] = ˆpˆpˆpˆp ˆxˆxˆxˆx 2 ˆx i ˆpˆpˆx i [ˆx i, [ˆx i, ˆpˆp]] i= ˆpˆpˆpˆp ˆxˆxˆxˆx 2, (55) [x x x x]ϕ = ( 2z)ϕ, (56) he ˆpˆp = x x ad f(ˆx) = x x are self-adjoi operaors acig i H = L 2 (R, dx). Seig defied o If f(ˆx) is local i H, he Eq. (57) is dese ad Hermiia. Ĥ = ˆpˆp f(ˆx), (57) D(ˆpˆp) D(f(ˆx)). (58) Theorem 2. The BBM Hamiloia operaor i Eq. (2) is esseially self-adjoi, give ha f(ˆx) a ˆx b. The BBM Hamiloia operaor i Eq. (2) is real-valued o he posiive half lie R, afer beig reduced o Eq. (56). From f(ˆx) a ˆx b we have Le us examie he uiqueess. f(ˆx) ˆxˆx b ˆx 2 cˆxˆx d. (59) Proof. As show i [6], if Ĥ is Hermiia, ad Ô is a posiive self-adjoi operaor, he C is a core of Ô such ha C D(Ĥ). As such, (ˆpˆp f(ˆx))ϕ 2 a (ˆpˆp ˆxˆx)ϕ 2 b ϕ 2, (6) where ϕ S. Sice ( ˆxˆx)ϕ L 2, f(ˆx)ϕ L 2. Therefore, S D(Ĥ). Moreover, sice f(ˆx)2 rˆxˆxˆxˆx s, f(ˆx)ϕ 2 r ˆxˆxϕ 2 s ϕ 2. (6) As such, from Eq. (54), Eq. (6) is saisfied. If ϕ S, he (f(ˆx)ϕ) L 2. Sice, as quadraic forms o C, we hus have ±i[ĥ, Ô] cô (62) for cosa c. ±i[ĥ, Ô] = ±i{[ˆpˆp, ˆxˆx] [f(ˆx), ˆpˆp]} = ±{2(ˆp ˆx ˆx ˆp) (ˆp f(ˆx) f(ˆx) ˆp)} 2(ˆpˆp ˆxˆx) ˆpˆp ( f(ˆx)) 2 2(ˆpˆp ˆxˆx) ˆpˆp 2(a 2ˆxˆx b 2 ) cô, (63)

9 9 D. Secod Quaizaio We begi wih he Beder-Brody-Müller (BBM) Schrödiger equaio i d dz = ˆ ˆxˆp ˆ ψ(x, z) ˆ ˆpˆx ˆ ψ(x, z), (64) where ˆ is give by Eq. (7), ˆx = x, ˆp = i x, =, x R, ad z C. Furhermore, le be he soluio of ψ (x) = ζ(z, x ) = (x ) z (65) = ( ˆ ˆxˆp ˆ ˆ ˆpˆx ˆ ) ψ (x) = E ψ (x), (66) where z are he orivial zeros of he Riema zea fucio give by Eq. (27), R(z) >, ad R(x ) >. Nex, we wrie ψ(x, z) = b (z)ψ (x). (67) From Eq. (64) we fid d dz b (z) = i E b (z). (68) We ow fid a Hamiloia ha yields Eq. (68) as he equaio of moio. Hece, we ake [ Ĥ = ψ (x, z) ˆ ˆxˆp ˆ ˆ ˆ ] ˆpˆx ψ(x, z) dx (69) R as he expecaio value. Upo subsiuig Eq. (67) io Eq. (69) ad usig Eq. (66) we obai he harmoic oscillaor Ĥ = E b (z)b (z). (7) Takig b (z) as a operaor, ad b (z) as he adjoi, we obai he usual properies: From he aalogous Heiseberg equaios of moio, i d dz ˆb = [ˆb, Ĥ] = m = m = m [ˆb, ˆb m ] = [ˆb, ˆb m] =, [ˆb, ˆb m] = δ m. (7) E m (ˆbˆb mˆbm ˆb mˆb mˆb ) E m ( δ mˆbm ˆb mˆb ˆbm ˆb mˆb mˆb ) E m ( δ mˆbm ˆb mˆb mˆb ˆb mˆb mˆb ) = E ˆb. (72) The eigevalues of Ĥ are Ĥ = E N, (73)

10 where N =,, 2, 3,...,. Sice, E = i(2z ), we ca rewrie Eq. (73) as Ĥ = i (2z )N. (74) However, from Eq. (72) i ca be see ha As such, i d dz ˆb = i(2z )ˆb. (75) d dz ˆb = (2z )ˆb. (76) Theorem 3. The eigevalues of he Hamiloia E. PT -symmeric Beder-Brody-Müller Hamiloia iĥ = are imagiary, where ˆp = i x, =, ad ˆx = x. i e iˆp (ˆxˆp ˆpˆx)( e iˆp ) (77) Corollary 3.. [] Soluios o he equaio iĥψ = Eψ are give by he Hurwiz zea fucio ψ z (x) = ζ(z, x ) = (x ) z (78) = o he posiive half lie x R wih eigevalues i(2z ), ad z C, for he boudary codiio ψ z () =. Moreover, R(z) >, ad R(x ) >. As ψ z () is he Riema zea fucio, i.e., Eq. (), his implies ha z belogs o he discree se of zeros of he Riema zea fucio. Proof. Le ψ be a eigefucio of Eq. (77) wih a eigevalue λ = i(2z ): iĥψ = λψ. (79) The we have he relaio Leig i e iˆp (ˆxˆp ˆpˆx)( e iˆp )ψ = λψ. (8) ϕ z (x) = [ exp( x )]ψ z (x), = ˆ ψ z (x), (8) where ˆ ψ z (x) = ψ z (x) ψ z (x ), ad iserig Eq. (8) io Eq. (8) wih ˆp = i x, =, ad ˆx = x, we obai The we have [x x x x]ϕ z (x) = λϕ z (x). (82) (x x ϕ z (x)) ϕ z (x)dx R ( x xϕ z (x)) ϕ z (x)dx = λ R ϕ z(x)ϕ z (x)dx. R (83) As ϕ z (x ), ex we iegrae he firs erm o he LHS of Eq. (83) by pars o obai x R ϕ z (x) x ϕ z(x)dx = ϕ z(x)ϕ z (x)dx ϕ z(x)x d R R dx (ϕ z(x))dx, (84)

11 ad he secod erm o he LHS of Eq. (83) by pars o obai xϕ z(x) x ϕ z (x)dx = ϕ z (x)ϕ z(x)dx ϕ z (x)x d R R R dx (ϕ z(x))dx. (85) Upo subsiuig Eqs. (84) ad (85) io Eq. (83), we obai ϕ z(x)x d R dx (ϕ z(x))dx ϕ z (x)x d R dx (ϕ z(x))dx = (λ 2)N, (86) where Nex, we spli ϕ z (x) io real ad imagiary compoes, such ha N = ϕ z(x)ϕ z (x)dx. R (87) ϕ z (x) = R(ϕ z (x)) ii(ϕ z (x)), (88) ad subsiue Eq. (88) io Eq. (86) such ha R(ϕ z (x))x d R dx R(ϕ z(x))dx I(ϕ z (x))x d R dx I(ϕ z(x))dx N = λ N. (89) 2 Upo seig λ = i(2z ), Eq. (89) ca be wrie R(ϕ z (x))x d R dx R(ϕ z(x))dx R I(ϕ z (x))x d dx I(ϕ z(x))dx N = I ca be see ha all erms o he LHS of Eq. (89) are real, hereby verifyig Theorem 3. Q.E.D. i( 2z) N. (9) 2 III. CONCLUSION I his sudy, we have discussed he domai ad eigevalues of he BBM Hamiloia. Moreover, a secod quaizaio procedure was performed for he BBM Schrödiger aalogue equaio. Fially, a closed-form expressio for he orivial zeros of he Riema zea fucio was obaied, ad a covergece es for he closed-form expressio was performed. [] Beder, C.M., Brody, D.C. ad Mller, M.P., 26. Hamiloia for he zeros of he Riema zea fucio. arxiv prepri arxiv: [2] Riema, B., O he Number of Prime Numbers less ha a Give Quaiy.(Ueber die Azahl der Primzahle uer eier gegebee Grösse.). [3] Berry, M.V. ad Keaig, J.P., 999. H= xp ad he Riema zeros. I Supersymmery ad Trace Formulae (pp ). Spriger US. [4] Coes, A., 999. Trace formula i ocommuaive geomery ad he zeros of he Riema zea fucio. Seleca Mahemaica, New Series, 5(), pp [5] Odlyzko, A.M., 2. The -d zero of he Riema zea fucio. Dyamical, Specral, ad Arihmeic Zea Fucios: AMS Special Sessio o Dyamical, Specral, ad Arihmeic Zea Fucios, Jauary 5-6, 999, Sa Aoio, Texas, 29, p.39. [6] Faris, W.G. ad Lavie, R.B., 974. Commuaors ad self-adjoiess of Hamiloia operaors. Commuicaios i Mahemaical Physics, 35(), pp [7] Gm, J. ad Jaffe, A., 972. The λϕ 24 Quaum Field Theory wihou Cuoffs. IV. Perurbaios of he Hamiloia. Joural of Mahemaical Physics, 3(), pp