On A Rapidly Converging Series For The Riemann Zeta Function

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by  Alois Pichler

On A Rapidly Converging Series For The Riemann Zeta Function
AloisPichler
DepartmentofStatisticsandOperationsResearch,UniversityofVienna,Austria,Universitätsstraße5,1010Vienna
2 Abstract
1
0 To evaluateRiemann’s zetafunction isimportant for manyinvestigations relatedto the areaof
2 number theory, and to have quickly converging series at hand in particular. We investigate a
class of summation formulae and find, as a special case, a new proof of a rapidly converging
n
a seriesfortheRiemannzetafunction. Theseriesconvergesintheentirecomplexplane, itsrate
J of convergence being significantly faster than comparable representations, and so is a useful

1 basisforevaluationalgorithms. Theevaluationofcorrespondingcoefficientsisnotproblematic,
3 andpreciseconvergenceratesareelaboratedindetail. Thegloballyconvergingseriesobtained

allow to reduce Riemann’s hypothesis to similar properties on polynomials. And interestingly,
]
T Laguerre’spolynomialsformakindofleitmotifthroughallsections.
N
Keywords: RiemannZetafunction,Kummerfunction,Laguerrepolynomials,Fourier
.
h transform,Riemannhypothesis.
t
a
m
[

1. IntroductionandDefinitions
1
v
1.1. ConfluentHypergeometricFunctions
8
3
Confluenthypergeometricfunctionsarespecialhypergeometricfunctions,sometimescalled
5
alsoKummer’sfunctionoffirstandsecondkind. TheyarelinearindependentsolutionsofKum-
6
mer’sdifferentialequation
.
1
0 zy(cid:48)(cid:48)(z)+(b−z) y(cid:48)(z)−ay(z)=0.
2
1
Thefirstsolution,Kummer’sfunctionofthefirstkind,isusuallygivenasagloballyconverging
:
v
powerseries
i (cid:16) (cid:17)
X (cid:88)∞ a+i−1 zi
r M(a;b;z):= (cid:16)b+ii−1(cid:17)i! (1.1)
a i=0 i
(Kummer’sfunctionofthefirstkind),andKummer’sfunctionofthesecondkindisoftengiven
by
Γ(1−b) Γ(b−1)
U(a;b;z):= M(a;b;z)− z1−bM(a−b+1;2−b;z) (1.2)
Γ(1−b+a) Γ(a)
Emailaddress:[email protected](AloisPichler)
Preprintsubmittedtonowhere February1,2012
1 INTRODUCTIONANDDEFINITIONS 1.2 Laguerre’sPolynomials
Kummer’s transformation states that M(a;b;z) = ezM(b−a;b;−z), which subsequently
leadstotheidentityU(a;b;z)=z1−bU(1+a−b;2−b;z).
Anothersolutionofthisdifferentialequation–whichturnsouttobeidenticaltoU andthus
issimplyanotherrepresentationofU –isobtainedasanintegralrepresentationbytheLaplace
transform
ˆ
1 ∞
U(a;b;z)= e−ztta−1(1+t)b−a−1dt
Γ(a)
0 ˆ
z1−b ∞ ta−b
= e−zt dt. (1.3)
Γ(1+a−b) (1+t)a
0
Some particular and frequently used confluent hypergeometric functions are the upper and
lowerincompleteGammafunction,whichhavetherepresentations
ˆ
∞
Γ(s,z):= ts−1e−tdt
z
=e−zU(1−s,1−s,z)=e−zzsU(1,1+s,z) (1.4)
and
ˆ
z (cid:88)(−1)k zs+k
γ(s,z):= ts−1e−tdt=
k! s+k
0 k=0
zs zs
= M(s,s+1,−z)= e−zM(1,s+1,z). (1.5)
s s
1.2. Laguerre’sPolynomials
ForaanegativeintegerthedefiningseriesforMreducestoapolynomial,whichturnsoutto
becloselyrelatedtoLaguerre’spolynomials: Theexplicitrepresentationis
(cid:32)i+α(cid:33) (cid:88)i (cid:32)i+α(cid:33)zj
L(α)(z)= M(−i,α+1,z)= (−1)j . (1.6)
i i i− j j!
j=0
Inviewof (1.2)Laguerre’spolynomialmaybegivenbyKummer’ssecondfunctionaswell,
that is L(α)(z) = (−1)iU(−i,α+1,z). This somehow suggests that Laguerre’s polynomial are
i i!
somewhat in between of both solutions M and U of Kummer’s differential equation. This is
centralinourinvestigationsandreflectedintheresultsofthenextsections.
Inadditiontothatitiswell-knowthatLaguerre’spolynomialsareorthogonalwithrespectto
theweight-functionzαe−z;morepreciselywefindthat
ˆ 
∞zαe−zL(α)(z)L(α)(z)dz=0(cid:16) (cid:17) i(cid:44) j (1.7)
i j  i+α Γ(α+1) i= j,
0 i
Laguerre’spolynomialsthusareorthogonalwithrespecttotheinnerproduct
ˆ
∞ zαe−z
(cid:104)g| f(cid:105):= g(z) f (z)dz.
Γ(α+1)
0
As a result of the classical theory on Hilbert spaces we may expand a function f in a series
(cid:112)
withrespecttothisorthogonalbasis,providedthat(cid:107)f(cid:107) := (cid:104)f| f(cid:105) < ∞. Thefunctionhasthe
2
2
2 FOURIERSERIES
expansion f (z) = (cid:80)i=0 fi(α)Li(α)(z)(andisthereforeintheclosedhullofallLi(α)),itscoefficients
takingtheexplicitform
ˆ
∞ L(α)(z) zαe−z
fiα = (cid:16)ii+α(cid:17) Γ(α+1)f (z)dz (1.8)
0
i
(cf. (1.7)). Conversely, the norm can be recovered from the function’s coefficients, as (cid:107)f(cid:107)2 =
(cid:80)i=0(cid:16)i+iα(cid:17)(cid:12)(cid:12)(cid:12)fiα(cid:12)(cid:12)(cid:12)2. 2
Anelementaryexampleofsucharepresentationinexplicittermsis
1 (cid:88) (cid:18) t (cid:19)i
e−tz = L(α)(z) , (1.9)
(1+t)α+1 i 1+t
i=0
ascanbeverifiedstraightforwardbyevaluatingtherespectiveintegralsforthecoefficients(1.8).
Notably,thisseriesconvergespoint-wiseift>−1 ((cid:107)f(cid:107) <∞),evenifα≤−1.
2 2
Togiveareferenceoftheseclassicalingredientsaggregatedinthissectionabovewewould
liketorefertothestandardwork[AS64].
2. FourierSeries
Kummer’sfunctionsallowsomeexplicitrepresentationasseriesofLaguerrepolynomials.
Theorem1(ExpansionofKummer’sfunctionsintermsofLaguerrepolynomials). Supposethat
Re(b−a)>0andRe(α−2b)>−5,then
2
(cid:16) (cid:17) (cid:16) (cid:17)
b−1 −a
(cid:88)
M(a;b;z)= (cid:16) a (cid:17) (−1)iL(β−i)(z)(cid:16) i (cid:17)
b−β−1 i β−b
a i=0 i
and
(cid:16) (cid:17)
(1+α−b)! (cid:88) −a
U(a;b;z)= (1+α−b+a)! Li(α)(z)(cid:16)b−a−iα−2(cid:17); (2.1)
i=0 i
moreover,
(cid:88) L(α)(z)
Γ(s;z)=zse−z k(cid:16) (cid:17) (2.2)
(k+1) k+1+α−s
k=0 k+1
(cid:16) (cid:17)
forRe s− α < 1.
2 4
Proof. As for the proof notice first that zi = (−1)i(cid:80)i (cid:16)−β(cid:17)L(β−j)(z), which is a kind of con-
i! j=0 i−j j
verse of (1.6) and verified straight forward by comparing the coefficients of respective powers
of z. Substituting this into (1.1), interchanging the order of summation and employing Gauss’
3
2 FOURIERSERIES
hypergeometrictheorem(cf. [Koe98])gives
(cid:16) (cid:17)
(cid:88)∞ a+i−1 (cid:88)i (cid:32) −β (cid:33)
M(a;b;z)= (cid:16) i (cid:17)(−1)i L(β−j)(z)
b+i−1 i− j j
i=0 i j=0
(cid:16) (cid:17)
(cid:88)∞ (cid:88)∞ a+i−1 (cid:32) −β (cid:33)
= L(β−j)(z) (−1)i (cid:16) i (cid:17)
j b+i−1 i− j
j=0 i=j i
(cid:88) Γ(b)Γ(b−a−β)Γ(a+ j)
= L(β−j)(z)(−1)j
j Γ(a)Γ(b−a)Γ(b−β+ j)
j=0
(cid:16) (cid:17) (cid:16) (cid:17)
b−1 a+j−1
(cid:88)
= L(β−j)(z)(−1)j (cid:16) a (cid:17)(cid:16) j (cid:17)
j b−β−1 b−β−1+j
j=0 a j
(cid:16) (cid:17) (cid:16) (cid:17)
b−1 −a
(cid:88)
= (cid:16) a (cid:17) (−1)jL(β−j)(z)(cid:16) j (cid:17),
b−β−1 j β−b
a j=0 j
whichisthedesiredassertion. Asregardsconvergenceitfollowsfromtheelementaryrecursion
(cid:32) (cid:33)
L(β−j)(z)=− 1+ z−β−1 L(β−j+1)(z)− zL(β−j+2)(z) (2.3)
j j j−1 j j−2
that(−1)jL(β−j)(z)=O(cid:16)j−β−1(cid:17),amorethoroughanalysisevendisclosesthat L(jβ−j)(z) =ez(cid:16)1+O(cid:16)1(cid:17)(cid:17).
j (β) j
Hence,(−1)jL(β−j)(z) (−ja) = O(cid:18)j−β−1 ja−1 (cid:19) = O(cid:16) 1 (cid:17),andtheseriesconvergjes–irrespective
j (β−b) jb−β−1 jb−a+1
j
ofβ–ifRe(b−a)>0.
Toverifythesecondstatementnoticethat
ˆ
∞ L(α)(z) zαe−z
Uiα = (cid:16)ii+α(cid:17) Γ(α+1)U(a;b;z)dz
0
i
ˆ ˆ
∞ L(α)(z) zαe−z 1 ∞
= (cid:16)ii+α(cid:17) Γ(α+1)Γ(a) e−ztta−1(1+t)b−a−1dtdz
0 0
i
ˆ ˆ
∞ ta−1 ∞ L(α)(z) zαe−z
= Γ(a)(1+t)b−a−1 (cid:16)ii+α(cid:17) Γ(α+1)e−ztdzdt
0 0
ˆ i
1 ∞ ti
= ta−1(1+t)b−a−1 dt,
Γ(a) (1+t)α+i+1
0
whichisaconsequenceof(1.3)and(1.9). Thelatterintegralisabetafunction,wethuscontinue
andfindthecoefficient
4
2 FOURIERSERIES
(cid:16) (cid:17) (cid:16) (cid:17)
Algorithm1toevaluateL(α)(z),basedonL(α)(z)= 2+ α−1−z L(α) (z)− 1+ α−1 L(α) (z).
i i i i−1 i i−2
L1:=0; Laguerre:= 1
For j:= 1 to i
L0:= L1; L1:= Laguerre;
Laguerre:= ((2∗ j+ alpha− 1− z)∗ L1− (j+ alpha− 1)∗ L0)/ j ;
Next j
Return Laguerre
Algorithm2toevaluateL(β−i)(z),basedon(2.3)
i
L1:=0; Laguerre:= 1
For j:= 1 to i
L0:= L1; L1:= Laguerre;
Laguerre:= (( beta+ 1− j− z)∗ L1− z∗ L0)/ j ;
Next j
Return Laguerre
ˆ
1 ∞
Uα = ta−1+i(1+t)b−a−α−i−2dt
i Γ(a)
0
1 Γ(a+i)Γ(α+2−b)
=
Γ(a) Γ(i+a+α+2−b)
(cid:16) (cid:17)
(α+1−b)! a+i−1
= i
(i+a+α+1−b)!(cid:16)i+a+α+2−b(cid:17)
i
whichistherespectivecoefficientforU.
Convergence is more difficult compared to Kummer’s function of the first kind. However,
we will show below (Theorem 9) that limsup Li(α)(z) < ∞ and the series thus convergences if
i iα2−14
Re(α−2b)>−5.
2
Remark2. Identity(2.1)canbefoundforthespecialcaseα=b−1in[EMOT53].
To evaluate the series above using Laguerre polynomials it is necessary to have good eval-
uations of Laguerre polynomials at hand for given parameters α and z. Moreover, numerical
approximationsshouldbesufficientlygoodandthecomputationstable. Althoughthereareex-
plicitexpressionsorHorner’sschemeavailabletoevaluateLaguerrepolynomials,theresulting
algorithmsusuallybehaveunstableverysoon.
Wehavefoundthealgorithmsdescribedmuchbetter,adaptationsevenallowtoevaluateand
thensuccessivelystoreintermediaryresults. However, theyshouldnotbeinterchanged, asthis
willcausenumericalinstabilityagain.
5
3 CONTINUOUSFOURIERTRANSFORMANDPOISSONSUMMATIONFORMULA
3. ContinuousFourierTransformandPoissonSummationFormula
3.1. ContinuousFourierTransform.
It is well-known that Poisson’s summation formula provides an efficient tool to evaluate
sums, some authors dedicate entire chapters to these summation techniques, see for instance
[IK04]. Toapplytheseeffectivesummationidentitiesweneedtohaveagoodexpressionforthe
functionsinvolvedathand,whichinvolvethecontinuousFouriertransform.
InliteraturethereoccurafewvariantsforthecontinuousFouriertransformofafunction f,
whichareinterchangedfrequently;forourpurposesitismostconvenienttostate
ˆ
∞
fˆ(k):=F (f)(k):= f (z)e−2πikzdz
−∞
asadefinition.
We will sometimes abuse the notation just introduced and improperly write F f (z) shortly
forF (f (z))(z),thatistosayweusetheargumentzforbothfunctions– f andF (f)–synony-
mously.
ThefollowingresultestablishesthatLaguerre’spolynomials,aswellasKummer’sfunctions
areeachothersFouriertransforminthefollowingsense:
Theorem3(FouriertransformofKummer’sfunctions).
1. F e−z2πL(α)(cid:16)z2π(cid:17)=(−1)ie−z2πL(−i−α−12)(cid:16)z2π(cid:17),
i i
2. F e−z2πU(cid:16)a,b,z2π(cid:17)= Γ(23−b) e−z2πM(cid:16)a,a+ 3 −b,z2π(cid:17),
Γ(a+3−b) 2
(cid:16) (cid:17) 2 (cid:16) (cid:17)
3. F e−z2πM a,b,z2π = Γ(b) e−z2πU a,a+ 3 −b,z2π .
Γ(b−a) 2
Proof. Noticefirst,thate−z2πisaneigenfunctionoftheFouriertransformF,whichalreadycov-
(cid:16) (cid:17)
ersthedesiredstatementfori=0. Nowrecallthat–duetointegrationbyparts–F dj f (z) =
dzj
(2πiz)jF (f (z))andobservethatdifferentiatingthisfirsteigenfunctioninvolvesLaguerre’spoly-
nomialsagain,as d2j e−z2π =(−4π)j j!e−z2πL(−21)(cid:16)z2π(cid:17). Hence,
dz2j j
(cid:16)z2π(cid:17)j (−1)j (−1)j (cid:32) d2j (cid:33)
e−z2π = (2πiz)2jF =e−z2π F e−z2π
j! (4π)j j! (4π)j j! dz2j
= (−1)j F (cid:18)(−4π)j j!e−z2πL(−21)(cid:16)z2π(cid:17)(cid:19)=F e−z2πL(−12)(cid:16)z2π(cid:17).
(4π)j j! j j
Next,by(1.6)andtheidentityjustderived,
(−1)ie−z2πL(−i−α−21)(cid:16)z2π(cid:17)=e−z2π(cid:88)i (−1)i−j(cid:32)−α− 21(cid:33)(cid:16)z2π(cid:17)j
i i− j j!
j=0
=F e−z2π(cid:88)i (cid:32)α− 12 +i− j(cid:33)L(−12)(cid:16)z2π(cid:17)=F e−z2πL(α)(cid:16)z2π(cid:17),
i− j j i
j=0
6
4 APPLICATIONTORIEMANNZETAFUNCTION 3.2 PoissonSummationFormula.
thelatteridentitybeingaspecialcase(x=0,β=−1)ofthemoregeneralidentity
2
(cid:88)i
L(α)(x)L(β)(y)= L(α+β+1)(x+y).
i−j j i
j=0
Thisprovesthefirststatement.
TheotherstatementsareanimmediateconsequenceofthisfirstoneandTheorem1.
3.2. PoissonSummationFormula.
Given the notation introduced Poisson’s summation formula reads (cid:80)z∈Z f (z) = (cid:80)k∈Z fˆ(k)
(cf. [Zyg68,Pin02]),orabitmoregenerally
(cid:88) 1(cid:88) (cid:32)k+k (cid:33)
f ((z−z )x)e−2πi(z−z0)k0 = fˆ 0 e−2πikz0 (3.1)
0 x x
z∈Z k∈Z
whenappliedtothesamefunction f buttranslated(z ),contracted(x)andshiftedinphase(k ).
0 0
As alreadymentioned, thePoisson summation formulaoften transformsslowly converging
seriesintoveryrapidlyconvergingseries.Wewillexploitthisfacttoobtainthefollowingresults,
whichareanapplicationofPoisson’ssummationformulatotheFouriertransformselaboratedin
thelattersection.
4. ApplicationtoRiemannZetaFunction
It turns out that a particular application of summation techniques outlined above is a very
rapidlyconvergingseriesfortheRiemannzetafunction. Weareevenabletoprovethesefollow-
ingvariants,andRiemann’sfunctionequationfollowsasaby-product:
Theorem4. Foranys∈Cwehave
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
ζ(s)Γ s (cid:88)Γ s,k2π Γ 1−s,k2π
s(s−1) 2 =1−s(1−s) 2 + 2 . (4.1)
π2s k=1 (cid:0)k2π(cid:1)2s (cid:0)k2π(cid:1)1−2s
Moregenerally,foreveryarbitrarilychosenxsatisfyingRe(x)>|Im(x)|wefind
(cid:16) (cid:17)
ζ(s)Γ s
s(s−1) 2 =(1−s)xs+sxs−1+
π2s
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:88)Γ s,k2x2π Γ 1−s,k2π
+s(s−1) 2 + 2 x2 ;
k=1 (cid:0)k2π(cid:1)2s (cid:0)k2π(cid:1)1−2s
moreover
(cid:16) (cid:17)
(cid:16) (cid:17)ζ(s)Γ s
(2s−1) 1−21−s 2 =Υ (s)+Υ (1−s),
π2s x 1x
whereΥ (s)isthe2ndforwarddifference
x
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Γ s,(4k+1)2πx2 Γ s,(4k+2)2πx2 Γ s,(4k+3)2πx2
(cid:88) 2 4 2 4 2 4
Υ (s)= −2 + .
x (cid:18) (cid:19)s (cid:18) (cid:19)s (cid:18) (cid:19)s
k=0 (4k+1)2π 2 (4k+2)2π 2 (4k+3)2π 2
4 4 4
7
4 APPLICATIONTORIEMANNZETAFUNCTION
Remark5. Itshouldbestressedthatthesearegloballyconvergentseriesforafunctionanalyticin
theentireplane,converginginparticularfors=1. Moreover,bydel’Hôpital’srule,lim Γ(s,z) =
zs−1e−z
lim zs−1e−z =1,therateofconvergencethusisoforder
zs−1e−z−(s−1)zs−2e−z
 
O(cid:88)∞ e−k(cid:48)2π = O(cid:16)e−k2π(cid:17)
k(cid:48)=k
(cid:16) (cid:17)
= O 0.04321...k2 ,
whichisprettyquick.
As an additional result, Riemann’s function equation follows immediately from equation
(4.1).
Itseemsthatanidentitycloseto(4.1)alreadywasknowntoRiemannhimself,althoughthe
proof being basedon integrals involving the Jacobifunction(cid:16)θ3(cid:17)(z) := (cid:80)k∈Ze−k2πz, or ratherthe
function θ˜(z) := θ3(cid:48) (z)+ 32θ3(cid:48)(cid:48)(z): Both satisfy θ3(z) = √1zθ3 1z , but θ3(k)(z) → 0 as z → 0 and
z → ∞foranyk ∈ {0, 1, 2,...}. 1 Byexploitingthisfact(4.1)canberecoveredaswell–this
is for example demonstrated in the third method (out of 7) of proving the function equation in
[Tit86].
Remark 6. The famous, fast algorithm for multiple evaluations of the Riemann Zeta function
in[OS88]isbasedonanarepresentationforthezetafunction, whichisnotconvergingonthe
entirecomplexplane,andtherateofconvergencebeingslower,butallowingpreciseboundsand
estimates.
The representation given by [Son94], ζ(s) = 1−211−s (cid:80)n=1(cid:80)nk=0 ((−k1+)1k)−s1(cid:16)nk(cid:17), converges globally
aswell,however,therateofconvergenceissignificantlyslower.
Asimilaralgorithmisproposedin[Bor91]withexponentialconvergencerate.
SotosummarizeandcomparetherepresentationsdescribedinTheorem4arefaster,theonly
(cid:16) (cid:17)
downsideisthatΓ s isverysmallforhugeimaginaryparts,andincompleteGammafunctions
2
havetobeevaluated. However,thiscanbedoneveryquickly,aswillbefurtheroutlinedbelow.
Proof. Asfortheproofnoticefirstthat
(cid:88)∞ e−z2πx2M(cid:16)a,b,z2πx2(cid:17)= 1xΓ(Γb(−b)a) (cid:88)∞ e−zx22πU(cid:32)a,a+ 23 −b,zx22π(cid:33),
z=−∞ z=−∞
or,using(1.2),
1+2(cid:80)∞ e−z2πx2M(cid:16)a,b,z2πx2(cid:17)
= 1xΓ(Γzb=(−b1)a)(cid:18)Γ(Γb(−b−12−1)a) +2(cid:80)∞z=1e−zx22πU(cid:16)a,a+ 32 −b,zx22π(cid:17)(cid:19).
2
Wechoosea=1andb=1+ s. Thus,using(1.5)and(1.4),
2
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
s (cid:88)∞ γ s,z2πx2 s 2Γ 1+ s (cid:88)∞ Γ 1−s,z2π
1+22 z=1 (cid:0)z22πx2(cid:1)2s = (s−1)x + x Γ(cid:16)2s(cid:17)2 z=1 (cid:16)z22π(cid:17)1−2xs2 ,
x2
1Bytheway: Supposethatθ(z)=z−α−1·θ(cid:16)1z(cid:17),then(−1)i z1+i!α+iθ(z)=(cid:80)ij=0(cid:16)ii+−αj(cid:17)zj1j!θ(j)(cid:16)1z(cid:17),strikinglyreminding
totheexplicitrepresentationofLaguerrepolynomials(1.6).
8
4 APPLICATIONTORIEMANNZETAFUNCTION
or
(cid:16) (cid:17) (cid:16) (cid:17)
xs xs−1 (cid:88)∞ γ s,z2πx2 (cid:88)∞ Γ 1−s,z2π
+ + 2 = 2 x2 . (4.2)
s 1−s z=1 (cid:0)z2π(cid:1)2s z=1 (cid:0)z2π(cid:1)1−2s
Thisniceidentityathandwefindthat
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
ζ(s)Γ s (cid:88)γ s;k2πx2 +Γ s;k2πx2
s(s−1) 2 = s(s−1) 2 2
π2s k=1 (cid:0)k2π(cid:1)2s
=(1−s)xs+sxs−1
(cid:16) (cid:17) (cid:16) (cid:17)
(cid:88)Γ 1−s;k2π Γ s;k2πx2
−s(1−s) 2 x2 + 2 ,
k=1 (cid:0)k2π(cid:1)1−2s (cid:0)k2π(cid:1)2s
(cid:16) (cid:17)
whichisthesecondstatement.Notice,thattheconditionRe(x)>|Im(x)|insuresboth,Re x2 >
(cid:16) (cid:17)
0andRe 1 >0,whichisnecessaryforconvergence.
x2
Thefirststatementisobviousbythechoicex=1.
Asforthenextstatementrecallthat
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
(cid:16) (cid:17)ζ(s)Γ s (cid:88) γ s,k2πx2 +Γ s,k2πx2
1−21−s 2 = (−1)k−1 2 2 .
π2s k=1 (cid:0)k2π(cid:1)2s
Inordertogetridofslowlyconvergingγ(andreplaceitbyrapidlyconvergingFouriertransform
Γ)weapplyPoissonsummationformula(3.1)again,nowwithz = 1 andk =0. Hence,
0 2 0
(cid:32) (cid:33)
(cid:16) (cid:17)ζ(s)Γ(cid:16)s(cid:17) xs (cid:88)Γ 1−2s,(k−x122)2π Γ(cid:16)s,k2πx2(cid:17)
1−21−s 2 = − +(−1)k 2
π2s s k=1 (cid:18)(cid:16)k− 1(cid:17)2π(cid:19)1−2s (cid:0)k2π(cid:1)2s
2
(cid:18) (cid:19)
= xs − 1 (cid:88)Γ 1−2s,(2k−x21)2π +(−1)k Γ(cid:16)2s,k2π4x2(cid:17),
s2s 2s−1 k=1 (cid:16)(2k−1)2π(cid:17)1−2s (cid:0)k2π(cid:1)2s
as the statement holds for x as well. Combining the latter two identities and rearranging the
2
termsiscumbersome,butfinallygivestheassertion.
√
Remark 7. x := s is notably a possible choice if Im(s) > 1+ 2 ≈ 1.21, as in this case
s−1 2
Re(x) > |Im(x)|. Theadvantageofthisthisparticularchoiceisthattheterm(1−s)xs+sxs−1
vanishes.
Remark 8. It should be noticed that the identity (4.2) is central here. It allows to replace the
slowly converging series (which involves γ) by its Fourier transform, which is the very fast
convergingseries(whichinvolvesΓ).Thisisthekeystrategyinallimprovementsofconvergence
above.
9
5 ASYMPTOTICSOFTHELAGUERREPOLYNOMIALS
5. AsymptoticsoftheLaguerrePolynomials
Theorem9. (AsymptoticsoftheLaguerrePolynomial)LetRe(z)>0. Then,asi→∞,
L(α)(z)≈ iα2√−14 e2z cos(cid:18)2(cid:113)z(cid:16)i+ α+1(cid:17)− π(cid:16)α+ 1(cid:17)(cid:19)and
Li(α)(−z)≈ iπα2√−14zα2+e−412z exp(cid:18)2(cid:113)z(cid:16)i+2α+1(cid:17)(cid:19).2 2
i π zα2+14 2
Remark10. Wewrite f ≈g asi→∞toindicatethatlim fi =1.
i i i→∞ gi
Remark 11. To prove the statements of the theorem we will involve Bessel functions. Bessel
functions,offirstandsecondkind,are
(cid:88) (−1)m (cid:18)x(cid:19)α+2m (cid:88) 1 (cid:18)x(cid:19)α+2m
J (x):= andI (x):= .
α m!Γ(α+m+1) 2 α m!Γ(α+m+1) 2
Notice,that Jα(ix) = Iα(x) areentire,evenfunctions. Moreover,Kummer’ssecondformulalinks
(ix)α xα
Besselfunctionstoconfluenthypergeometricfunctions,as Jα(z) =e−zM(a+12,2a+1,2z).
(z)α Γ(a+1)
2
Proof. Westartwithidentity13.3.7from[AS64]whichstatesthat
(cid:16)√ (cid:17)
MΓ(a(,bb),z) =e2z ·(cid:88)n=0An·(cid:18)2z(cid:19)n (cid:16)Jb1−√1+2nz(b2−z2(ba)−(cid:17)b2−1a+)n, (5.1)
2
whereJistheBesselfunctionofthefirstkindandAsatisfiestherecursionA :=1,A :=0,A :=
(cid:16) (cid:17) 0 1 2
b and A := 1+ b−2 A − b−2aA . Although we refer to this Identity (5.1) without proof
2 n n n−2 n n−3
wementionthatitfollowsstraightforwardbysuccessivelycomparingtherespectivecoefficients
(cid:16)(cid:16) (cid:17)(cid:17)
of z in (5.1). The coefficients satisfy A = O a−b for b > 2a, as follows directly from the
n n
recursivedefinition.
Following[AW05],
(cid:114) (cid:32) (cid:32) (cid:33)(cid:33)
2 π 1 ex
J (x) ≈ cos x− α+ , I (x)≈ √ (5.2)
α πx 2 2 α 2πx
asx→∞.
Combining all those ingredients we see that the initial term (n = 0) dominates all other
summands in (5.1) when a → −∞, so we may neglect them for n ≥ 1 to identify the rate of
convergence. Butforn=0theassertionstatesthat
(cid:16)√ (cid:17)
MΓ(a(,bb),z) ≈e2z · (cid:16)Jb1−√12z(2bz−(b2−a)2(cid:17)ba−)1,
2
whichisausefulapproximationitself.
Thereisanother,similarapproach,whichisquiteuseful,whichwewanttogivehereaswell
for the sake of completeness and further reference: It starts with identity 13.3.8 from [AS64]
withparameterh= 1. Thisreads
2
(cid:16) √ (cid:17)
MΓ(a(,bb),z) =e2z ·(cid:88)CnznJb(cid:16)−√1+n 2(cid:17)b−−1+anz ,
n=0 −az
10