ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICS

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ZETA FUNCTIONS, TOPOLOGYAND QUANTUM PHYSICS

Developments in MathematicsVOLUME 14Series Editor:Krishnaswami Alladi, University of Florida, U.S.A.Aims and ScopeDevelopments in Mathematics is a book series publishing(i)Proceedings of conferences dealing with the latest researchadvances,(ii)Research monographs, and(iii)Contributed volumes focusing on certain areas of specialinterest.Editors of conference proceedings are urged to include a few surveypapers for wider appeal. Research monographs, which could be usedas texts or references for graduate level courses, would also besuitable for the series. Contributed volumes are those where variousauthors either write papers or chapters in an organized volumedevoted to a topic of speciaVcurrent interest or importance. Acontributed volume could deal with a classical topic that is onceagain in the limelight owing to new developments.

ZETA FUNCTIONS, TOPOLOGYAND QUANTUM PHYSICSEdited byTAKASHI AOKIKinki University, JapanSHIGERU KANEMITSUKinki University, JapanMIKIO NAKAHARAKinki University, JapanYASUO OHNOKinki University, Japana- springer

Zeta functions, topology, and quantum physics 1 edited by Takashi Aoki . [et al.].p. cm. - (Developments in mathematics ; v. 14)Includes bibliographicalreferences.ISBN 0-387-24972-9 (acid-freepaper) - ISBN 0-387-24981-8 (e-book)1. Functions, Zeta-Congresses. 2. Mathematical physics-Congresses. 3.Differential geometry-Congresses. I. Aoki, Takashi, 1953- 11.Series.AMS Subiect Classifications: 1 1Mxx. 35Qxx. 34Mxx. 14Gxx. 51PO5ISBN-10: 0-387-24972-9e-ISBN-10: 0-387-24981-8ISBN-13: 978-0387-24972-8e-ISBN-13: 978-0387-24981-0Printed on acid-free paper.O 2005 Springer Science Business Media, Inc.All rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher (Springer Science Business Media, Inc., 233 Spring Street,New York, NY 10013,USA), except for brief excerpts in connection with reviews or scholarlyanalysis. Use in connection with any form of information storage and retrieval, electronicadaptation, computer software, or by similar or dissimilar methodology now know or hereafterdeveloped is forbidden.The use in this publication of trade names, trademarks, service marks and similar terms, even ifthe are not identified as such, is not to be taken as an expression of opinion as to whether or notthey are subject to proprietary rights.Printed in the United States of America.987654321SPIN 11161400

ContentsPrefaceConference scheduleList of participantsGollnitz-Gordon partitions with weights and parity conditionsKrishnaswami Alladi and Alexander Berlcovich1Introduction2A new weighted partition theorem3Series representations4A new infinite hierarchyAcknowledgmentsReferencesPartition Identities for the Multiple Zeta FunctionDavid M. Bradley1Introduction2Definitions3Rational Functions4Stuffles and Partition IdentitiesReferencesA perturbative theory of the evolution of the center of typhoonsSergey Dobrolchotov, Evgeny Semenov, Brunello TirozziIntroductionD namics of vortex square-root type singularities and Hugonibtd s l o v chainsEquation for the smooth and singular part of the solutions3Cauchy-Riemann conditionsDerivation of the Hugonibt-Maslov chain using complex vari4ables and its integralsAcknowledgments12Referencesxixiixiv1

viiiZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSAlgebraic Aspects of Multiple Zeta ValuesMichael E. Hoffman1Introduction2The Shuffle Algebra3The Harmonic Algebra and Quasi-Symmetric Functions4Derivations and an Action by Quasi-Symmetric Functions5Cyclic Derivations6Finite Multiple Sums and Mod p Results51References71On the local factor of the zeta function of quadratic ordersMasanobu s involving the Hurwitz zeta-function valuesS. Kanemitsu, A. Schinzel, Y. Tanigawa1Introduction and statement of results2Proof of resultsReferences8991Crystal Symmetry Viewed as Zeta SymmetryShigeru Kanemitsu, Yoshio Tanigawa, Haruo Tsukada, Masami Yoshimoto1Introduction922Lattice zeta-functions and Epstein zeta-functions1033Abel means and screened Coulomb potential120References128Sum relations for multiple zeta valuesYasuo Ohno1Introduction2Generalizations of the sum formula3Identities associated with Arakawa-Kaneko zeta functions4Multiple zeta-star values and restriction on weight, depth, andheightAcknowledgment131142143References143The Sum Formula for Multiple Zeta ValuesOKUDA Jun-ichi and UENO Kimio1Introduction131133140

ixContentsAcknowledgment2Shuffle AlgebraMultiple Polylogarithms and the formal KZ equation3Mellin transforms of polylogarithms and the sum formula for4MZVsKnizhnik-Zamolodchikovequation over the configuration space5x3 (@)ReferencesZeta functions over zeros of general zeta and L-functionsAndre' Voros1Generalities2The first family { T ( s , x))3The second family ( 2(a,v))4The third family {3(a,y))Concrete examples5ReferencesHopf Algebras and Transcendental NumbersMichel WaldschmidtTranscendence, exponential polynomials and commutative lin1ear algebraic groups2Bicommutative Hopf algebrasHopf algebras and multiple zeta values3References218

PrefaceThis volume contains papers by invited speakers of the symposium"Zeta Functions, Topology and Quantum Physics" held at Kinki University in Osaka, Japan, during the period of March 3-6, 2003. Theaims of this symposium were to establish mutual understanding and toexchange ideas among researchers working in various fields which haverelation to zeta functions and zeta values.We are very happy to add this volume to the series Developmentsin Mathematics from Springer. In this respect, Professor KrishnaswamiAlladi helped us a lot by showing his keen and enthusiastic interest inpublishing this volume and by contributing his paper with AlexanderBerkovich.We gratefully acknowledge financial support from Kinki University.We would like to thank Professor Megumu Munakata, Vice-Rector ofKinki University, and Professor Nobuki Kawashima, Director of Schoolof Interdisciplinary Studies of Science and Engineering, Kinki University, for their interest and support. We also thank John Martindale ofSpringer for his excellent editorial work.Osaka, October 2004Takashi AokiShigeru KanemitsuMikio NakaharaYasuo Ohno

xiiZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSZ e t a Functions, Topology,andQ u a n t u m PhysicsKinki University, Osaka, Japan3 - 6 March20033 MarchM. Waldschmidt (Paris VI)How to prove relations between polyzeta values using automataH. Tsukada (Kinki Univ.)Crystal symmetry viewed as zeta symmetry(cowork with S. Kanemitsu, Y. Tanigawa and M. Yoshimoto)S. Akiyama (Niigata Univ.)Quasi-crystals and Pisot dual tilingK. Alladi (Florida)Insights into the structure of Rogers-Ramanujan type identities, somefrom physicsA. Voros (Saclay)Zeta functions for the Riemann zeros4 MarchY. Ohno (Kinki Univ.)Sum relations for multiple zeta valuesM. Hoffman (U. S. Naval Acad.)Algebraic aspects of multiple zeta valuesB. Tirozzi (Rome)Application of shallow water equation to typhoonsJ. Okuda (Waseda Univ.)Multiple zeta values and Mellin transforms of multiple polylogarithms(cowork with K. Ueno)D. Broadhurst (The Open Univ.)Polylogarithms in quantum field theoryHigh School Session (Two lectures for younger generation)(i) K. Alladi (Univ. Florida)Prime numbers and primality testing(ii) M. Waldschmidt (Univ. Paris VI)Error correcting codes

Conference schedulexiii5 MarchM. Kaneko (Kyushu Univ.)On a new q-analogue of the Riemann zeta functionK. F'ukaya (Kyoto Univ.)Theta function and its potential generalization which appear in Mirrorsymmetry6 MarchT . Ibukiyama (Osaka Univ.)Graded rings of Siege1 modular forms and differential operatorsD. Bradley (Maine)Multiple polylogarithms and multiple zeta values: Some results andconjecturesJ. Murakami (Waseda Univ.)Multiple zeta values and quantum invariants of knotsA. Schinzel (Warsaw)An extension of some formulae of LerchG. Lachaud (CNRS)Eisenstein series and the Riemann hypothesis

xivZETA FUNCTIONS, T O P O L O G Y AND QUANTUM PHYSICSList of participantsAkiyama, ShigekiAlladi, KrishnaswamiAoki, TakashiArakawa, TsuneoAsada, AkiraAsai, TsunenobuBradley, David M.Broadhurst, David J .Chinen, KojiFujita, KeikoFujiwara, HidenoriFukaya, KenjiHasegawa, HiroyasuHata, KazuyaHirabayashi, MikihitoHironaka, YumikoHoffman, Michael E.Ibukiyama, TomoyoshiIshii, TadamasaIzumi, ShuzoKaneko, MasanobuKanemitsu, ShigeruKawashima, NobukiKimura, DaijiKogiso, TakeyoshiKomatsu, TakaoKondo, YasushiKubota, YoshihiroKumagai, HiroshiKuribayashi, MasanoriLachaud, GillesMaruyama, FumitsunaMima, YukiMizuno, YoshinoriMunakata, MegumuMunemoto, TomoyukiMurakami, JunNagaoka, ShoyuNakagawa, KoichiNakagawa, NobuoNakahara, MikioNishihara, HideakiNiigata University, JapanUniversity of Florida, Gainesville, USAKinki University, JapanRikkyo University, JapanHyogo, JapanKinki University, JapanUniversity of Maine, USAOpen University, UKOsaka Institute of Technology, JapanSaga University, JapanKinki University, JapanKyoto University, JapanKinki University, JapanKinki University, JapanKanazawa Institute of Technology, JapanWaseda University, JapanU. S. Naval Academy, USAOsaka University, JapanKinki University, JapanKinki University, JapanKyushu University, JapanKinki University, JapanKinki University, JapanHiroshima University, JapanJosai University, JapanMie University, JapanKinki University, JapanThe University of the Air, JapanKagoshima National College of Technology, JapanOsaka University, JapanInstitut Mathhmatiques, Luminy, FranceToyo University, JapanKinki University, JapanOsaka University, JapanKinki University, JapanKinki University, JapanWaseda University, JapanKinki University, JapanHoshi University, JapanKinki University, JapanKinki University, JapanOsaka University, Japan

List of participantsOchiai, HiroyukiOhishi, RyokoOhno, YasuoOhyama, YousukeOkazaki, RyotaroOkuda, Jun-ichiOwa, ShigeyoshiSakuma, KazuhiroSato, FumihiroSchinzel, AndrzejSugiyama, KazunariSuzuki, MasatoshiTakahashi, HiroakiTakahashi, KoichiTakei, YoshitsuguTanaka, SatoshiTanaka, TatsushiTanigawa, YoshioTanimura, ShogoTazawa, ShinseiTerajima, HitomiTirozzi, BrunelloToda, MasayukiTohyama, MasakiTsukada, HaruoUchiyama, TadashiUeno, KimioUshio, KazuhikoVoros, AndreWaldschmidt, MichelWatanabe, MasashiYoshimoto, MasamiYuasa, ManabuNagoya University, JapanUniversity of Tokyo, JapanKinki University, JapanOsaka University, JapanDoshisha University, JapanWaseda University, JapanKinki University, JapanKinki University, JapanRikkyo University, JapanPolish Academy of Science, Institute of Mathematics, PolandTsukuba University, JapanNagoya University, JapanTakamatsu National College of Technology, JapanKinki University, JapanRIMS, Kyoto University, JapanKinki University, JapanKyushu University, JapanNagoya University, JapanKyoto University, JapanKinki University, JapanKobe University, JapanUniversity of Rome, La Sapienza, ItalyKinki University, JapanTokyo University of Science, JapanKinki University, JapanKinki University, JapanWaseda University, JapanKinki University, JapanCEA, Saclay, FranceInstitut Mathhmatiques, Paris, FranceKyushu University, JapanNagoya University, JapanKinki University, Japan

Zeta Functions, Topology and Quantum Physics, pp. 1-18T. Aoki, S. Kanemitsu, M. Nakahara and Y. Ohno, eds. 2005 Springer Science Business Media, Inc.

2ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSThe analytic representation of Theorem 1 iswhen i 1, andwhen i 3. In (1.1)) (1.2), and in what follows, we have used thestandard notationfor any complex number a, and03for 141 1. The products on the right in (1.1)) (1.2) are also equal toandrespectively, which have obvious interpretations as generating functionsof partitions into parts in certain residue classes (mod 8)) repetitionallowed. The equally well known Gollnitz-Gordon partition theorem is-Theorem 2. For i 1,3, the number of partitions into partsfi, 4(mod 8) equals the number ofpartitions into parts differing by 2 2, wherethe inequality is strict i f a part is even, and the smallest part is 2 i.The analytic representation of Theorem 2 iswhen i 1, and

Gollnitz-Gordon partitions with weights and parity conditions3when i 3. Actually (1.3) and (1.4) are equations (36) and (37) inSlater's famous list [9], but it was Gollnitz [6] and Gordon [7] who independently realized their combinatorial interpretation.By a reformulation of the (Big) Theorem of Gollnitz [6] (not Theorem 1) using certain quartic transformations, Alladi [l]provided a uniform treatment of all four partition functions Qi(n), i 0 , 1 , 2 , 3 interms of partitions into parts differing by 1 4, and with certain powersof 2 as weights attached. As a consequence, it was noticed in [I] thatQ2(n) and Qo(n) possess certain more interesting properties than theirwell known counterparts Ql (n) and Qs(n). In particular, Q2(n) aloneamong the four functions satisfies the property that for every positiveinteger k, Q2(n) is a multiple of 2k for almost all n which was proved byGordon in an Appendix to [I].Our goal is to prove Theorem 3 in 52 which shows that by attachingweights which are powers of 2 to the Gollnitz-Gordon partitions of n, andby imposing certain parity conditions, this is made equal to Q2(n). Hereby a Gollnitx-Gordon partition we mean a partition into parts differingby 2 2, where the inequality is strict if a part is even. There is asimilar result for Qo(n), and this is stated as Theorem 4 at the end of52. Theorems 3 and 4 are nice complements to Theorem 1 and to resultsof Alladi [I].A combinatorial proof of Theorem 3 is given in full in the next section.Theorem 4 is only stated, and its proof which is similar, is omitted.In proving Theorem 3 we are able to cast it as an analytic identity (see(3.2) in 53) which equates a double series with the product which is thegenerating function of Q2(n). It turns out that there is a two parameterrefinement of (3.2) (see (3.3) of 3) which leads to similar double seriesrepresentations for all four productsm O,m i (mod 4)for i 0,1,2,3. It will be shown in 53 that only in the cases i 1 , 3 dothese double series reduce to the single series in (1.1) and (1.2).Actually, the double series identity (3.2) is the case k 2 of a newinfinite hierarchy of identities valid for every k 1 1. In 4 we use alimiting case of Bailey's lemma to derive this hierarchy. We give a partition theoretic interpretation of the case k 1 and state without proof adoubly bounded polynomial identity which yields our new hierarchy asa limiting case. This polynomial identity will be investigated in detailelsewhere.

42.ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSA new weighted partition theoremNormally, by the parity of an integer we mean its residue class(mod 2). Here by the parity of an odd (or even) integer we mean itsresidue class (mod 4).Next, given a partition n into parts differing by 2, by a chain x inn we mean a maximal string of parts differing by exactly 2. Thus everypartition into parts differing by 2 2 can be decomposed into chains.Note that if one part of a chain is odd (resp. even), then all parts of thechain are odd (resp. even). Hence we may refer to a chain as an oddchain or an even chain. Also let X(X) denote the least part of a chain xand X(n) the least part of n.Note that in a Gollnitz-Gordon partition, since the gap between evenparts is 2, this is the same as saying that every even chain is of length1, that is, it has only one element.Finally, given part b of partition T, by t(b; n ) t(b) we denote thenumber of odd parts of n that are b. With this new statistic t we nowhave Theorem 3. Let S denote the set of all special Gollnitz-Gordon partitions, namely, Gollnitx-Gordon partitions n satisfying the parity condition that for every even part b of nb 2t(b)Decompose each IT E S into chains4x1 2,if(mod 4).(2.1)x and define the weight w(x) as-x is an odd chain, X(X) 5,and X(X) 11, otherwise. 2t(X(x))(mod 4),(2.2)The weight w(n) of the partition n is defined multiplicatively asthe product over all chainsx of T.W e then havewhere a ( n ) is the sum of the parts of T. Proof: Consider the partition n : bl b2 - - . bN, n E S, wherecontrary to the standard practice of writing parts in descending order, we bN. Subtract 0 from bl, 2 from b2, . . . , 2 N - 2now have bl b2

5Gollnitz-Gordon partitions with weights and parity conditionsfrom bN, to get a partition n*. We call this process the Euler subtraction.Note that in n* the even parts cannot repeat, but the odd parts can.Let the parts of n* be bT 5 ba 5 . . . 5 b&.Now identify the parts of n which are odd, and which are the smallestparts of chains and satisfy both the parity and low bound conditions in(2.2). Mark such parts with a tilde at the top. That is, if bk is such apart, we write bk bk for purposes of identification. Let bk yield bi b iafter the Euler subtraction.Next, split the parts of n* into two piles nT and ng, with nT consistingonly of certain odd parts, and n; containing the remaining parts. In thisdecomposition we adopt the following rule:(a) the odd parts of n* which are not identified as above are put in7rT.(b) the odd parts of n* which have been identified could be put ineither nT or ng.Thus we have two choices for each identified part.Let us say, in a certain given situation, after making the choices, wehave n l parts in nT and n2 parts in ng. We now add 0 to the smallestpart of n;, 2 to the second smallest part of ng, ., 2n2 - 2 to the largestpart of n;, 2n2 to the smallest part of nT, 2n2 2 to the second smallestpart of nT, ., 2(nl n2) - 2 2 N - 2 to the largest part of nT. Wecall this the Bressoud redistribution process. As a consequence of thisredistribution, we have created two partitions(out of nT) and 7r2 (outof n;) satisfying the following conditions:(i) n1 consists only of distinct odd parts, with each odd part beinggreater than twice the number of parts of n2.(ii) Since both the even and odd parts of n; are distinct, the partsof n2 differ by 2 4. Also since the odd parts of n; are chosen from thesmallest of parts of certain chains in n, the odd parts of .rm actually differby 2 6, and each such odd part is 2 5.In transforming the original partition n into the pair (nl, n2), we needto see how the parity conditions of n given by (2.1) and (2.2) transformto parity conditions in nl and 2 .First observe that since the parity conditions on n are imposed only.the identified odd parts of n , the transformedon the even parts of 7 andparity conditions (to be determined below) will be imposed only on 7r2and not on nl. Thus nl will satisfy only condition (i) above.Suppose bk is an even part of n and that t(bk;n ) t, that is there aret odd parts of n which are less than bk. Now bk becomes- - -

6ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSafter the Euler subtraction. Notice that t(b ;n*) t(bk;n) t. Nowsuppose that from among the t odd parts of n* less than b , r of themare put in n; and the remaining t - r odd parts are put in n . Thenb becomes the ( k - r ) - t h smallest part in n . SO in the Bressoudredistribution process, 2(k - r ) - 2 is added to b; making it a new evenpart ek-r in n2. ThusWe see from (2.1) and (2.3) thatek E 2t- 2r 2(t - r ) 2t(ek-,;n2)(mod 4 )(2.4)and so the parity condition (2.1) on the even parts does not change whengoing to 2 Thus.we may write (2.4) in short ase 2t(e)(2.5)(mod 4 )for any even part in n2.Now we need to determine the parity conditions on the odd parts in7r2 which are derived from some of the identified odd parts of T . To-this end- suppose that is an identified odd part of n which becomesisb; bk - (2k - 2 ) in n* due to the Euler subtraction, and thatplaced in r; Let t k ; T ) t. Notice thatzk&,Suppose that from among the t odd parts of n* which arer of themare placed in a; and the remaining t - r are placed in T;. Thenbecomes the ( k - r ) - t h smallest part in n . Thus under the Bressoudredistribution, 2(k - r ) - 2 is added to it to yield the part fk given byas in (2.3). Therefore the parity condition (2.2) yieldsfkE1 2t- 2r 1 2(t - r )(mod 4 ) .But t ( f k n2); t - r . So this could be expressed in short asfor any odd part of 2 Thus.the pair of partitions ( n l ,n2)is determinedby condition (i) on nl, and conditions (ii) and the parity conditions (2.5)and (2.6) on n2.

Gollnitz-Gordon partitions with weights and parity conditions7In going from 7r to the pair (7r1, 7r2) we had a choice of deciding whetheran identified part of 7r would end up in 7rl or 7r2. This choice is preciselythe weight w ( x ) 2 associated with certain chains X. The weight ofthe partition 7r is computed multiplicatively because these choices areindependent. So what we have established up to now is:Lemma 1. The weighted count of the special Gollnitz-Gordon partitionsof n equals the number of bipartitions (7rl, 7r2) of n satisfying conditions(i), (ii), (2.5) and (2.6).Next, we discuss a bijective mapwhere 7r3 is a partition into distinct multiples of 4 andinto distinct odd parts such that71-4 isa partitionHere by U(T) we mean the number of parts of a partition 7r and by A(7r)the largest part of 7r.To describe the map (2.7) we represent .rra as a Ferrers graph withweights 1 , 2 or 4, at each node. We construct the graph as follows:1) With each odd (resp. even) part f (resp. e) of ./ra we associate arow of 3 f:2t(f)(resp. Te 2t(e)) nodes.2) We place a 1 at end of any row that represents an odd part of 7r2.3) Every node in the column directly above each 1 is given weight 2.4) Each remaining node is given weight 4.Every part of 7r2 is given by the sum of weights in an associated row.It is clear from these weights, that the partition represented by thisweighted Ferrers graph satisfies precisely the conditions (ii), (2.5) and(2.6) that characterize 7r2.Next we extract from this weighted Ferrers graph all columns with a 1at the bottom, and assemble these columns as rows to form a 2-modularFerrers graph as shown below.

8ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSClearly this 2-modular graph represents a partition 7r4 that satisfiescondition (2.9).After this extraction, the decorated graph of 7r2 becomes a 4-modulargraph (in this case a graph with weight 4 at every node). This graph 7r3clearly satisfies (2.8).It is easy to check that (2.7) is a bijection. Thus Lemma 1 can berecasted in the formLemma 2. The weighted count of the special Gollnitz-Gordon partitionsof n as in Theorem 3 is equal to the number of partitions of n i n the)form (TI, 7r3, 4 where(iii) 7r3 consists only of distinct multiples of 4,(iv) 7r4 has distinct odd parts and A(7r4) 2v(r3),(v) 7rl has distinct odd parts and X(7rl) 2v(r3),Finally, observe that conditions (iv) and (v) above yield partitionsinto distinct odd parts (without any other conditions). This togetherwith (iii) yields partitions counted by Q2(n), thereby completing thecombinatorial proof of Theorem 3.In a similar fashion, we can obtain the following representation forQo(n) with weights and parity conditions imposed on the Gollnitz-Gordonpartitions:Theorem 4. Let S* denote the set of all special Gollnitz-Gordon partitions, namely, Gollnitz-Gordon partitions 7r satisfying the parity condition that for every even part b of ITDecompose each T E S* into chainsW(X xand define the weight w(x) as2, if x is an odd chain, X(X) 2 3,and X(X) 2t(X(x)) - 1 (mod 4),1, otherwise.(2.11)

Gollnitz-Gordon partitions with weights and parity conditionsThe weight w(n-) of the partition n- is defined multiplicatively asthe product over all chainsx of n-.We then havewhere a ( n ) is the s u m of the parts ofn-.3.Series representationsIf we let u(nl) nl and u(n-2) n2, then (2.7) and conditions (iii),(iv), and (v) of Lemma 2 imply that the generating function of all suchtriples of partitions (nl, n-3, n4) isIf the expression in (3.1) is summed over all non-negative integers n land n2, it yieldsBy just following the above steps we can actually get a two parameterrefinement of (3.2), namely,

10ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PHYSICSOne may view (3.2) as the analytic version of Theorem 3. In reality,the correct way to view (3.2) is that, if the summand on the left isdecomposed into three factors as (3.1), then (3.2) is the analytic versionof the statement that the number of partitions of an integer n into thetriple of partitions (7r1, 7r3, 7r4) is equal to Q2(n). This is of course onlythe final step of the proof given above. and (3.2), which is quite simple,is equivalent to it.The advantage in the two parameter refinement (3.3) is that by suitable choice of the parameters we get similar representations involvingQi(n) for i 0,1,3. For example, if we replace w by wq-2 in (3.3) wegetwhich is the analytic representation of Theorem 4 above.Next, replacing z by zq and w by wq-l in (3.3) we getNow choose z 1 in (3.5). Then the double series on the left becomesIf we now put n n lthe form n2 and j n2, then (3.6) could be rewritten inwhich is the single series identity (1.1) in a refined form.

Differential geometry-Congresses. I. Aoki, Takashi, 1953- 11. Series. . Topology and Quantum Physics" held at Kinki Uni- versity in Osaka, Japan, during the period of March 3-6, 2003. The . Mikio Nakahara Yasuo Ohno . xii ZETA FUNCTIONS, TOPOLOGY AND QUANTUM PH

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