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Polylogarithms and Riemann's Function

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Polylogarithms and Riemann's Function PHYSICAL REVIEW E VOLUME 56, NUMBER 4 OCTOBER 1997 Polylogarithms and Riemann’s z function M. Howard Lee Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602 ~Received 12 March 1997! Riemann’s z function has been important in statistical mechanics for many years, especially for the under- standing of Bose-Einstein condensation. Polylogarithms can yield values of Riemann’s z 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 z 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 z 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. @S1063-651X~97!01510-9# PACS number~s!: 05.90.1m, 02.90.1p I. INTRODUCTION standard methods. It also lends an interesting insight into the nature of the values of Riemann’s z function. Riemann’s z 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 @1#. Subsequently, this function has played an To show their relationship to Riemann’s z 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 @6#, defined by tion ~BEC!@2#. More recently, this function together with the z 1 dt Mellin transform has become a powerful tool for the analysis s21 Lis~z!5 @log~1/t!# ,Imt50, ~1! of the thermodynamic potentials @3,4#. It would be no sur- G~s! E0 12zt prise to find fruitful applications of Riemann’s z function in other areas of today’s theoretical physics @5#. whenever this integral converges, i.e., Res.0, Rez,1, 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 @6#. cut from z51to`. Also if s52, the standard expression for There is richness that this unified picture reveals, such as the the dilog is recovered @10#. The above equation ~1! bears anomalous physics in null dimension @7#, the Fermi-Bose resemblance to an integral representation for Riemann’s z reflection in d>3 @8#, and the Fermi-Bose equivalence in d function z(s) of a complex number s @12#. 52 @9#. 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. ~1!. polylog of integral order becomes Riemann’s z function of ~a! Lis(z51)5z(s). the same order when its argument attains unity @10#. Thus ~b! If s5n52,3,4,..., Lin(z) are classical polylogs, Riemann’s z 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.! 21 way of evaluating Riemann’s z function, which is presented ~c! limz 0z Lis(z)51. There is a trivial fixed point at in this work. the origin. → The classical theory of polylogs begins with Euler’s dilog ~d! 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 ~z!5Li ~z!. polylog is related to physical dimensions d. Thus polylogs of dz s11 s integral order lower than the dilog have been conceived, such as the nil-log and the monolog for the physics in d ~e! Duplication relation 50 and 2 @6#. There have been suggestions that negative 12s 2 dimensions can be of theoretical interest @11#. They require Lis~z!1Lis~2z!52 Lis~z !. the polylogs of still lower orders than the nil-log, departing from the direction of the classical theory of polylogs. What If s5n52 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 @10#. simpler relationship between the polylogs and Riemann’s z ~f! uLis(z521)u,` since the function is analytic at z5 function. We can use this relationship to evaluate Riemann’s 21. z function very simply, perhaps more simply than by most ~g! If uzu,1 and s5n>1 ~also uzu51 included if n>2!, 1063-651X/97/56~4!/3909~4!/$10.0056 3909 © 1997 The American Physical Society 3910 M. HOWARD LEE 56 ` 2 3 4 7 Li26~z!5z~11z!~1156z1246z 156z 1z !/~12z! 50, k n Lin~z!5 ( z /k . k51 2 3 Li27~z!5z~11120z11191z 12416z ~h! 11191z41120z51z6!/~12z!8 z~n! if .1 517/16, Li ~1!5 n H ` if n<1. 2 3 4 Li28~z!5z~11z!~11246z14047z 111572z 14047z The property ~h! is the basis for the existence of BEC if 5 6 9 d>3 and for the absence if d<2 @1#. It should be noted that 1246z 1z !/~12z! if s5n.1, there is thus no difference in Lin(1) between the 50. one given in ~h! and the other given in ~a!.Ifn,1, however, by Lin(1) we shall mean simply the integral ~1!, which then The numerical values given on the right-hand side of Eq. becomes undefined as shown above because of the singular- ~4! are the polylogs evaluated at z521. There seems to be ity in Lin(z)atz51. However, one can obtain Lis(1), s5 no general closed form expression recognizable from these 2m, m.0, by analytic continuation in the manner of z(s lower order polylogs. One can, however, obtain the polylog 52m,0). 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 z(s) in place of Lis(1) considered all but complete. These polylogs of order n50, whenever analytic continuation is implied. Since z521is 21,22,... have shed log character. We might call them not a singular point of Lis(z), this kind of distinction need polypseudologs, e.g., Li22(z) the dipseudolog. This distinc- not be made for Lis(21). 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 ~i! The recurrence relation ~d! is satisfied. ~ii! The duplication relation ~e! is satisfied. ~iii! They are finite at z If z51 in the duplication relation ~e!, we obtain together 521. See ~f!. ~iv! If uzu,1, the expansions of polyp- with ~a!, seudologs are also given by ~g!. Hence the condition s5n 12s 21 >1 given therein may be relaxed to s5n. In addition, the z~s!5~2 21! Lis~21!. ~2! polypseudologs Li2n(z), n.0 show the following proper- ties. ~v! They have a pole of order n11atz51. See ~h!. ~vi! Thus it is possible to evaluate z(s)ifLi(z) can be given at s They are factorable by z @see ~c!# and also by (z11) if n is z521. If s5n52, for example, we recover the familiar an even number. ~vii! The numerical coefficients add up to result z(2)5p2/6, given that Li (21)52p2/12. See the 2 n!. ~viii! Evidently Li (21)50ifn52,4,6,..., and Appendix for Li (21), n52,4,6,...,obtained purely from 2n n some numbers relatable to Bernoulli’s numbers if n the inversion property of classical polylogs. Hence it is el- 51,3,5,... .These results may be used to obtain the values ementary to evaluate z(s5n)ifn52,4,6,... .Toevaluate of Riemann’s z function z(s5n), n,1 to any desired lower z(s5n) when n,1, we need to know first the form of Li (z) s order. for s5n51,0,21,22, etc. These lower order polylogs can be obtained from any one of the higher order ones through repeated differentiation. See the recurrence relation ~d!. IV. REIMANN’S z 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 s5n51 in Eq. ~1!, we immediately obtain in Eq. ~2!, → the form for the monolog, 12s 21 z~s 1!5 lim~2 21! Li1~21!. ~5! Li1~z!52log~12z!, zÞ1. ~3! → s 1 → 12s By applying the recurrence relation to the above form of the From Eq. ~3!,Li1(21)52log 2. Also, (2 21) 2(s monolog and repeating it over and again, we have obtained 21)log2 as s 1. Hence z(s 1)51/(s21), s 1,→ a well- → 1 → → the lower order polylogs given below to order n528: known result. If s5 2 in Eq. ~2!, Li0~z!5z/~12z!521/2, ~4! 1 z~ 2 !5~&11!Li1/2~21!521.460..., ~6! 2 Li21~z!5z/~12z! 521/4, where by ~g! Li1/2(21)52111/&21/)1 ...5 20.6048..., aslow but converging series 13,14 .
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