On the Riemann-Siegel formula

Transcription

1 On he Riemann-Siegel formula A. Kuznesov Dep. of Mahemaical Sciences Universiy of New Brunswick Sain John, NB, EL L5, Canada Curren version: June 6, 7 Absrac In his aricle we derive a generalizaion of he Riemann-Siegel asympoic formula for he Riemann zea funcion. By subracing he singulariies closes o he criical poin we obain a significan reducion of he error erm a he expense of a few evaluaions of he error funcion. We illusrae he efficiency of his mehod by comparing i o he classical Riemann-Siegel formula. Keywords: Riemann zea funcion, Riemann-Siegel formula, asympoic expansion, incomplee Gamma funcion Published in Proc. R. Soc. A 6, doi:.98/rspa The auhor would like o hank anonymous referees for many helpful commens.

2 Inroducion The Riemann-Siegel RS asympoic formula is a very efficien mehod o compue he Riemann zea funcion ζ + i for large and i has been used exensively in he las seveny years o compue he nonrivial zeros [ ] of he zea funcion see Odlyzko 99. The RS formula conains he main sum of N = erms and he asympoic correcion erms which allow reducion of he error. The funcion being approximaed is acually Z, which is defined as Z = ζ + i e iθ, where θ = Im { ln Γ + } i lnπ. The funcion Z is real for real his follows from he funcional equaion for ζ and Z = ζ + i. Below we presen he RS formula: π Z = N n= cosθ logn n + N π [ m j= π j C j + O ] m, where he firs hree correcion erms are C = Ψτ = cos π τ τ 6, cosπτ and τ = series: π C = 96π Ψ τ, C = 8π Ψ 6 τ + 6π Ψ τ, N. To compue θ for large one can use he asympoic expansion derived from he Sirling θ = ln π π In he las fifeen years, several new asympoic expansions for he Riemann zea funcion have appeared. In Berry & Keaing 99 he auhors use he Cauchy inegral formula o derive he asympoic expansion for Z, where he leading erm Z, K is Z, K = Re expi[θ lnn] ξn, Erfc, n n QK, where ξn, = lnn θ, Q K, = K iθ and K is a free real parameer. Noe ha he leading erm is similar o he main sum of he RS, wih he cuoff a n = N being smoohed by he complemenary error funcion, hus he approximaion o Z is a smooh funcion, unlike he RS. Auhors show ha increasing K gives a beer accuracy and ha for suiable K he leading erm Z, K always gives beer accuracy han he RS main sum. As K increases o infiniy Z also conains a leas he firs correcion erm C. Higher correcion erms Z j, K also provide a significan increase in accuracy. In Paris 99 he auhor uses he Poisson summaion formula and he uniform asympoic expansion for he incomplee Gamma funcion o derive an asympoic expansion for Z. This expansion also involves he complemenary error funcion, alhough in a differen manner compared o he Berry & Keaing 99 approximaion. The free parameers in his approximaion can be chosen o decrease he error significanly hough again a he expense of compuing error funcions, see Paris 99 for deails.

3 In his aricle we propose anoher mehod of approximaing Z for large. The resul is a generalizaion of he RS formula wih an addiional free parameer δ: in essence we replace δ highes erms in he main sum by δ erms involving he incomplee Gamma funcion and he funcion Ψτ in he correcion erms is replaced by a similar funcion Ψτ, δ. This approximaion is obained by removing δ poles of he inegrand around he saionary poin and hen performing asympoic expansion. We find ha he form of he correcion erms is he same as in RS, and when δ = we recover he classical RS formula as a special case. Combined wih he asympoic expansion for he incomplee Gamma funcion derived in Temme 979 see also Dunser e al. 998 we obain a simple and efficien mehod of reducing he error in he RS formula see secion for he numerical resuls. Derivaion of he asympoic formula We sar wih he following inegral represenaion for he Riemann zea funcion, which Riemann used o prove he funcional equaion see Tichmarsh 986, page 7: π s s Γ ζs = Υs + Υ s, where Υs = e πis s π s s Γ πi+e πi R e iw π w s sinh w dw. 5 Noe ha he inegral in equaion 5 converges absoluely for all complex s. Throughou his aricle we assume ha [ s = + i and ha is large and posiive. To find he saionary poin of he inegral in 5 we solve ] d iw + i lnw =, hus w = π and he saionary poins are ±i π. We can no move he conour dw π of inegraion o w = i π because of he branch poin a w =, hus we choose he saionary poin w = i π. Remark : Noe ha equaion w = π has wo real soluions when is negaive, however we do no obain new asympoic formulas. We canno move he conour of inegraion o w = π because of he branch poin a, and if we move he conour of inegraion o w = π we would in fac obain he same asympoic expansion as by choosing posiive and w = i π. In order o obain he asympoic represenaion for he inegral in equaion 5 we need o move he conour of inegraion o pass hrough he saionary poin w = i π, expand funcion e iw π w s in Taylor series around w and inegrae erm by erm. However he funcion {sinh w } always has poles near he criical poin, and his will affec he accuracy of he approximaion. Thus we do he following: we fix an ineger number δ and subrac δ singulariies of {sinh w } around he criical poin. Thus we define a funcion F N,δ w as F N,δ w = sinh w N+δ n=n+ δ n w πin. 6

4 Now we move he conour of inegraion o w and we obain he following decomposiion for Υs: Υs = Υ s + Υ s + Υ s = 7 N δ [ ] = e πis s π s s e iw π w s Γ πi Res w=πin sinh + w n= N+δ + e πis s π s s Γ n e iw π w s w πin dw + n=n+ δ αi+r + e πis s π s s Γ e iw π w s F N,δ wdw. Firs we simplify Υ s by compuing he residues: Υ s = π s s Γ w +e πi R N δ Nex, we express Υ s in erms of he incomplee Gamma funcions Υ s = π s Γ N+δ s n=n+ δ n s. 8 n= n s [ Q s ; πin + signnq s ; πin], 9 where Q a; x = Γa,x is he normalized incomplee Gamma funcion see Gradsheyn & Ryzhik. If Γx n = he erm in he sum should be replaced by π s πi iθ+ e s /Γ s. Equaion 9 was derived using he following inegral ɛi+r e x x s x + bi ] dx = πie b + πis ˆb ib [ˆbQ s s ; b Q s; b, where he argumen of ib is beween π; π] and ˆb = signreb. To obain we separae x + bi ino real and imaginary pars and use Gradsheyn & Ryzhik o evaluae each inegral. Now we have o approximae Υ s. Firs we perform a change of variables, w = πix + w, and obain Υ s = s s e i s Γ πi where funcion g, x is defined as g, x = e x ix + x +i = e x i R e x g, xf N,δ πix + w dx, ix+ +i ln + x i Expanding ln + x i in a Taylor series we find ha for every x g, x as and we obain he following asympoic expansion g, x = e x i x x i x +O = + x + x i + 8 x 5 x + 8 x6 i + O. =.

5 As a nex sep we subsiue ino and inegrae erm by erm. To achieve his we need o compue funcions f n τ, δ = e x x n F N,δ πix + w dx, where τ = N. We follow he seps of he π R derivaion of he RS formula in Tichmarsh 986 and compue he exponenial generaing funcion for f n τ, δ f n τ, δ u n = e x +xu F N,δ πix + w dx = n! n = R R e x +xu sinh πix + w dx τ + u, δ πi = N πie u Ψ where he funcion Ψτ, δ is defined as πi e 8 sinπτ e Ψτ, πiτ τ δ = cosπτ and Φx = π k= δ N+δ n=n+ δ n R e x +xu πix + w πin dx = δ k e [sign ] πiτ k k + Φ π τ k, i x e d is he error funcion see Gradsheyn & Ryzhik. The firs inegral in is one of he Mordell inegrals see Ramanujan 95, Mordell 9, Kuznesov 6 and can be compued using he mehod described in Tichmarsh 986. The second inegral can be found in Gradsheyn & Ryzhik. Taking n-h derivaive of boh sides of we obain f n τ, δ = N πi 8π n e πin s Γ [ n ] k= n! k!n k! πik Ψn k τ, δ. 5 Now we use 5, and o obain he following asympoic formula for Υ s: Υ s = N s s [ e i S + π S + π where he correcion erms are S = Ψ τ, δ S = Ψ τ, δ 96π S = i Ψ τ, δ + Ψ τ, δ + Ψ 6 τ, δ. 96π 6π 8π Finally, we combine, 7, 8, 9 and 6 o obain he following expression for Z: N δ Z = + Re k= { N+δ ] S + O, 6 cosθ lnn n + 7 n=n+ δ + N π Re iθ ln n e [ Q + i; πin + signnq } + i; πin] + n { e iθ i πe [ S + π 5 S + π ]} S + O.

6 We have also used he fac ha π s Γ s = e Γ iθ which follows from he definiion of θ see. s The correcion erms in he above equaion 7 can be simplified if we use he asympoic formula o rewrie he facor e iθ πe i as e iθ i = e πi 8 i 8 7i 576 πe { Thus we inroduce he new funcion Ψτ, δ = Re e πi 8 Ψτ, δ }, use o simplify he expression for Ψτ, δ and presen all he resuls in he following heorem: [ Theorem. Le be a real posiive number. Define N = δ. Then N δ Z = + Re k= { N+δ π ] and τ = π N. Fix an ineger number cosθ lnn n + 8 n=n+ δ + N π iθ ln n e [ Q + i; πin + signnq } + i; πin] + n [ m j= where he firs hree correcion erms are: and Ψτ, δ = δ cos π τ τ 6 cosπτ π j S j + O ] m, S = Ψ τ, δ 9 S = 96π Ψ τ, δ S = 8π Ψ 6 τ, δ + 6π Ψ τ, δ. cosπδτ + { δ k Re k= δ e πi 8 πiτ k Φ Here Qa, z = Γa,z Γa is he normalized incomplee Gamma funcion and Φx = π funcion. If δ > N he n = erm in he second sum in equaion 8 should be replaced by π i τ k }. x e d is he error. π e πi 8 π + i Γ +. i Remark : Noe ha he correcion erms 9 have he same form as in he classical RS formula see equaion. This can be explained as follows: he coefficiens in fron of derivaives of Ψτ, δ are obained from he expansion of g, x and hus do no depend on δ. However, when δ = equaion 8 mus give us he classical RS formula, hus hese coefficiens mus be he same for all δ. 6

7 Remark : Noe ha using he asympoic expansion of g, x is no he only way o approximae he erm Υ s. One could prove ha if a consan a saisfies < a, hen g, x can be expanded in convergen series in he Hermie polynomials H n ax. In his way we would obain a convergen asympoic series for Z, where he correcion erms would also involve linear combinaions of he derivaives of Ψτ, δ. However we found ha such an expansion is cerainly more complicaed and is no as accurae as 8. Remark : When δ + we see ha he firs sum in 8 vanishes, funcion Ψ τ, δ his follows from equaion and he fac ha F N,δ w, hus 8 reduces o he following expansion: Z = Re { e iθ n n s Q s, πin π e πis s Γ s }, s = + i, which was used in Paris & Cang 997 o derive an asympoic expansion for ζ + i. Numerical resuls Before we can use formula 8 we need o be able o compue efficienly he incomplee Gamma funcion Q σ + i; πin, where σ =, and N + δ n N + δ. In applicaions, especially when is large, we will be ineresed in he case when δ N. Bu hen we find ha πin, hus we need o approximae σ+ i Qa, z in he region z. Forunaely we have an excellen approximaion o Qa, z in his region derived a in Temme 979, see also Dunser e al Here we presen an approximaion o Qa, z of order a 5 which is enough for our purposes for all he deails see Temme 979. Proposiion. Define µ = z a and η = µ ln + µ = µ µ µ 7 5 µ Then Qa, z = Erfc a η + e a η πa [ c + c a + O a ], where coefficiens c, c are given by c = η + µ = µ η c = η + µ + µ µ = µ 7 5 µ +... µ η 5 7 µ +... Remark 5: Noe ha coefficiens c k have removable singulariies a µ =, hus when is close o πn and parameers µ and η are close o we have o use he second se of equaions in, which do no involve subracing large numbers. Below we presen he numerical resuls. An approximaion o Z given by equaion8 wih m correcion erms and fixed δ will be denoed as RS[m, δ]. The classical RS formula will be denoed as RS[m, ]. As we will see, for differen choices of parameers m and δ hese approximaions have differen shapes of he error Z m,δ Z as a funcion of : someimes he error is smaller for τ while for oher choices of m and δ he error is smaller a he endpoins τ and τ. Thus i is hard o compare he efficiency of approximaion RS[m, δ] a a single poin and insead we will presen he error graphically for he range of. 7

8 Firs, we compare RS[, ] wih RS[, ], see figure. We find ha for < < 7 approximaion RS[, ] is acually beer and for < < 6 boh of hese approximaions have comparable accuracy. For even larger RS[, ] becomes a beer approximaion, since i has an error erm of he order O 7 while RS[, ] is O. Bu we find ha even a large we could increase δ and make RS[, δ] as good as RS[, ]: for example a 5 < < 7 we find ha δ = 6 is enough for his purpose. x RS: m= RS: m=, δ= x x 7 RS: m= RS: m=, δ= x Figure : The error for [, 7] N and [, 6] 5 N 9. 8 x 5 RS: m=, δ= RS: m=, δ= 6 6 x x 7 RS: m=, δ= RS: m=, δ= x x x Figure : The error for [5, 7] 9 N 5. m = and δ increases from o. 8

9 RS: m=, δ= RS: m=, δ= x 5 6 x 6 RS: m=, δ= RS: m=, δ= x x RS: m=, δ= RS: m=, δ= x x RS: m=, δ= RS: m=, δ= x Figure : The error for [5, ] 5 N 8. m increases from o, δ = in he op row and δ = in he boom row. Second, we examine he effecs of increasing δ while keeping m = fixed see figure in he region 5 < < 7. We see ha increasing δ by decreases he error roughly by a facor of. Also noe ha he shape of he error he shape of he graph of he nex correcion erm becomes more linear as δ increases. This is easy o explain if we remember ha δ is he number of subraced singulariies, hus for larger δ funcion Ψτ, δ and is derivaives become less oscillaory. This fac means ha insead of compuing Ψτ, δ and is derivaives we can efficienly approximae he correcion erm by jus a few erms of is expansion in a Taylor series or Chebyshev polynomials. Finally, we examine he effecs of increasing he number of correcion erms, while keeping δ fixed see figure. In he op boom row we plo he absolue error of RS[m, ] RS[m, ] when m increases from o. Noe he decrease of 5 orders of magniude as m goes from o in he second row δ =, while in he op row δ = we have a decrease of orders of magniude. Afer his fas iniial decrease we gain roughly order of magniude in he boom row and orders in he op row. This seems o be a general rend: he error decreases faser wih he increase in m for small δ compared o large δ. Noe again ha in he boom 9

10 row δ = he correcion erms are smooher compared o he op row δ = x Figure : The effec of using a smooh cuoff funcion. [, 7] N, m = and δ = Conclusion In his aricle we presen a generalizaion of he Riemann-Siegel asympoic formula, which allows o obain a significan increase in he accuracy wihou a lo of exra compuaional effor. This approximaion has an exra free ineger parameer δ, which corresponds o half of he number of he singulariies removed around he criical poin. In general increasing δ resuls in he decrease of he error. In his aricle we compared his new approximaion scheme o he classical RS formula, and found ha he new formula consisenly gives beer accuracy even for small δ. We did no compare our approximaion o oher resuls, such as approximaions by Berry & Keaing 99 or by Paris 99: while i seems ha our approximaion can achieve he same accuracy, he quesion is a wha compuaional cos. To answer his quesion one would have o opimize approximaion schemes. Noe ha in our scheme here are several hings ha can be done o reduce he compuaional cos o jus δ evaluaions of he error funcion:. In equaion 8 we have wo incomplee gamma funcions Q σ + i; πin wih σ =,, bu when is large we can approximae hem wih jus a few erms of he Taylor series around σ =.. The almos linear form of he graphs of correcion erms see figures and suggess an approximaion by polynomials: one could use jus a few erms of eiher he Taylor series around τ = or he Chebyshev series o approximae S j. An advanage of he Berry & Keaing 99 asympoic formula is ha he approximaion [ ] erms are smooh funcions of ; in he approximaion by Paris 99 one can also choose N and hus have a smooh π approximaion a he ransiion poins. There is also a simple way o do i in our approximaion: insead of compleely removing δ singulariies around he criical poin =, we can assign o every singulariy a π

11 weigh, depending on he disance from he criical poin. For example, Theorem could be obained using he following weighs: if he singulariy is wihin he disance of πδ from he criical poin, i is assigned he weigh of and is removed compleely, oherwise he weigh is see equaion 6. Noe ha his scheme is equivalen o using a sharp cuoff funcion wih jumps a ±πδ see figure, and hese jumps creae disconinuiies in he RS formula. However, one could also use a smooh cuoff funcion, where he weigh of each singulariy is if i is close o he criical poin and he weigh would decrease smoohly o as he disance o increases see figure. The resul of using a smooh cuoff funcion is ha he error a he ransiion poins πn is cerainly smaller, bu a he same ime he compuaional complexiy is increased: noe ha on figure he sharp cuoff conains jus ransiion poins n = and n =, while he smooh cuoff also conains n = and we will need more evaluaions of he error funcion. Anoher undesirable feaure of using he smooh cuoff is ha he correcion erms become dependen on and no jus on τ as wih he sharp cuoff. References [] Berry, M.V. & Keaing, J.P. 99 A new asympoic represenaion for ζ + i and quanum specral deerminans. Proc. Roy. Soc. London, A7, No. 899, 5-7. [] Dunser, T.M., Paris, R.B. & Cang S. 998 On he high-order coefficiens in he uniform asympoic expansion for he incomplee gamma funcion. Mehods Appl. Anal. 5, -7. [] Gradsheyn I.S. & Ryzhik I.M. Tables of inegrals, series and producs. 6h edn, Academic Press. [] Kuznesov A. 7 Inegral represenaions for he Dirichle L-funcions and heir expansions in Meixner- Pollaczek polynomials and rising facorials. Inegral Transforms and Special Funcions o appear. [5] Mordell L.J. 9 The definie inegral -6. e a +b d and he analyic heory of numbers. Aca Mah. 6, e c +d [6] Odlyzko A.M. 99 Analyic compuaions in number heory. Mahemaics of Compuaion 9-99: A Half-Cenury of Compuaional Mahemaics, W. Gauschi ed., Amer. Mah. Soc., Proc. Symp. Appl. Mah. 8, pp [7] Paris R.B. 99 An asympoic represenaion for he Riemann zea funcion on he criical line. Proc. Roy. Soc. London, A6, No. 98, 99, [8] Paris R.B. & Cang S. 997 An asympoic represenaion for ζ + i. Mehods Appl. Anal.,,997, 9-7. [9] Ramanujan S. 95 Some definie inegrals conneced wih Gauss sums. Messenger of Mahemaics, XLIV, [] Temme N.M. 979 The asympoic expansion of he incomplee gamma funcions. SIAM J.Mah.Anal., [] Tichmarsh E.C. 986 The heory of he Riemann zea-funcion. nd edn, Oxford Universiy Press.