THE IMPORTANCE OF BEING STRANGE

Transcription

1 THE IMPORTANCE OF BEING STRANGE ROBERT SCHNEIDER Abstract. We discuss infinite series similar to the strange function F (q) of Kontsevich which has been studied by Zagier Bryson Ono Pitman Rhoades and others in connection to quantum modular forms. We show that a class of strange alternating series that are well-defined almost nowhere in the complex plane can be added to classical infinite products (but with some ambiguity as to ± signs depending on how we compute limits) to produce convergent q-hypergeometric series of a shape that specializes to Ramanujan s mock theta function f(q) and other interesting number-theoretic objects. Let us begin this philosophical essay on q-series by considering the ancestor of all power series expansions which is of course the infinite geometric series G(q) : + q + q 2 + q q for q <. We need not advertise this enormously useful and foundational object considered even in antiquity although one would have good reason to feel enthusiasm for it. It is the sense of enigma also associated with G(q) since classical times that we wish to invoke here. Recall Zeno s paradox of the runner who can never reach the finish line: it is as if an infinite sequence of invisible purely conceptual half-way points should exert a resisting force on the athlete and therefore on all moving objects a very bizarre physical scenario. In light of calculus now we interpret Zeno s mind-bending thought experiment to be equivalent to evaluating the series /2 + /4 + / G(/2) modulo some understandable confusion with regard to instantaneous velocity. The geometric series G(q) borders other famously bizarre arithmetic phenomena as well. The boundary case G() : lim G(q) q is the first example of infinity that taunts the imaginations of school-children. At the other side of the domain of convergence the sum () G( ) : lim G(q) q + is the archetype of a divergent alternating series; the partial sums oscillate between the values 0 and. Euler claimed this sum is equal to /2 lim q +( q) [5] which seems like an outrageous statement as the partial sums are not even close to this value. On the other hand G( ) certainly feels more finite than G() does (and the statement G( ) /2 can indeed be justified by Abel summation). We began with these classical examples with the hope of exciting a feeling of strangeness in the reader to set the stage for Fields medalist Maxim Kontsevich a student of Don Zagier. In a 997 lecture at the Max Planck Institute for Mathematics Kontsevich

2 2 ROBERT SCHNEIDER discussed an almost nonsensical q-hypergeometric series [] (2) F (q) : (q; q) n where the q-pochhammer symbol is defined by (a; q) 0 : (a; q) n : n j0 ( aqj ) and (a; q) : lim n (a; q) n for a q C q <. This series F (q) is often referred to in the literature as Kontsevich s strange function and has since been studied deeply by Zagier (it was one of his prototypes for quantum modularity)[ 0] as well as by Bryson Ono Pitman Rhoades [4] and other authors [ 8] in connection to quantum modular forms unimodal sequences and other topics. There are many reasons to say the series (2) is strange. For the sake of this elementary essay let us merely note that as n (q; q) n converges on the unit disk is essentially singular on the unit circle (except at roots of unity where it vanishes) and diverges when q >. Thus n 0 (q; q) n is divergent at almost every point q in the complex plane; at its tail at least F (q) looks very much like (3) (q; q) ( ). But unlike (3) above which is nowhere a meaningful statement at q ζ m an mth order root of unity F does make sense: because (ζ m ; ζ m ) n 0 for n m then as q ζ m radially F (ζ m ) : lim q ζm F (q) is just a polynomial in Z[ζ m ]. Now let us turn our attention to the alternating case F (q) : ( ) n (q; q) n a summation studied by Cohen [4] that is similarly strange : it doesn t converge anywhere in C except at roots of unity. Once again at the tail this is basically ±(q; q) ( +...). We observe that F (0) is exactly the divergent equation (); also like +... the series F (q) feels more manageable than the non-alternating case as we see below. At this point we recall that Zagier provided the series q n(n+)/2 (4) σ(q) : + ( ) n q n+ (q; q) n from Ramanujan s lost notebook the right-hand side of which is due to Andrews [3] as another of his prototypes for quantum modularity (when multiplied by q /24 ) [0]. Then using telescoping series to find that ( (q; q) (q; q) n ( q n+ ) ) q n+ (q; q) n and combining this functional equation with the right side of (4) above gives σ(q) (q; q) 2 q 2n+ (q; q) 2n.

3 THE IMPORTANCE OF BEING STRANGE 3 On the other hand manipulating symbols heuristically suggests we can rewrite ( F (q) ((q; q) 2n (q; q) 2n+ ) (q; q) 2n ( q 2n+ ) ) q 2n+ (q; q) 2n which agrees with computation of the coefficients if by convergence on the left we mean the limit as N of the partial sums 2N ( )n (q; q) n. (We note throughout that we might also choose the alternate coupling of summands to similar effect e.g. considering here the partial sums + N n [(q; q) 2n (q; q) 2n ] (q; q) 2N as N ; this dichotomy gives rise to a ± sign ambiguity in the following propositions.) Combining the above equations leads to a somewhat paradoxical statement. Proposition. For 0 < q < and at q we claim σ(q) ±(q; q) + 2F (q) where the ± sign is positive if we mean convergence through lim 2N N on the righthand side and is negative if instead we use lim 2N N. The nearly-quantum-modular form on the left is the by-product of a strange interaction with the nearly-modular form (q; q) on the right. A similar case involves Ramanujan s prototype f(q) for a mock theta function f(q) : q n2 ( q; q) 2 n n ( ) n q n the right-hand side of which is due to Fine (see (26.22) in [7] Ch. 3). Then following the same formal approach to that taken above we can use ( ) q n ( q; q) + q n+ and a related alternating strange series that also turns into +... at q 0 ( ) n ( ) q 2n+ ( q; q) 2n + q 2n+ ( q; q) 2n+ which fails to converge for 0 < q < on the left-hand side but makes sense using a modified definition of convergence as above to suggest the following identity that is borne out by computational evidence. Proposition 2. For 0 < q < and at q we claim ( ) n f(q) ± + 2 ( q; q) where the ± sign is positive if we mean convergence through lim 2N N on the righthand side and is negative if instead we use lim 2N N. Morally one gets the feeling f(q) inherits a little modularity and a little strangeness from its parents on the right-hand side. Propositions and 2 typify a general phenomenon one can observe formally as well as computationally: the combination of an alternating Kontsevich-style strange function n

4 4 ROBERT SCHNEIDER with a related infinite product is apparently a convergent q-series. Let us introduce a few notations in order to discuss this succinctly. As usual we write (a a 2... a r ; q) n : (a ; q) n (a 2 ; q) n (a r ; q) n along with the limiting case (a a 2... a r ; q) as n. Associated to the sequence a a 2... a r of complex coefficients we will define a polynomial α r (X) by the relation thus ( a X)( a 2 X) ( a r X) : α r (X)X (a q a 2 q... a r q; q) n n ( α r (q j )q j ) and we follow this convention in also writing ( b X)( b 2 X) ( b s X) : β s (X)X. Then precisely the same heuristic path that led us to Propositions and 2 i.e. manipulating and comparing telescoping-type series points to the following general formula which agrees with computational examples. Proposition 3. For 0 < q < we claim ± (a q a 2 q... a r q; q) (b q b 2 q... b s q; q) + 2 n j ( ) n (a q a 2 q... a r q; q) n (b q b 2 q... b s q; q) n ( ) n q n (α r (q n ) β s (q n )) (a q a 2 q... a r q; q) n (b q b 2 q... b s q; q) n where the ± sign is positive if we mean convergence through lim 2N N on the left and is negative if instead we use lim 2N N. Remark. Proposition is the case a a i b j 0 for all i > j. Proposition 2 is the case b a i b j 0 for all i j >. Remark. A more rigorous interpretation of the propositions above is as follows: the N th partial sum of an alternating strange series oscillates asymptotically as N between 2 (S(q) + ( )N P (q)) where S is an Eulerian infinite series and P is an infinite product as given in Proposition 3. Aside from the formulas having been verified in finite computations the heuristic claims about divergent series in this essay read like a fantasy. Yet the ephemeral strange functions almost seem to enter into mathematics as beautifully 2 as their better-behaved (but still eccentric) relatives mock theta functions. Of course there are the known connections of Kontsevich s F (q) to numerous important objects of study to support this claim. Consider as further evidence that the strange left-hand side of Proposition 3 is formally simpler and more transparent combinatorially than the convergent right side which specializes to mock theta functions quantum modular forms and other interesting shapes as a generating function to compute the sequence of coefficients. We leave the q case open depending on the a i b j. We also note that as with F (q) and F (q) these strange identities have finite radial limits at applicable roots of unity (see [9] for details). 2 To misapply Ramanujan s words.

5 THE IMPORTANCE OF BEING STRANGE 5 Remark. It follows from Euler s continued fraction formula [6] that alternating strange functions have representations such as ( ) n ( ) n (q; q) n. + ( q; q) n + q+ +q q 2 + +q2 q q+ q2 q 2 + q3 These continued fractions equal /2 at q 0 (which is consistent with Abel summation and Euler s evaluation) take the correct values at q and also diverge on 0 < q <. Now the value /2 doesn t satisfy the claimed equalities in the propositions so we cannot extend the domains to include q 0. But for q 0 we can substitute strange continued fractions for the Kontsevich-style summations in the propositions for example to write 2 f(q) ± + ( q; q) + q+ +q q 2 + +q2 if we change the wording to read:...where the ± sign is positive if we mean convergence through successive even convergents and is negative if instead we use odd convergents. Acknowledgements The author is thankful to George Andrews for a discussion of divergent series that sparked this study and to Olivia Beckwith for comments that clarified the exposition. References [] K. Bringmann A. Folsom and R. C. Rhoades Unimodal sequences and strange functions: a family of quantum modular forms Pacific Journal of Mathematics 274. (205): -25. [2] G. E. Andrews Ramanujan s lost notebook V: Euler s partition identity Advances in Mathematics 6.2 (986): [3] G. E. Andrews J. Jiménez-Urroz and K. Ono q-series identities and values of certain L-functions Duke Mathematical Journal 08.3 (200): [4] J. Bryson K. Ono S. Pitman and R. C. Rhoades Unimodal sequences and quantum and mock modular forms Proceedings of the National Academy of Sciences (202): [5] W. Dunham Euler: The master of us all The Dolciani Mathematical Expositions vol. 22 Mathematical Association of America Washington D.C [6] L. Euler An essay on continued fractions Mathematical Systems Theory 8.4 (985): [7] N. J. Fine Basic hypergeometric series and applications With a foreword by George E. Andrews Mathematical Surveys and Monographs 27. American Mathematical Society Providence RI 988. [8] L. Rolen and R. Schneider A strange vector-valued quantum modular form Archiv der Mathematik 0. (203): [9] R. Schneider Jacobi s triple product mock theta functions unimodal sequences and the q-bracket Preprint (206). [0] D. Zagier Quantum modular forms Quanta of maths (200): [] D. Zagier Vassiliev invariants and a strange identity related to the Dedekind eta-function Topology 40 (200) no Department of Mathematics and Computer Science Emory University 400 Dowman Dr. W40 Atlanta Georgia address: robert.schneider@emory.edu