ON THE BÁEZ-DUARTE APPROACH TO THE NYMAN-BEURLING CRITERION FOR THE RIEMANN HYPOTHESIS. Daniel Juncos. B.S. Mathematics, West Chester University, 2003

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1 ON THE BÁEZ-DUARTE APPROACH TO THE NYMAN-BEURLING CRITERION FOR THE RIEMANN HYPOTHESIS by Daiel Jucos B.S. Mahemaics, Wes Cheser Uiversiy, 3 Submied o he Graduae Faculy of School of Ars ad Scieces i parial fulfillme of he requiremes for he degree of Maser of Sciece Uiversiy of Pisburgh

2 UNIVERSITY OF PITTSBURGH School of Ars ad Scieces This disseraio was preseed by Daiel Jucos I was defeded o April, ad approved by Aleader Borisov, Assisa Professor of Mahemaics Kiumars Kevah, Assisa Professor of Mahemaics Jeffrey Wheeler, Assisa Isrucor of Mahemaics Disseraio Advisor: Aleader Borisov, Assisa Professor of Mahemaics ii

3 ON THE BÁEZ-DUARTE APPROACH TO THE NYMAN-BEURLING CRITERION FOR THE RIEMANN HYPOTHESIS Daiel Jucos, M.S. Uiversiy of Pisburgh, Copyrigh by Daiel Jucos iii

4 ON THE BÁEZ-DUARTE APPROACH TO THE NYMAN-BEURLING CRITERION FOR THE RIEMANN HYPOTHESIS Daiel Jucos, M.S. Uiversiy of Pisburgh, The Nyma-Beurlig Crierio paraphrases he Riema Hypohesis as a closure problem i a Hilber space. The simples versio of his, due o Baez-Duare, saes ha RH is equivale o oe paricular eleme i a Hilber space beig i he closure of a spa of couably may oher elemes. We ivesigae his umerically ad aalyically. I paricular, we esablish ew formulas for he ier producs of he vecors ivolved. iv

5 TABLE OF CONTENTS PREFACE... VII PRELIMINARIES... RESULTS AND DISCUSSION... 9 BIBLIOGRAPHY... 5 v

6 LIST OF FIGURES Figure. The Orhogoal Projecio oo V()... Figure. Cojecure a j= vi

7 PREFACE Acowledgemes: The auhor wishes o epress sicere appreciaio o Professor Borisov for his guidace ad assisace i he preparaio of his hesis. I addiio, special has o Chioo Pael whose masery i compuer programmig was helpful durig he programmig porio of his uderaig. vii

8 PRELIMINARIES I 859, Berhard Riema posied a heory regardig he disribuio of prime umbers. To his day, his proposal remais wha may cosider o be he mos impora ope problem i he whole of pure mahemaics. I is oe of he Clay Mahemaics Isiue s Milleium Prize Problems - he usolved remas of he wey-hree problems proffered i 9 by David Hilber. Hilber believed hese problems o be he mos impacig ad iflueial problems i mahemaics, ad ha hey should occupy he aeio of mahemaicias hroughou he followig ceury. There have bee several differe reformulaios of Riema s cojecure culivaed hroughou ha h ceury, ad eve some reformaio as lae as 6. We will ivesigae he hypohesis i he vei of some of he mos rece developmes ha have bee made. Le us begi wih some geeral defiiios, as well as perie fucios ad heorems. Firs ad foremos, we defie he fucio ad cojecure ha cosiue he cru of his paper: Defiiio: (Riema Zea Fucio) The Reima Zea fucio is defied o he half-plae s C wih Re s > by ζ s s = The Zea fucio has log bee a subjec of grea ieres due o is may ieresig qualiies. Chief amog hese qualiies is is relaio o he disribuio of prime. umbers. Usig he formula for a geomeric series, ad he covergece of ζ s as a Dirichle series, we ca obai he ideiy s = = p p s, for Re s >

9 over all primes, p. This ideiy yields aoher oeworhy propery of ζ i is relaio o he followig fucio: Defiiio: (Möbius Fucio) For ay Nwih prime facorizaio = p a p a p a, he From (), we have ζ s = μ, if a = a = = a =, oerwise p = = + p s s 3 s 5 s s 3 s 6 s 5s. From his i becomes clear ha we ca derive he ideiy ζ s = Cojecure: (Riema Hypohesis) = μ() s, for Re s > All o-rivial zeros of ζ s have real par equal o. (RH) I is impora o oe ha zeros of he zea fucio are o limied o hose of he fucio as i is defied above. Here, we are referrig o he zeros of he zea fucio as depiced by is aalyic coiuaio. This is a echique used o eed he domai of a aalyic fucio defied o a ope subse of he comple plae o a larger ope subse. Whe he aalyic coiuaio, F, of a fucio, f, is defied over coeced subse, Ω C, he F is uique o fo Ω. coiuaio of ζ. Defiiio: (Melli Trasform) Le us defie a few ey fucios ivolved i he aalyic For ayf:, R, he Melli rasform of fis give by Mf s φ s = The iverse rasform is he give by Defiiio: (Gamma Fucio) M φ f = πi s f d a+i φ s s a i The Gamma fucio is defied o s C wih Re s > by ds

10 Γ s e s d. The Gamma fucio has several ieresig properies ha are perie o our ivesigaio. I saisfies he fucioal equaios: Γ s + = sγ s, 3 ad π Γ s Γ s = si πs. 4 I is well-ow ha he fucio Ζ s π s Γ s ζ s is he aalyic coiuaio of ζ oo he whole comple plae, ecep a where ζ has a simple pole. Oe mehod Riema used o show his was via he Melli rasform of he Jacobi hea fucio θ z e iπ z, defied o he upper half of he comple plae. Usig he Poussi summaio formula, i is sraighforward o show ha θ saisfies he fucioal equaio θ z = θ z iz. The, we ca employ he chage of variables π i he gamma fucio, yieldig s π s Γ s = e π s d. Summig boh sides over N, we ca swich he summad wih he iegral, sice boh sides coverge uiformly, ad we achieve Sice θ iz = Γ s π s s = e iπ z = + + e iπ z = π s Γ s ζ s = = = s d e π. = θ i s d θ i s d, he we have + θ i s d From aoher chage of variables i he porio of he iegral from o, we have = θ i s d + θ i s d Sice θ z = θ z iz, he 3

11 = = d s θ i + θ i s d s + + = s s θ i s d θ i s + s d s s d I is clear ow ha he righ had side is aalyic ad defied o he whole comple plae ecep a he simple poles s = ad s =. However, his fucio is symmeric abou he chage of variables s s. So, chagig from s o s, we have ha π s Γ s ζ s is he aalyic coiuaio oo C. Now, we cocer ourselves wih he zeros of he zea fucio as defied by is s aalyic coiuaio, Ζ s ; i.e. ζ s = Ζ s π. From (3) ad (4), we ca derive Γ s ha Γ z π =. Give his ideiy, i becomes clear ha zγ z si πz Γ s = for all s, 4,. This se is commoly referred o as he rivial zeros of he Riema zea fucio. Le us defie a few more umber-heoreical fucios, ad he we will rigorously prove a very impora heorem. Defiiio: (Prime Couig Fucio) The prime couig fucio π, N is give eplicily by Defiiio: (Magold Fucio) π e umber of primes less a or equal o For ay ieger, he Λ N R by Λ l p, if = pm for some prime p ad some m, oerwise Defiiio: (Chebychev ψ-fucio) The Chebychev fucio ψ, N is give by ψ 4 Λ

12 Theorem: (Prime Number Theorem) Proof: Taig he logarihm of (), we have l ζ s = l p s ; Ne, fi a prime, p. The = s Λ p ps = Λ s d d l ζ s = ds = = p = π lim l = PNT l p ps p ds l p s = l(p) p l(p) ps p s s 5 = ps = p ps = p l p l p p s p s = l(p) p s = d ζ s l ζ s = ds ζ s Sice Λ is arihmeic, he Perro s formula, which employs he Melli rasform, yields So, we have = Λ s s ζ = ζ s ψ = πi Λ a+i a i = πi ζ s ζ s a+i a i s s ds ζ s ζ s s s ds We ow ha a ay, he iegral of his coour is evaluaed as a sum of he opposies of he residues of each sigulariy of he iegrad. We ow ha ζ, as defied by is aalyic coiuaio over he comple plae, has a simple pole a s =, a se of rivial zeros a all eve egaive iegers, ad a se of orivial zeros. Combiig hese sigulariies of ζ wih he sigulariy s = of he iegrad, we achieve ψ = = ρ ρ ρ ζ ζ Sice he covergece of boh series is uiform for >, he we have ψ d = + = ρ ρ + ρ ρ + ζ ζ + C

13 ψ d So we have ψ d ε < ψ d = = ψ d ρ as ρ ρ ρ + ζ ζ + C ~. Fi λ >. The for ε >, X such ha X < + ε ε λ ε < Sice ψ is icreasig, we have λ ψ λ λ λ λ ad ε < ψ d < + ε λ ψ d ψ d < + ε λ + ε (5) ψ d λ ψ λ (6) Combiig (5) ad (6) yields iequaliies idepede of he iegral: ψ + ε λ + + ε λ, ad ε λ + λ ε λ λ ψ λ λ We ca choose λ close eough o ad ε close eough o zero ha for sufficiely large X we have i.e. ψ ~. Fially, we have ψ ψ = Also, for < y <, is wihi ay arbirarily small eighborhood of for all X; π = π y + The, for y = y<p Λ l, we have = l p p π y + l l p l p l y y<p p < y + l y l. y<p = π l. l p y + l y ψ 6

14 Thus π π l l + l l l l + l l l l ψ ψ So, sice ψ ψ π l l + l l as, he hese esimaes reveal ha ψ π l as as well. Currely, he Prime Number Theorem is he bes esimae of he disribuio of primes ha we have. Ad while i is a impressive posulae ha sheds ligh o he behavior of prime umbers, i is sill o ha grea a esimaio. This brigs us o he mai cosequece of he Riema Hypohesis. If RH is rue, his would provide a sroger esimae of he prime couig fucio: π = d log + O log. I his 95 hesis, Beril Nyma made a asoudig breahrough. He proved ha RH is equivale o a problem cocerig closure i a Hilber space. Le us defie he space of fucioals L p, f:, C p f p d <. Subsequely, we defie a subspace of L, as N, g L, g = c θ =, < θ ad = c θ =, where, for all R, he fracioal par of is give by, ad deoes he ieger par of (i.e. = + ). Theorem: (Nyma): RH is rue if ad oly if N, is dese i L,. Five years laer, Are Beurlig formulaed a geeralizaio of Nyma s heorem. Today, i is geerally referred o as he Nyma-Beurlig Crierio for he Riema Hypohesis. Theorem: (Beurlig): 7

15 For < p <, he followig are equivale: (NB) i. ζ s has o zeros i he half-plae Re s > p ii. N, is dese i L p, iii. χ, N,, where χ, is he characerisic fucio o,. I, Luis Báez -Duare modified NB io a problem regardig a couable se. Le us defie ρ a = a se ρ a a N. The we have Theorem: (Báez-Duare), ad B be he space of all liear combiaios of he RH is rue if ad oly if χ, B. (BD) 8

16 RESULTS AND DISCUSSION Le V deoe he l -space of sequeces = i i= such ha R ad i i= coverges. For ay, y V, we defie he ier-produc, y = i i+ i y i i= i i+. (Noe: Geerally, his space is defied for sequeces over he comple field wih he ier-produc i y i i= i i+, bu we resriced our ivesigaio o he real umbers.) We he ca defie U, a orhogoal basis for V, as U u, u,, where u i =,..,,,,.., he i he i posiio. The, for each j N, we defie he eleme v (j) V as v (j) i= i (mod j ) u, ad a cosa vecor v V as v i= u i. Fially, le V () be he spa of he se v (),, v (). Uder he framewor of his paricular oaio, we ca rephrase Báez-Duare s crierio: The Riema Hypohesis is rue if ad oly if vis i he closure of = V (). (7) I is evide ha he cosa vecor v cao lie i ay V (). However, sice V is a ierproduc space, he a each, here is a o-egaive miimal disace from v o V (). I oher words, a each, here is a error vecor e () ha is orhogoal o V () ad ha he differece e () v is i V (). 9

17 s v e V Figure. Subsequely, a each, here is a uique se of scalars A () = α (),, α () ha serve as coefficies o he spaig se v (),, v () for which e () v is a liear combiaio. Hece e (), v (j) = for each v (j) v (),, v (), ad his yields he -dimesioal sysem v, v j = = α v (), v j. Assumig he orm,., iduced from he ierproduc, we ca oce agai resae BD: The Riema Hypohesis is rue if ad oly if e () as. (8) We ca ow rely o some basic liear maipulaio o solve for he se of scalars A (). Le s call C () he mari give by C () = c j,, where c j, = v (j), v. I is evide ha C () is boh real ad symmeric. The v, v j = = α cj,. Also, le B () be he iverse of C (), wih eries b,j (Noe: The superscrip is ecessary i b,j, sice he, j ery of he iverse is depede o, as opposed o c j, ). The we have fiie sums. α = j = b,j v (j ), v. Now, we will employ a fucio ha will le us epress v, v j ad v (j ), v as Defiiio: (Digamma Fucio) The Digamma fucio is defied o C\ N by

18 Ψ d Γ l Γ = d Γ Noe: Though he oaio for his fucio, Ψ, is similar o he Chebychev fucio,ψ, besides he subleies of ialics ad lower-case, udersad ha hese sigify wo separae fucios; his is he commo omeclaure. followig: Lemma: v, v j = l j Proof: We have he ideiy Ψ z + = γ + Wih his fucio, we ca show he z = +z cosa. Through some maipulaio, we achieve he followig: The, sice v, v j = = (r + d + )(r + d) i (mod j ) i i+ = Ψ d+ r, where Ψ = γ is he Euler r Ψ d r i=, ad sice r + d (mod r) = d (mod r), (i.e. he umeraors have period j), he, by rearragig erms, his sum ca be epressed as By (9), we have j v, v j = = Ψ + j j Ψ j j = = (j + + )(j + ) 9 = j Ψ j Ψ j + Ψ j Ψ 3 j + + j Ψ j j j Ψ j j = j Ψ j + Ψ j + Ψ j j + Ψ j j Ψ j j j Ψ = j j = Ψ j + Ψ I is well-ow ha j Ψ = j = j γ + l j, ad Ψ = γ. Hece v, v j = j j(γ + l j γ = l j

19 Now a each, α = j = b,j l j. Ne, we eamie v, v j = i (mod j ) i (mod ) i=. Here, he period of he umeraors is j. So, i(i+) agai, by rearragig erms, ad by (9), we have v, v j = j c j = j = j = (mod j ) (mod ) Ψ + j j Ψ + j Ψ j Ψ j Le us ow refer o v e (), he orhogoal projecio of v oo V (), as s (). Sice s () = α () = v, he he i compoe of s () is s i () = each, α () = (mod ) = i α () ; ha is s () i = e () i. By Lemma, we also have for j = b j () l(j). Sice RH is equivale o e () edig o zero, he clearly if here is ay i such ha s i () does o coverge o as, he RH does o hold. I is aural, he, o loo for a limi α o which each α () coverges, so ha s i () a each i. Lemma: Proof: Assume α () s i () = lim s () = v if ad oly if lim α () μ = = i α () i = α () i lim s i () = i μ as. The a each i, we have mod = α () i i = α () i i = α () i + α () i =i+ i + α () i lim α () i + α () i + lim i = α () i =i+ α () i α () i,. + α () i =i+

20 s i = Sice i μ = i i s i = + (s ) i μ i + i lim =, i, we have α () If s, he s i diverges. Heces =, ad, subsequely s i =, i ; i.e. lim s = v. Coversely, assume lim s = v; i.e. lim s i (), i. We have s () = α () mod + α () mod + + α () mod = α () + + α (), ad s () = α () mod + α () mod + + α mod = α 3 () + + α s () s = α () s s = = α α = = μ () Now we employ srog iducio o ; assume α () α = The μ (), for {,, i }. s i () = α () i mod + α () i mod + + α () i mod i = α () i = lim = lim i = lim mod i i = lim i + α () i α () i α () α () i i α () i mod mod lim = lim α () i i + lim = lim i mod =i+ α () i α () α i mod lim i, ad lim =i+ α i + α i i α i 3

21 = lim α i μ i i i μ i as. + i lim α i = i μ i + i lim α i Thus, our fial rephrasig of BD: For α () = Cojecure: () j = l(j), he RH is rue if ad oly if α () b j RH is rue if ad oly if The followig is a graph of vs. μ μ, N. = c j = l j () μ = c 5 l Figure. 4

22 BIBLIOGRAPHY [] Aposol, Iroducio To Aalyic Number Theory, Spriger, 976 [] Ash, Comple Variables, olie oes hp:// [3] Beurlig, A Closure Problem Relaed o he Riema Zea Fucio, Proc. Nal. Acad. Sci.4 (955) 3 34 [4] Báez-Duare, A Sregheig of he Nyma Beurlig Crierio for he Riema Hypohesis, Ai Acad. Naz. Licei 4 (3) 5 [5] Balazard, Saias, The Nyma-Beurlig Equivale Form for he Riema Hypohesis, Epos. Mah. 8 () 3-38 [6] Bober, Ieger Raios of Facorials, Hypergeomeric Fucios, ad Relaed Sep Fucios, Submied o he Lodo Mahemaical Sociey [7] Bombieri, The Riema Hypohesis - The Milleium Prize Problems, Cambridge, MA: Clay Mahemaics Isiue (6) 7-4 [8] Borisov, Quoie Sigulariies, Ieger Raios of Facorials, ad he Riema Hypohesis, Ieraioal Mahemaics Research Noices, Vol. 8 [9] Caldero,The Riema Hypohesis, Moografías de la Real Academia de Ciecias de Zaragoza. 6 (4) 3 [] Che, Disribuio Of Prime Numbers, Imperial College, Uiversiy Of Lodo, Chaper - Elemeary Prime Number Theory, 98 [] Howell, The Prime Number Theorem, The Riema Zea Fucio Ad The Riema-Siegel Formula, Submied o Durham Uiversiy [] Nyma, O he Oe-Dimesioal Traslaio Group ad Semi-Group Icerai Fucio Spaces, Thesis, Uppsala, 95 5

23 [] Vasyui,O a Biorhogoal Sysem Associaed wih he Riema Hypohesis, Algebra i Aaliz,Volume 7,Issue 3, (995) 8 35 [3] Weigarer, O a Quesio of Balazard ad Saias Relaed o he Riema Hypohesis, Advaces i Mahemaics,8 (7)