POLYLOGARITHMS, MULTIPLE ZETA VALUES, AND THE SERIES OF HJORTNAES AND COMTET

Transcription

1 POLYLOGARITHMS, MULTIPLE ZETA VALUES, AND THE SERIES OF HJORTNAES AND COMTET Noes by Tim Jameso December 9 edied by Graham Jameso, wih refereces added, March 4 Irodcio The polylogarihm fcio is defied for < ad ay real s by Li s s. For pariclar vales of s, i ca he be eeded aalyically o a wider rage of real or comple. I is simlaeosly a power series i ad a Dirichle series i s. The firs pblished sdy of he fcio was by A. Joqière i 889, ad i is someimes called Joqière s fcio. Noe ha polylogarihmic fcios meas somehig qie differe! These oes are a smmary of some resls o polylogarihms ha I have see saed i varios places, sch as Wikipedia ad [BBC], largely wih proofs ha I have worked o for myself. The mehods are mosly elemeary. We resric o ieger s k, ad sally real. However, i places he reader is ivied o accep ha formlae esablished for real also apply for comple ; I ca spply a rigoros jsificaio if pressed. We sar wih a few immediae facs. Firsly, Li, eedig meromorphically o he whole plae, ad Li log, eedig o all real < or comple ecldig real. For <, or wih k >, we have Li k + Li k k Li k. For k >, Li k ζk, 3 I pariclar, Li ζ ad Li ζ3. Li k k ζk. 4

2 For k ad, we have from he series Li k ζk ad Li k. Frher, Li k Li k also, Li k ad Li k Li k d. 5 Termwise iegraio of he series is jsified by iform covergece for <, ad he by coiiy of Li k a ad. We will work o varios special vales of Li ad Li 3, which were fod by Lade as log ago as 78. I pariclar, we evalae Li 3 φ, where φ + 5 is he golde raio. This is sed o prove Hjoraes series epressio 35 for ζ3. My proof of Come s correspodig sm for ζ4 ivolves qaiies like Li 4 e πi/3. The dilogarihm The dilogarihm Li is someimes called Spece s fcio, i ribe o a pioeerig sdy of i by W. Spece i 89. Beware of he fac ha some comper algebra sysems deoe Li by dilog. By ermwise iegraio of he series log we obai he ideiy log Li d, which we ow se o defie Li for all. Noe ha ad for >, Li ζ log d log d log log + Li d d. Clearly, Li log /, ad hece Li is sricly icreasig for all <. For < <, we have Li ζ + ζ + log d log d.

3 Replacig by, his says Li ζ log d [ log log ] + log log Li, so we have he followig ideiy, kow as Eler s reflecio formla: log d Li + Li ζ log log. 7 I pariclar, sig he well-kow ideiy ζ π /, we have We ca rewrie 8 as a epressio for ζ: Li ζ log π log. 8 ζ log +. Takig log as kow, his ca be regarded as a series for ζ ha coverges mch more rapidly ha /. where We ow give wo frher ideiies for Li. Firsly, for >, we have Li Li + F ζ + F F log + Combied wih he same saeme for /, his gives d. Li + Li ζ + F + F. B so we have F / / log + d log + log F log, Li + Li ζ log. 9 d 3

4 So for >, we have he followig series epressio for Li i powers of : Li log ζ +, ad we ca dedce Li log ζ + r, where < r /. For < <, we have also Li log d log / d log log + v v Li log. d dv log Applyig his also o /, we have he followig ideiy for all > : Li + Li log. Usig, 7, 9 ad, we ca redce he compaio of Li for ay < o compaio of vales of a mos a few vales of Li y wih y. Now cosider φ. Noe ha he fdameal eqaio saisified by φ may be wrie: φ φ +, φ + φ, φ + φ. Pig φ i,, 7 respecively, we obai hree liear eqaios relaig Li φ, Li φ, ad Li φ : Sbracig from gives Now sbracig 4 from 3, we have Li φ + Li φ Li φ, Li φ + Li φ log φ. Li φ + Li φ ζ log φ. 3 Li φ 3 Li φ log φ. 4 5 Li φ ζ 5 log φ, 4

5 i.e. Now 3 or 4 gives Li φ 5 ζ log φ π 5 log φ. 5 while gives Li φ 3 5 ζ log φ π log φ, Li φ 5 ζ + log φ π 5 + log φ, 7 hece fially 9 gives Li φ 3 5 ζ log φ π log φ. 8 If we iclde Li, he we have fod eigh special vales of Li for real. I have see i saed ha hese are he oly oes kow o be epressible i his kid of way. We ow briefly cosider comple argmes. Sbsiio i he series gives Li i ζ + ig, Li 8 i ζ ig, 8 where G is Caala s cosa. Assmig valid for comple argmes, ake + i, so ha i ad + i. We obai + i Li 8 ζ + ig For e iθ, we have RLi e iθ 5π 9 8 log + i cos θ πi log + 4 G π 8 log. π B θ, π where B is he Berolli polyomial B + ad B B {}. Also, ILi e iθ which is called he Clase fcio i [BBC]. si θ, There are may frher formlae ivolvig Li : see, for eample, Wikipedia, [Lew] ad [Ma]. Oe is Abel s ideiy, which eqaes a combiaio of five Li vales o log log y. 5

6 The rilogarihm For <, we have by 5 Li 3 Li d. 9 Sice Li has bee defied by for all, we ca se 9 o defie Li 3 for all sch. So we have Li 3 Li d log d d log d d log log d, valid regardless of he sig of. For < <, sbsiio of ow gives Li 3 log log d + Li log, which ca also be derived direcly from he series epressio. Frher, for >, Li 3 log log d log + log d log log + d + Li log. I pariclar, ζ3 Li 3 log log d. 3 Iegraig by pars i, we obai for, I pariclar, Li 3 [ log log ] log d + Li log, log d + Li log + log log. 4 log d ζ3.

7 Usig 9, we ca derive a correspodig saeme for Li 3. [This has bee added o Tim s origial versio, i which he followig ideiy was dedced from below.] For >, wrie By, Wih replaced by, his says F log log + d. Li 3 F F + Li log. Li 3 F F Li log. Combiig hese epressios ad sig 9, we have Li 3 Li 3 F F log [ Li + Li ] Sbsiig /, we have ad we coclde ha F F F + ζ log + log3. / / log log + 3 log3 + F 3 log3 + F, d log [log + log ] d Li 3 Li 3 ζ log + log3. 5 I follows ha for >, Li 3 log3 ζ log r, where < r. We ow prove aoher ideiy relaig Li 3, Li 3 ad Li 3. I agai goes back o Lade arod 78. [The proof give here was fod amog Tim s hadwrie papers; i replaces a more complicaed proof i Tim s yped versio.] Le < < ad S Li 3 + Li 3 + Li 3. We epress everyhig i erms of iegrals o [, ]. The deails are kep simpler by epressig he iegrads i erms of Li raher ha log erms. Firsly, sig 9 ad 7, we 7

8 have Secodly, Li Li 3 d Li ζ3 d ζ3 [ζ log log Li ] d ζ3 + ζ log + Li 3 Thirdly, wih obvios sbsiios, Combiig he hree erms, we have where G Li log log d Li 3 / / S ζ3 + ζ log + log log + Li d + Li d Li v dv v Li d Li d. G d, Li d. + [Li + Li ]. Now by, we have G log log log + d [ ] d log log log3, d. so G d log3 log log. So for < <, we have he ideiy Li 3 + Li 3 + Li 3 ζ3 + ζ log + log3 log log. 8

9 Oe ca dedce 5 by wriig his wih i place of, akig he differece ad pig for. Pig i gives so ha Li 3 Li 3 + ζ3 + ζ log log log + log3 3 4 ζ3 + ζ3 ζ log + log3 log3 7 4 ζ3 ζ log + 3 log3, Li 3 7 π ζ3 log + 8 log3. 7 This gives a series covergig rapidly o ζ3. A more direc proof of 7 is give i [Mel]. Pig φ ad sig he ideiies relaig φ, φ ad φ, we ge Li 3 φ + Li 3 φ + Li 3 φ ζ3 ζ log φ log φ log φ + log3 φ ζ3 ζ log φ + log 3 φ log3 φ ζ3 ζ log φ + 5 log3 φ. B by, Li 3 φ + Li 3 φ 4 Li 3φ, hece Li 3 φ 4 π ζ3 log φ log3 φ. 8 I seems ha here are o kow sch special vales ivolvig φ for higher k, alhogh here are impora relaios bewee varios vales. This is o do wih polylogarihm ladders irodced by Leoard Lewi, which are impora i K-heory ad algebraic geomery, ad ca be sed i cojcio wih he BBP algorihm for compig varios cosas. A proof of ζ π / ad some power series relaed o arcsi For <, he sbsio gives d + d + arca arcsi. 9 Sice also d d arcsi arcsi, 9

10 we ow have arcsi arcsi y y dy y d dy y + y y dy d y + y [ log y + y ] y y d log + d. 3 I pariclar, o show ha ζ π /, we have π 4 arcsi log d log d 3 ζ. + Aleraively, we ca eqae he iegral o Li Li by. This proof has similariies wih he oe give by Nick Lord [Lo]. [Tim s proof has ow appeared i Mah. Gazee [Jam].] To derive a power series from 9, observe ha for < ad, ad d So we have for <, + + π/ cos θ dθ arcsi. 3 This ca also be see as a case of he hypergeomeric series F, ; 3 ;. For eample, he case gives π 3 3.

11 Formlae like his form he basis for varios programs which se a spigo algorihm o calclae π, wih ime roghly proporioal o he sqare of he mber of digis reqired e.g. oe by Wier ad Flimmekamp. Now by iegraig, or by similar reasoig from 3, we have eqivalely for <, The case gives arcsi, 3 arcsi π Eiher by he ideiy sih y i si iy, or by similar reasoig wih replaced by, we have also Hjoraes series for ζ3 sih. 34 We give a proof of his ideiy sig 34 ad he vales of Li φ ad Li 3 φ. The coecio wih φ arises from he fac ha sih log φ. Le The by 34, S S 3. 4 sih d sih log φ log φ log φ log φ 4 sih cosh d sih coh d e + e d v e v + dv e v v ev + e v dv

12 log φ 4 3 log3 φ log3 φ log3 φ + ζ3 v + log φ φ φ e v dv v e v dv log d log φ d log d log3 φ. v log Now sbsiig from 4, ad he he vales of Li φ ad Li 3 φ from 5 ad 8, we have S ζ3 Li 3 φ + Li φ log φ + log φ log φ log3 φ ζ3 Li 3 φ 4Li φ log φ + 4 log φ log φ log3 φ ζ3 Li 3 φ 4Li φ log φ 8 3 log3 φ ζ3 Li 3 φ 4 5 ζ log φ log φ 8 3 log3 φ ζ3 Li 3 φ 8 5 ζ log φ log3 φ ζ3 4 5 ζ3 4 5 ζ log φ + 3 log3 φ 8 5 ζ log φ log3 φ 5 ζ3, by a ea cacellaio. So ζ This series was fod by Hjoraes i 953 [Hj]. I has someimes bee wrogly aribed o Apéry, becase i was sed i his proof ha ζ3 is irraioal [Ap]. However, i is o sed i he simpler proof of Apéry s heorem by Bekers [Be]. A proof of 35 ha does o ivolve polylogarihms is olied i [VDP]; i is reprodced i [BB, p ]. A geeralized versio is proved i [AG]. Some series ivolvig H We deoe he harmoic sm by H r. For <, we have easily r log j j k k k H.

13 Similarly, i he prodc log j j /j k k /k, he coefficie of, for, is so j j j j log j + H j, Sice H H + also for wih H, we dedce Iegraio of 3 gives hece Iegraig agai, we obai H log + Li. H H H. 3 log d, 37 log d + Li 3 H 3 log log log d d d d log d, 38 which we ll se wih e πi/3 laer. Some mliple zea vales We defie he mliple zea vale fcio as ζs,..., s k s s k,..., k < < < k Aoyigly his is someimes wrie ζs k,..., s isead! Here I will oly cosider ζs,. We have ζs, m m s m+ m s m k. m +. 3

14 Also, reversig he order i he firs epressio, we have i pariclar, ζs, ζ, m m s. H. 39 Mos obviosly, we have for Rs > ζs, s m s m, m< m s m, m m,m s s ζs ζs. 4 The case s gives ζ, π4 3 π4 9 3 ζ Takig i 37 ad applyig 4, we have ζ, ζ3. H log d a ideiy already kow o Eler. Laer I fod he followig direc proof avoidig iegrals. [Noe added by Graham Jameso: his mehod ca be see, for eample, i [BBr, p. 7 8], where i is aribed o R. Seiberg]. Sice we have ζ, ζ, m m m mm +. m + m + mm +. 4

15 Now ad by cacellaio so mm + m m m + m m + + m m H m, ζ, m H m m ζ3 + m ζ3 + ζ,. H m m A similar mehod delivers a ideiy for ζ, 3, which will be sed for Come s series. [Tim s origial mehod was by maiplaio of he series for Li ; he followig is slighly shorer ad more like he previos proof.] ζ, 3 m m m + 3 m + mm + ad mm + m m + m + m m + m m + m 3 m + m m +, so, as before, ζ, 3 m ζ4 + H m m 3 m m m m + H m m 3 ζ, ζ4 + ζ, 3 ζ,. Hece, by 4, ζ, 3 ζ4 ζ, ζ

16 Come s series for ζ4 For his, [BBC] gives a referece o Come s book [Com], b I have o see his or ay oher proof. Here is my proof compleed Perhaps i cold be sbsaially simplified! Le From 33, we have so S y y z z π π π y S 3 4. y 4 arcsi z dz dy z arcsi z dy z y dz arcsi z log z dz 4 arcsi z dz z si θ θ log si θ cos θ dθ z si θ θ co θ log si θ dθ θ d dθ log si θ dθ 4 [ θ log si θ ] π π + 4 θ log si θ dθ π 8 θ log si θ dθ. Now epad as follows maybe somehig circlar is happeig here?: e iθ e iθ log si θ log θ π i + log e iθ. i so π S 8 θ θ π + i θ π log e iθ + log e iθ dθ 8I + I + 8I 3, where I π θ θ π dθ

17 π I i i i i i i ad by 3 π θ 3 πθ + π 4 θ dθ π π π4 4 4, θ θ π log e iθ dθ π θ πθ e iθ dθ [ θ πθ ] π e iθ π θ π e iθ i i dθ π [ 3 π e πi/3 i θ π ] π e iθ + i π ie πi/3 π 8 3 π e πi/3 4 + π π i 3 e πi/3 π 4 e πi/3 + π 8 π 3 e πi/3 + π 4 + i 4 3 e πi/3 πi 4 3 e πi/3 πi e πi/ , e iθ i dθ [ e iθ i 3 ] π I 3 π θ log e iθ dθ H H H H π θe iθ dθ [ ] π π θ e iθ e iθ i i dθ [ ] π πi e iθ e πi/3 i πi e πi/3 + 4 e πi/3 H πi e πi/3 + 3 e πi/3 3. 7

18 Hece S π π 9 e πi/3 + πi 3 3 e πi/3 πi e πi/ H 4πi 3 e πi/ e πi/3 4 3 π4 πiζ3 + 4ζ4 4ζ, π 9 + πi 3 4 e πi/ πi H e πi/3. 3 By 4, 4ζ, 3 ζ4, so he firs for erms eqae o π4 + 3ζ4 πiζ π 4 πiζ π 4 πiζ π4 πiζ3 53π πiζ3. So or formla becomes S 53π πiζ3 + 4π 9 + πi 3 4 e πi/ The imagiary par of S is zero, so we have proved 4π π πζ3 si π π cos π si 4 3 4π π + H cos π si πi H e πi/3. 3 Icideally, I cao evalae he separae coribios of ay of hese erms. The real erms give S 53π π π cos π 3 + 4π π H si π cos 3 3 π si π cos The wo sms here will r o o give a oal of ζ4. We ow evalae he firs sm. The coribio of each of he hree erms is of he form CB k, by he Forier series for 8

19 he periodified Berolli polyomial where B k B k {} k! e πi πi k k! πi k F k, + k F k,, F s, α Li s e πiα e πiα s. is he Lerch zea fcio. Spliig accordig o pariy gives ad cos π k The Berolli polyomials p o B 4 are k πk k! B k, si π k πk+ k+ k +! B k+. B, B, B +, B , B , so we fid cosπ/3 π B π 3, siπ/3 3 π3 3! B 3 π π π3 3 3, cosπ/3 4 π4 4! B 4 9

20 π π π So he firs sm i 43 is 4π 9 cosπ/3 + π 3 siπ/3 4 cosπ/ π 9 π + π 3 π π π π π π π Sice ad , we have proved S We ow defie By 38, we have so ha 44 becomes 9π π siπ/3 H 3 f f H log log log + cosπ/ d. 45 H 3, S 9π Rfeπi/3. 4 The obvios iegraio by pars gives a reflecio relaio bewee f ad f : f [ log log ] log log d log log log log + d log log log log + f d.

21 By 39 ad 4, f ζ, 3 ζ4. Also, so log log d. log log d f, f + f ζ4 + log log. 47 Sice fz fz his occrs whe f is aalyic ad real o he real ais, he case e πi/3 gives s he vale we wa: Iserig his io 4 gives so fially we have Come s series Rfe πi/3 fe πi/3 + fe πi/3 fe πi/3 + f e πi/3 ζ4 + πi 3 π 4 ζ4 + 3 π4 S π πi π π ζ4, ζ We have had wo occreces of ζ, 3 ζ4 i or calclaio of S. These did o simply 4 cacel o: he firs occrrece was 4ζ, 3, while he secod was f ζ, 3. Refereces added by Graham Jameso [AG] G. Almkvis ad A. Graville, Borwei ad Bradley s Apéry-like formla for ζ4 + 3, Eperimeal Mah , [Ap] R. Apéry, Irraioalié de ζ e ζ3, Asérisqe 979, 3. [Be] F. Bekers, A oe o he irraioaliy of ζ ad ζ3, Bll. Lodo Mah. Soc. 979, 8 7. [BB] Joaha M. Borwei ad Peer B. Borwei, Pi ad he AGM, Joh Wiley 987.

22 [BBr] [BBC] Joaha M. Borwei ad David M. Bradley, Thiry-wo Goldbach variaios, I. J. Nmber Theory,, 5 3. Joaha M. Borwei, David M. Bradley ad Richard E. Cradall, Compaioal sraegies for he Riema zea fcio, J. Comp. Appl. Mah., [Com] Lois Come, Advaced Combiaorics, Reidel, Dordrech 974. [Hj] [Jam] M. M. Hjoraes, Overførig av rekke k /k3 il e besem iegral, Proc. h Cog. Scad. Mah. Ld, 953, Ld 954. Tim Jameso, Aoher proof ha ζ π / via doble iegraio, Mah. Gazee 97 3, [Lew] Leoard Lewi, Polylogarihms ad associaed fcios, Norh Hollad 98. [Lo] N. Lord, Ye aoher proof ha / π /, Mah. Gazee 8, [Ma] [Mel] [VDP] [Zag] Leoard Maimo, The dilogarihm fcio for comple argme, Proc. Royal Soc. Lodo 459 3, Joh Melville, A simple series represeaio for Apéry s cosa, Mah. Gazee 97 3, Alfred va der Poore, A proof ha Eler missed: Apéry s proof of he irraioaliy of ζ3, Mah. Ielligecer 979, Do Zagier, The dilogarihm fcio, mahs.dr.ac.k/dma hg/dilog.pdf pdaed wih mior correcios, November 8