Polylogarithms and Riemann's Function

Home , Analytic continuation, Polylogarithm

PHYSICAL REVIEW E VOLUME 56, NUMBER 4 OCTOBER 1997

Polylogarithms and Riemann’s ␨ function

M. Howard Lee Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602 ͑Received 12 March 1997͒ Riemann’s ␨ function has been important in statistical mechanics for many years, especially for the understanding of Bose-Einstein condensation. Polylogarithms can yield values of Riemann’s ␨ function in a special limit. Recently these polylogarithm functions have uniﬁed the statistical mechanics of ideal gases. Our par- ticular concern is obtaining the values of Riemann’s ␨ function of negative order suggested by a physical application of polylogs. We ﬁnd that there is an elementary way of obtaining them, which also provides an insight into the nature of the values of Riemann’s ␨ function. It relies on two properties of polylogs—the recurrence and duplication relations. The relevance of the limit process in the statistical thermodynamics is described. ͓S1063-651X͑97͒01510-9͔ PACS number͑s͒: 05.90.ϩm, 02.90.ϩp

I. INTRODUCTION standard methods. It also lends an interesting insight into the nature of the values of Riemann’s ␨ function. Riemann’s ␨ function perhaps first appeared in statistical mechanics in 1900 in Planck’s theory of the blackbody ra- II. POLYLOGS AND THEIR PROPERTIES diation and then in 1912 in Debye’s theory of the specific heats of solids ͓1͔. Subsequently, this function has played an To show their relationship to Riemann’s ␨ function, we important role in the statistical theory of the ideal Bose gas, shall introduce a convenient integral representation for poly- especially for the understanding of Bose-Einstein condensa- logs Lis(z) of complex numbers s and z ͓6͔, defined by tion ͑BEC͓͒2͔. More recently, this function together with the z 1 dt Mellin transform has become a powerful tool for the analysis sϪ1 Lis͑z͒ϭ ͓log͑1/t͔͒ ,Imtϭ0, ͑1͒ of the thermodynamic potentials ͓3,4͔. It would be no sur- ⌫͑s͒ ͵0 1Ϫzt prise to find fruitful applications of Riemann’s ␨ function in other areas of today’s theoretical physics ͓5͔. whenever this integral converges, i.e., ResϾ0, RezϽ1, and Recently it was found that the statistical thermodynamics elsewhere by analytic continuation. It is understood that of ideal gases can be given a unified picture through polylogs log(1/t) has its principal value. Evidently there is a branch defined in terms of the fugacity z and dimensions d ͓6͔. cut from zϭ1toϱ. Also if sϭ2, the standard expression for There is richness that this unified picture reveals, such as the the dilog is recovered ͓10͔. The above equation ͑1͒ bears anomalous physics in null dimension ͓7͔, the Fermi-Bose resemblance to an integral representation for Riemann’s ␨ reflection in dу3 ͓8͔, and the Fermi-Bose equivalence in d function ␨(s) of a complex number s ͓12͔. ϭ2 ͓9͔. These physical results are consequences of some We shall now state a few useful properties for our pur- special properties of polylogs. It has been long known that a poses which follow directly from Eq. ͑1͒. polylog of integral order becomes Riemann’s ␨ function of ͑a͒ Lis(zϭ1)ϭ␨(s). the same order when its argument attains unity ͓10͔. Thus ͑b͒ If sϭnϭ2,3,4,..., Lin(z) are classical polylogs, Riemann’s ␨ function can enter into the unified theory of the known, respectively, as the dilog, trilog, quadrilog, etc. statistical thermodynamics via the polylogs quite naturally. ͑Throughout this work we shall reserve n to denote real in- Interestingly, we find that this formulation shows another tegers, both positive and negative.͒ Ϫ1 way of evaluating Riemann’s ␨ function, which is presented ͑c͒ limz 0z Lis(z)ϭ1. There is a trivial fixed point at in this work. the origin. → The classical theory of polylogs begins with Euler’s dilog ͑d͒ Recurrence relation and Landen’s trilog, and extends to higher order polylogs such as the quadrilog. In physical applications the order of a d z Li ͑z͒ϭLi ͑z͒. polylog is related to physical dimensions d. Thus polylogs of dz sϩ1 s integral order lower than the dilog have been conceived, such as the nil-log and the monolog for the physics in d ͑e͒ Duplication relation ϭ0 and 2 ͓6͔. There have been suggestions that negative 1Ϫs 2 dimensions can be of theoretical interest ͓11͔. They require Lis͑z͒ϩLis͑Ϫz͒ϭ2 Lis͑z ͒. the polylogs of still lower orders than the nil-log, departing from the direction of the classical theory of polylogs. What If sϭnϭ2 and 3, one recovers Euler’s formula for the dilog we find is that in these circumstances there exists even a and Landen’s for the trilog, respectively ͓10͔. simpler relationship between the polylogs and Riemann’s ␨ ͑f͒ ͉Lis(zϭϪ1)͉Ͻϱ since the function is analytic at zϭ function. We can use this relationship to evaluate Riemann’s Ϫ1. ␨ function very simply, perhaps more simply than by most ͑g͒ If ͉z͉Ͻ1 and sϭnу1 ͑also ͉z͉ϭ1 included if nу2͒,

ϱ 2 3 4 7 LiϪ6͑z͒ϭz͑1ϩz͒͑1ϩ56zϩ246z ϩ56z ϩz ͒/͑1Ϫz͒ ϭ0, k n Lin͑z͒ϭ ͚ z /k . kϭ1 2 3 LiϪ7͑z͒ϭz͑1ϩ120zϩ1191z ϩ2416z ͑h͒ ϩ1191z4ϩ120z5ϩz6͒/͑1Ϫz͒8 ␨͑n͒ if Ͼ1 ϭ17/16, Li ͑1͒ϭ n ͭ ϱ if nр1. 2 3 4 LiϪ8͑z͒ϭz͑1ϩz͒͑1ϩ246zϩ4047z ϩ11572z ϩ4047z The property ͑h͒ is the basis for the existence of BEC if 5 6 9 dу3 and for the absence if dр2 ͓1͔. It should be noted that ϩ246z ϩz ͒/͑1Ϫz͒ if sϭnϾ1, there is thus no difference in Lin(1) between the ϭ0. one given in ͑h͒ and the other given in ͑a͒.IfnϽ1, however, by Lin(1) we shall mean simply the integral ͑1͒, which then The numerical values given on the right-hand side of Eq. becomes undefined as shown above because of the singular- ͑4͒ are the polylogs evaluated at zϭϪ1. There seems to be ity in Lin(z)atzϭ1. However, one can obtain Lis(1), sϭ no general closed form expression recognizable from these Ϫm, mϾ0, by analytic continuation in the manner of ␨(s lower order polylogs. One can, however, obtain the polylog ϭϪmϽ0). To denote this latter case— and to avoid the of any desired lower order. Thus our results may perhaps be possible confusion we shall use ␨(s) in place of Lis(1) considered all but complete. These polylogs of order nϭ0, whenever analytic continuation is implied. Since zϭϪ1is Ϫ1,Ϫ2,... have shed log character. We might call them not a singular point of Lis(z), this kind of distinction need polypseudologs, e.g., LiϪ2(z) the dipseudolog. This distinc- not be made for Lis(Ϫ1). tion will be found useful. We can easily verify that these polypseudologs satisfy III. POLYLOGS OF NEGATIVE INTEGRAL ORDER: several important functional properties of polylogs stated in POLYPSEUDOLOGS Sec. II ͑i͒ The recurrence relation ͑d͒ is satisfied. ͑ii͒ The duplication relation ͑e͒ is satisfied. ͑iii͒ They are finite at z If zϭ1 in the duplication relation ͑e͒, we obtain together ϭϪ1. See ͑f͒. ͑iv͒ If ͉z͉Ͻ1, the expansions of polyp- with ͑a͒, seudologs are also given by ͑g͒. Hence the condition sϭn

1Ϫs Ϫ1 у1 given therein may be relaxed to sϭn. In addition, the ␨͑s͒ϭ͑2 Ϫ1͒ Lis͑Ϫ1͒. ͑2͒ polypseudologs LiϪn(z), nϾ0 show the following properties. ͑v͒ They have a pole of order nϩ1atzϭ1. See ͑h͒. ͑vi͒ Thus it is possible to evaluate ␨(s)ifLi(z) can be given at s They are factorable by z ͓see ͑c͔͒ and also by (zϩ1) if n is zϭϪ1. If sϭnϭ2, for example, we recover the familiar an even number. ͑vii͒ The numerical coefficients add up to result ␨(2)ϭ␲2/6, given that Li (Ϫ1)ϭϪ␲2/12. See the 2 n!. ͑viii͒ Evidently Li (Ϫ1)ϭ0ifnϭ2,4,6,..., and Appendix for Li (Ϫ1), nϭ2,4,6,...,obtained purely from Ϫn n some numbers relatable to Bernoulli’s numbers if n the inversion property of classical polylogs. Hence it is el- ϭ1,3,5,... .These results may be used to obtain the values ementary to evaluate ␨(sϭn)ifnϭ2,4,6,... .Toevaluate of Riemann’s ␨ function ␨(sϭn), nϽ1 to any desired lower ␨(sϭn) when nϽ1, we need to know first the form of Li (z) s order. for sϭnϭ1,0,Ϫ1,Ϫ2, etc. These lower order polylogs can be obtained from any one of the higher order ones through repeated differentiation. See the recurrence relation ͑d͒. IV. REIMANN’S ␨ FUNCTION Hence it is sufficient to know one polylog function in closed We shall first consider one or two special cases. If s 1 form. By setting sϭnϭ1 in Eq. ͑1͒, we immediately obtain in Eq. ͑2͒, → the form for the monolog, 1Ϫs Ϫ1 ␨͑s 1͒ϭ lim͑2 Ϫ1͒ Li1͑Ϫ1͒. ͑5͒ Li1͑z͒ϭϪlog͑1Ϫz͒, zÞ1. ͑3͒ → s 1 → 1Ϫs By applying the recurrence relation to the above form of the From Eq. ͑3͒,Li1(Ϫ1)ϭϪlog 2. Also, (2 Ϫ1) Ϫ(s monolog and repeating it over and again, we have obtained Ϫ1)log2 as s 1. Hence ␨(s 1)ϭ1/(sϪ1), s 1,→ a well- → 1 → → the lower order polylogs given below to order nϭϪ8: known result. If sϭ 2 in Eq. ͑2͒,

Li0͑z͒ϭz/͑1Ϫz͒ϭϪ1/2, ͑4͒ 1 ␨͑ 2 ͒ϭ͑&ϩ1͒Li1/2͑Ϫ1͒ϭϪ1.460..., ͑6͒ 2 LiϪ1͑z͒ϭz/͑1Ϫz͒ ϭϪ1/4, where by ͑g͒ Li1/2(Ϫ1)ϭϪ1ϩ1/&Ϫ1/)ϩ ...ϭ Ϫ0.6048..., aslow but converging series 13,14 . This re- 3 ͓ ͔ LiϪ2͑z͒ϭz͑1ϩz͒/͑1Ϫz͒ ϭ0, sults, Eq. ͑6͒, cannot be obtained by the reflection formula of Riemann ͓12͔. 2 4 LiϪ3͑z͒ϭz͑1ϩ4zϩz ͒/͑1Ϫz͒ ϭ1/8, From Eq. ͑2͒ we see that the coefficient standing before Lis(Ϫ1) is always finite if s is a negative real number. 2 5 LiϪ4͑z͒ϭz͑1ϩz͒͑1ϩ10zϩz ͒/͑1Ϫz͒ ϭ0, Hence from Eq. ͑4͒ we obtain at once that

2 3 4 6 LiϪ5͑z͒ϭz͑1ϩ26zϩ66z ϩ26z ϩz ͒/͑1Ϫz͒ ϭϪ1/4, ␨͑Ϫ2m͒ϭ0, mϭ1,2,... . ͑7͒ 56 POLYLOGARITHMS AND RIEMANN’S ␨ FUNCTION 3911

For the others we can evaluate one by one. By using Eq. ͑4͒ spectively, for the massive Bose and Fermi gases, where z 1 1 in Eq. ͑2͒, we obtain at once ␨(0)ϭϪ 2 , ␨(Ϫ1)ϭϪ 12 , denotes the fugacity and d the number of physical dimen- 1 1 1 ␨(Ϫ3)ϭ 120 , ␨(Ϫ5)ϭϪ 252 , ␨(Ϫ7)ϭ 240 , etc. According sions. One finds in the Bose gas in lower dimensions a con- to Titchmarsh’s book ͓15͔, ␨(Ϫ2m)ϭ0 and ␨(1Ϫ2m)ϭ dition equivalent to taking the d/2 1 limit first. If z 1, m → → (Ϫ1) Bm /(2m), mϭ1,2,..., where Bm are the Bernoulli then the divergence of Lid/2(zϭ1), dр2, implies that BEC 1 1 1 1 5 numbers, e.g., B1ϭ 6 , B2ϭ 30 , B3ϭ 42 , B4ϭ 30 , B5ϭ 66 , etc. does not occur in lower dimensions. But for the Fermi gas Insofar as we have determined, we recover the values of zϭ1 is not a singular point of ϪLid/2(Ϫz). Hence the two Riemann’s ␨ function very simply. limits may be exchanged harmlessly, indicating the regular- Observe the remarkable difference between the two ness of the Fermi thermodynamics. In higher dimensions, classes of numbers, ␨(n), nϭϪ1,Ϫ3,Ϫ5,... and n where there are no divergent singular points, these gases at ϭ2,4,6,... . Theformer are rational numbers, whereas the zϭ1 have a special significance, being a source of reflection latter are not, containing even powers of ␲. This difference symmetry. At this point the chemical potentials vanish and can be traced to polypseudologs having lost log character but both gases ͑excluding kinematical factors͒ have the same polylogs retaining it. Also see the Appendix for the source of universal entropy, a constant made up of Riemann’s ␨ func- ␲. tion. When the chemical potential of a gas vanishes, it implies of course that the gas has no control over the flow of V. CONCLUDING REMARKS particles when in contact with a particle reservoir. These fundamental properties of the ideal gases are manifested in Riemann’s ␨ function of a negative number is not ordi- Riemann’s ␨ function, reached through the special limits of nary, being defined only through analytic continuation. This polylogs. condition no doubt limits the number of possible avenues of approach to it. As far as we know, there is only one direct ACKNOWLEDGMENTS method of obtaining the values of ␨(nϽ0), given in the literature. It is by applying Cauchy’s theorem of residues to Riemann’s integral representation of Hurwitz’s generalized ␨ I wish to thank Professor L. Lewin for his critical reading function ͓12,15͔. Although not as general, our method of of this work and Professor R. L. Anderson for his helpful solution by comparison is elementary. It relies on two basic suggestions. I also wish to thank Professor N. E. Frankel for properties of polylogs—the recurrence and duplication rela- drawing my attention to Euler’s work ͑Ref. ͓16͔͒. This work tions. Polylogs have two independent parameters and special has been supported in part by NATO ͑Grant No. CRG values of polylogs are obtained in certain limit processes 921268͒. peculiar to polylogs, one of which corresponds to Riemann’s …␨ function. APPENDIX : EVALUATION OF Lin„؊1 We have alluded that if siрs0ϭ1, the order of two pos- FROM THE INVERSION RELATION sible limiting processes in polylogs ͑s s and z 1͒ may i There are only two general relations known for polylogs, not be exchanged. We can illustrate the→ difference→ through the duplication and inversion formulas, the former given be- the following examples. Consider Li (z) taken in two differ- s fore ͓see ͑e͔͒ and the latter to be given below. It is possible to ent orders of s and z, when siϭ1 and 0: obtain the values of Lin(Ϫ1), nϭ2,4,... from the inversion formula, hence purely from properties of classical poly- lim limLis͑z͒ϭ␨͑s 1͒ϭ1/͑sϪ1͒, s 1 ͑8a͒ s 1z 1 → → logs. To our knowledge, these values are ordinarily obtained → → from Riemann’s ␨ function through ͑a͒ and Eq. ͑2͒, hence but indirectly. See p. 188 of Lewin’s book ͓10͔. The inversion relation is known—it is not difficult to es- lim limLis͑z͒ϭϪlog͑1Ϫz͒, z 1. ͑8b͒ tablish it using the integral representation ͑1͒. Instead of z 1s 1 → → → writing down a general form ͑see Lewin’s book ͓10͔ or Ref. Also, ͓6͔͒, it will suffice for our purpose to express it as follows:

1 n n nϩ1 lim limLis͑z͒ϭ␨͑0͒ϭϪ 2 , ͑9a͒ Linϩ1͑Ϫ1/z͒ϭ͑Ϫ1͒ Linϩ1͑Ϫz͒ϩ͑Ϫ1͒ a /⌫͑nϩ2͒ s 0z 1 → → ϩKnϩ1͑a͒, nϭ1,2,... ͑A1͒ but where aϭlogz and Knϩ1 is a polynomial in a in which lim limLi ͑z͒ϭ1/͑1Ϫz͒, z 1. ͑9b͒ s Lin(Ϫ1), LinϪ2(Ϫ1), . . . are its coefficients. The first few z 1s 0 → → → are listed below: In both cases we obtain quite different results depending on the order of limits taken. This difference extends to all lower K2ϭ2Li2͑Ϫ1͒, ͑A2a͒ order polylogs ͑i.e., polypseudologs͒ since Lis(z), sϭnϭ Ϫ1,Ϫ2,..., all have poles of order Ϫnϩ1atzϭ1, K3ϭϪ2aLi2͑Ϫ1͒, ͑A2b͒ whereas ␨(n) has been found finite. In the s-z plane, there is 2 a line of singularity for sϭnр1atzϭ1. K4ϭ2͓Li4͑Ϫ1͒ϩ͑a /2͒Li2͑Ϫ1͔͒, ͑A2c͒ This difference in the limit process is important to the 3 statistical thermodynamics, formulated in ϮLid/2(Ϯz), re- K5ϭϪ2͓aLi4͑Ϫ1͒ϩ͑a /3!͒Li2͑Ϫ1͔͒, ͑A2d͒ 3912 M. HOWARD LEE 56

2 4 K6ϭ2͓Li6͑Ϫ1͒ϩ͑a /2͒Li4͑Ϫ1͒ϩ͑a /4!͒Li2͑Ϫ1͔͒, In the classical theory of polylogs the values of Li2n ͑A2e͒ (Ϫ1), nϭ1,2,...,arealso obtained by combining the inversion relation with the duplication relation. See pp. 172 3 5 K7ϭϪ2͓aLi6͑Ϫ1͒ϩ͑a /3!͒Li4͑Ϫ1͒ϩ͑a /5!͒Li2͑Ϫ1͔͒, and 173 of Lewin’s book ͓10͔. As a result, the ease and ͑A2f͒ transparency of our approach are not easily seen, which is based on the inversion property alone. etc. Observe that if zϭϪ1 and nϭ2,4,6,... i.e., even ͑ Perhaps the simplest method of obtaining the values of numbers͒, Eq. ͑A1͒ becomes ␨(2n), nϭ1,2,... isthe following one, evidently due to nϩ1 Euler ͓16͔. Since the zeros of sin x are xϭϮk␲, a /⌫͑nϩ2͒ϩKnϩ1͑a͒ϭ0, ͑A3͒ kϭ0,1,2,... onecanwrite 3 where now a(zϭϪ1)ϭi␲.Ifnϭ2, we have a /3!ϩK3 3 2 ϭa /3!Ϫ2aLi2(Ϫ1)ϭ0. Hence Li2(Ϫ1)ϭϪ␲ /12 and 2 ϱ ϱ x 2 ␨(2)ϭ␲ /6 by Eq. ͑2͒. m 2mϩ1 5 sinxϭ ͚ ͑Ϫ͒ x /͑2mϩ1͒!ϭx ͟ 1Ϫ . If nϭ4, we have a /5!ϩK5ϭ0. Hence using Eq. ͑A2d͒ mϭ0 kϭ1 ͫ ͩ k␲ͪ ͬ 4 4 we find that Li4(Ϫ1)ϭϪ7␲ /6!, recovering ␨(4)ϭ␲ /90. ͑A4͒ 6 Similarly if nϭ6, we obtain Li6(Ϫ1)ϭϪ31␲ /7!6, recovering ␨(6)ϭ␲6/945. In this way it is possible to obtain any 2mϩ1 desired values of Lin(Ϫ1) and hence also ␨(n), n If the coefficients of x , mу1 are now equated, the val- ϭ2,4,6,... directly from the two general properties of poly- ues of ␨(2m) follow immediately. It is certainly much sim- logs. Also observe that the source of ␲ in ␨(n), n pler than the standard method of obtaining them by contour ϭ2,4,6,... is log(zϭϪ1). It is interesting to note that we integration ͓17͔. More generally one can obtain ␨(n), n can obtain in this way Riemann’s ␨ function of only positive Ͼ1, by Riemann’s self-reflection formula since ␨(1Ϫn) can even integral order. be evaluated through the Hurwitz function ͓12,15͔.

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