Srinivasa Ramanujan: Going Strong at 125, Part II

Home , Almost prime, Rank of a partition

Krishnaswami Alladi, Editor

Ken Ono

Modular Forms, Partitions, and Mock Theta Functions

Ramanujan’s work on modular forms, partitions, and mock theta functions is rich with tantalizing examples of deeper mathematical structures. In- deed, his tau-function, his partition congruences, and his mock theta functions are prototypes that have helped to shape the modern mathematical landscape. Ramanujan was fascinated by the coefficients of the function ∞ ∞ X n Y n 24 (1) ∆(z) = τ(n)q := q (1 − q ) n=1 n=1 = q − 24q2 + 252q3 − 1472q4 + · · · , where q := e2πiz and Im(z) > 0. This function is a Ramanujan passport photo. weight 12 modular form. In other words, ∆(z) is a function on the upper half of the complex plane The 125th anniversary of the birth of Srinivasa such that Ramanujan was on December 22, 2012. To mark az + b = (cz + d)12 (z) the occasion, the Notices is publishing a feature ∆ cz + d ∆ article of which this is the second and final install- a b ment. This installment contains pieces by Ken Ono, for every matrix c d ∈ SL2(Z). He conjectured Kannan Soundararajan, Robert Vaughan, and Ole (see p. 153 of [12]) that Warnaar on various aspects of Ramanujan’s work. τ(nm) = τ(n)τ(m), The first installment appeared in the December for every pair of coprime positive integers n and 2012 issue, and contained an introductory piece m, and that by Krishnaswami Alladi plus pieces by George An- drews, Bruce Berndt, and Jonathan Borwein on τ(p)τ(ps ) = τ(ps+1) + p11τ(ps−1), Ramanujan’s work. for primes p and positive integers s. Mordell proved these conjectures, and in the 1930s Hecke later Krishnaswami Alladi is professor of mathematics at the University of Florida. His email address is alladik@ufl. Ken Ono is the Asa Griggs Candler Professor of Mathematics edu. at Emory University. His email address is ono@mathcs. DOI: http://dx.doi.org/10.1090/noti926 emory.edu. He thanks the NSF for their generous support.

10 Notices of the AMS Volume 60, Number 1 developed a framework for the theory of modular is n, and the number of partitions of n is denoted forms and L-functions in which such properties by p(n). The generating function for p(n) has two play a fundamental role. convenient representations: Ramanujan was also very interested in the (3) size of the numbers τ(n), and for primes p he ∞ ∞ X Y 1 conjectured (see pp. 153–154 of [12]), but could p(n)qn = 1 − qn not prove, that n=0 n=1 11 ∞ n2 | | ≤ 2 X q τ(p) 2p . = 1+ . (1−q)2(1−q2)2 ··· (1−qn)2 This speculation is the first example of the n=1 Ramanujan-Petersson Conjectures, among the The infinite product representation is analogous to deepest problems in the analytic theory of auto- the infinite product for ∆(z) in (1), which explains morphic forms. This inequality was triumphantly the important role that modular forms play in confirmed by Deligne [8] as a corollary to his proof the study of p(n), while the other representation of the Weil Conjectures, work which won him a as a “basic hypergeometric series” foreshadows Fields Medal in 1978. Although Ramanujan did Ramanujan’s last work: his discovery of “mock not anticipate the Weil Conjectures (the Riemann theta functions”. Hypothesis for varieties over finite fields), he In groundbreaking work, Ramanujan proved correctly anticipated the importance of optimally that bounding coefficients of modular forms. Ramanujan is also well known for his work on p(5n + 4) ≡ 0 (mod 5), the congruence properties of his tau-numbers. For p(7n + 5) ≡ 0 (mod 7), example, he proved that (see page 159 of [12]) p(11n + 6) ≡ 0 (mod 11). X (2) τ(n) ≡ d11 (mod 691). He also conjectured corresponding congruences, d|n known as Ramanujan’s congruences, modulo pow- About forty years ago, Serre [13] and Swinnerton- ers of 5, 7, and 11. Extending his ideas, Atkin and Dyer [14] wrote beautiful papers interpreting such Watson proved [3], [15]: congruences in terms of certain two-dimensional a b c `-adic representations of Gal(Q/Q), the Galois If δ = 5 7 11 and 24λ ≡ 1 (mod δ), group of the algebraic closure of Q. Deligne had a b b c+1 c then p(δn + λ) ≡ 0 (mod 5 7 2 11 ). just proved that such representations encode the coefficients of certain modular forms as “traces These congruences have inspired much further of the images of Frobenius elements”. Serre and work. Swinnerton-Dyer interpreted Ramanujan’s tau- In the 1940s Dyson, seeking a combinatorial congruences, such as (2), as the first nontrivial explanation for the congruences, defined [9] the examples of certain exceptional representations. rank of a partition to be the largest summand For the prime ` = 691, there is a (residual) Galois minus the number of summands. He conjectured representation that the partitions of 5n + 4 (resp. 7n + 5) are divided into 5 (resp. 7) groups of equal size when ρ ,691 : Gal(Q/Q) -→ GL2(Z/691Z) ∆ sorted by their ranks modulo 5 (resp. 7), thereby which, for primes p ≠ 691, satisfies providing a combinatorial explanation for the ! 1 ∗ congruences mod 5 and 7. This conjecture was ρ (Frob(p)) = , ∆,691 0 p11 proved in the 1950s by Atkin and Swinnerton-Dyer [5]. Dyson was unable to offer a combinatorial where Frob(p) ∈ Gal(Q/Q) denotes the “Frobenius explanation for the mod 11 congruence, and he element at p”. Congruence (2) is then implied by conjectured the existence of a statistic, which he Deligne’s prescription that, for primes p ≠ 691, referred to as the “crank”, which would suitably one has explain the congruences mod 5, 7, and 11. In the = ≡ + 11 τ(p) Tr(ρ∆,691(Frob(p))) 1 p (mod 691). 1980s Andrews and Garvan [2] finally found the The theory of modular `-adic Galois representa- elusive crank. tions has subsequently flourished and famously In addition to finding a combinatorial expla- is the “language” of Wiles’s proof of Fermat’s Last nation for Ramanujan’s original congruences, Theorem. researchers have looked for further congruences. Ramanujan’s work on tau-congruences is inti- In the 1960s Atkin [4] found congruences, though mately related to his work on partition congruences. not so systematic, with modulus 13, 17, 19, 23, A partition of the natural number n is any nonin- 29, and 31. About ten years ago the author and creasing sequence of natural numbers whose sum Ahlgren [1], [10] found that there are partition

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Loosely forms. Maass harmonic weak 1 x 2πi 16 Z 1) (q), f and ∂x n ∂ k = r ht ucin,adh proved he and functions, theta are + − i 2 ∞ and aigi a it making , z :priin and partitions ]: M(z) 2 = = oua omadwihalso which and form modular iy .Hr the Here 0. 1 − and (q) f 1 ogunepoete for properties Congruence Ono, K. 8 (g z,F (z), + √ z,G (z), G(z) k p(n) ∈ 16) 14–32. (1967), Some Swinnerton-Dyer, F. P. H. 2q 1 −iz ∂y ζ yo’ rn fa of crank Dyson’s Garvan, F. τ,g (τ), ∂ amncMasform Maass harmonic 0 0 0 24 −1 H on 2 n t companion its and 1 3 rc a.Aa.Si U.S.A. Sci. Acad. Nat. Proc. 2 , ω(q 2 ζ with ! rc odnMt.Soc. Math. London Proc. 0 0 satisﬁes Volume · rc odnMt.Soc. Math. London Proc. p (z)) 3 2 H (z)) −i(τ + ζ 0 0 3 0 0 0 0 0 1 1 0 hc rnfrsas transforms which yeblcLaplacian hyperbolic 2 1 (τ), iky ! T ,y x, ), etrvle real vector-valued (N.S.) T H(z) −1 + 2q

q ∈ −g z) Number 60, ∂x sre,moon- -series, ∂ H(z), 1 3 18 R ω(−q 2 sgvnby given is , + (τ)) o (1988), 2 no. , i ∂y ∂ T 1 2 ω(q) ! )) dτ, sa is . T 66 18 98 , 1 , , [6] B. C. Berndt and R. A. Rankin, Ramanujan: Let- ters and Commentary, History of Mathematics, vol. 9, Amer. Math. Soc., Providence, RI, 1995. [7] J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), 45–90. [8] P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math., no. 43 (1974), 273–307. [9] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. [10] K. Ono, Distribution of the partition function modulo m, Ann. Math. 151 (2000), 392–307. courtesy of Krishnaswami Alladi. [11] , Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory, 2008 Harvard- Photo MIT Current Developments in Mathematics, Int. Press, Somerville, MA, 2009, pp. 347–454. Kannan Soundarajan receiving the First SASTRA [12] S. Ramanujan, Collected Papers of Ramanujan, Ramanujan Prize from Dr. Aurobinda Mitra, Cambridge Univ. Press, Cambridge, 1927. director of the Department of Science and [13] J.-P. Serre, Congruences et forms modulaires (d’pre`sH. Technology, India, on December 20, 2005. Manjul P. F. Swinnerton-Dyer), Sèminaire Bourbaki, 24e année Bhargava (second from left), also a winner of the (1971/1972), Exp. No. 416, Lecture Notes in Math., vol. prize that year, looks on. Also in the picture are 317, Springer, 1973, pp. 319–338. Krishnaswami Alladi, chair of the prize [14] , On `-adic representations and congruences committee (next to Soundararajan), and R. for coefficients of modular forms, Modular Functions Sethuraman, vice-chancellor of SASTRA (between of One Variable, III (Proc. Internat. Summer School, Bhargava and Mitra). Univ. Antwerp, 1972), Lecture Notes in Math., vol. 350, Springer, 1973, pp. 1–55. [15] G. N. Watson, Ramanujan’s Vermutung über Zer- fällungsanzahlen, J. Reine Angew. Math. 179 (1938), Ω(n), which counts the prime factors of n with 97–128. multiplicity. [16] D. Zagier, Ramanujan’s Mock Theta Functions and The Hardy-Ramanujan result was refined by Their Applications [d’après Zwegers and Bringmann- Erd˝osand Kac [9], who showed that (ω(n) − Ono], Sem. Bourbaki, 60ème année, 2006-2007, log log n)/plog log n has an approximately normal no. 986. [17] S. P. Zwegers, Mock theta functions, Ph.D. thesis, distribution with mean 0 and variance 1. The Erd˝os- Universiteit Utrecht, 2002. Kac theorem is in relation to the Hardy-Ramanujan theorem as the Central Limit Theorem is to the Law of Large Numbers. These two results formed K. Soundararajan the impetus for the development of probabilistic ideas in number theory, and we refer the reader to Ramanujan and the Anatomy of Integers Elliott [7], Tenenbaum [27], and Kubilius [17] for accounts of related work. The story of Hardy and Ramanujan and the taxicab Hardy and Ramanujan proved their theorem by number 1729 is well known. It may be less familiar means of an ingenious induction argument that that taxicab numbers also formed the motivation establishes a uniform upper bound for π (x), the for a famous paper of Hardy and Ramanujan on k number of integers below x with exactly k prime the number of prime factors of integers (see paper factors. The prime number theorem tells us that 35 and announcement 32 in [22]). They note that π(x) = π (x), the number of primes below x, is “anybody who will make a practice of factoris- 1 approximately x/ log x. Landau generalized this to ing numbers, such as the numbers of taxi-cabs” show that, for any fixed k and large x, πk(x) is ap- would verify that “round numbers”—integers com- k−1 proximately x (log log x) . In probabilistic terms, posed of a considerable number of comparatively log x (k−1)! small factors—“are exceedingly rare.” Hardy and this suggests that the number of prime factors Ramanujan explain this phenomenon by show- of an integer has an approximately Poisson distri- ing that almost all numbers n below x (in the bution with parameter log log x. Since a Poisson sense of density) are formed of about log log n distribution with large parameter λ approximates a prime factors. More precisely, if ω(n) denotes normal distribution with mean and variance equal the number of distinct prime factors of n, then to λ, this interpretation is in keeping with the p Hardy-Ramanujan and Erd˝os-Kactheorems. It is |ω(n) − log log n| ≤ Φ(n) log log n for almost all therefore natural to wonder whether the uniform n ≤ x, where Φ(n) is any function tending to infinity with n, and the same result holds also for upper bound of Hardy and Ramanujan could be refined to an asymptotic formula, which would K. Soundararajan is professor of mathematics at Stanford be a considerable strengthening of the Erd˝os-Kac University. His email address is [email protected]. theorem. This problem was highlighted by both

January 2013 Notices of the AMS 13 Hardy and Ramanujan and inspired a great deal that d(n) > d(k) for all 1 ≤ k < n, where d(n) of work over the years by many authors (notably counts the number of divisors of n. Ramanujan Pillai [20], Erd˝os[8], Sathe [25], and Selberg [26]), described (see paper 15 in [22]) the anatomy of such culminating in the definitive work of Hildebrand highly composite integers and also made a careful and Tenenbaum [15]. study of the extremal order of d(n). Ramanujan was The Hardy-Ramanujan theorem, furthered by also interested in finding large values of the iterated the insight of Erd˝os,led to a host of problems functions d(d(n)) (whose exact order is determined (see, for example, [13]) designed to understand the in the surprisingly recent paper [3]) and d(d(d(n))). multiplicative structure of a random integer—or Closely related to highly composite numbers are as Granville [4] has termed it, “the anatomy of the superabundant numbers: these are the integers (random) integers”. Interestingly, the anatomy of a n for which σ (n)/n is larger than σ (k)/k for random integer is strikingly similar to the anatomy all 1 ≤ k ≤ n − 1, and where σ (n) denotes the of random polynomials over finite fields and to the sum of the divisors of n. Superabundant numbers anatomy of the cycle decompositions of random were studied extensively by Alaoglu and Erd˝os permutations; see [2] and the forthcoming graphic [1], who were directly motivated by Ramanujan’s novel(!) [12] for accounts of these similarities. More- work. Interestingly, Ramanujan himself had studied over, understanding the anatomy of integers has these classes of numbers, in notes related to his proven valuable in many problems of algorithmic long paper on highly composite numbers, but number theory, for example, in the development owing to space considerations these were not of factorization algorithms such as the quadratic included in the published work. These manuscript sieve [21]. notes were discovered as part of Ramanujan’s Out of the many beautiful theorems in the “Lost Notebook” [23] and were published with area, let us mention the striking recent work of annotations by Nicolas and Robin in [24]. While Ford [10] in understanding the distribution of interesting, these problems seem far from the integers having a divisor of a given size. Ford’s mainstream, and so it may come as a surprise work resolved a longstanding problem of Erd˝os: that the Riemann Hypothesis is equivalent to the How many distinct integers are there in the estimate σ (n) ≤ Hn + exp(Hn) log Hn (where Hn is Pn N × N multiplication table? The Hardy-Ramanujan the “harmonic number” Hn = j=1 1/j), and the theorem, Erd˝osobserved, gives a quick proof that difficult cases of this inequality are precisely when the number of distinct integers in the multiplication n is a superabundant number; see Lagarias [18] for table is o(N2). A typical integer of size N has a charming account of this connection. about log log N prime factors, and so the product We end by mentioning an unpublished and of two such integers has about 2 log log N prime largely unknown fragment of Ramanujan on a factors. Thus a typical element appearing in the central topic in the anatomy of integers, namely, multiplication table has 2 log log N prime factors, on “smooth” (or “friable”) numbers. A number whereas a typical integer of size N2 has only is called y-smooth if all its prime factors are about log log(N2) = log log N + log 2 prime factors. at most y. Smooth numbers appear prominently In other words, elements in the multiplication in algorithmic number theory (e.g., in factoring table are quite atypical and thus few in number. algorithms) and are also useful in seemingly Refining a result of Tenenbaum [28], Ford’s work unrelated questions such as Waring’s problem; see establishes that true order of magnitude for [16] and [11] for excellent surveys on these integers the number of integers in the multiplication without large prime factors. A basic question is to 2 δ 3 table is N /((log N) (log log N) 2 ), with δ being estimate Ψ(x, y), which counts the number of y- the esoteric constant 1 − (1 + log log 2)/ log 2 = smooth integers below x; a precise understanding 0.086 ... . In the study of random permutations, of Ψ(x, y) uniformly in a wide range is equivalent there arises an analogous problem: Given n and to the Riemann Hypothesis [14]. The first published 1 ≤ k ≤ n − 1, the permutation group Sn acts work on smooth numbers in 1930 was by Dickman, naturally on the k-element subsets of {1, . . . , n}. who showed that if x = yu, u > 0 is considered Now given a random element in Sn, what is the fixed, and y tends to infinity, then Ψ(x, y) ∼ ρ(u)x, probability that there exists some k-element subset where ρ (the Dickman function) is defined as the left fixed by this permutation? Partial results on unique continuous solution to the differential- this problem were established by Luczak and Pyber difference equation uρ0(u) = −ρ(u − 1) for u > 1, [19] and have interesting applications [5]. Using and with initial condition ρ(u) = 1 for 0 ≤ u ≤ 1. Ford’s work on integers as a guide to the anatomy Given the significance of smooth numbers, it may of permutations, one can obtain a more refined be interesting to record that Ramanujan appears understanding of this question [6]. to have made a study of this problem. On page 337 u We now turn to Ramanujan’s work on “highly of [23] one finds formulas for Ψ(y , y) in the range composite numbers”. These are numbers n such 1 ≤ u ≤ 5, which give inclusion-exclusion type

14 Notices of the AMS Volume 60, Number 1 expressions for ρ(u) in terms of iterated integrals [21] C. Pomerance, A tale of two sieves, Notices Amer. Math. Soc. 43 (1996), 1473–1485. and with a clear indication of how the pattern is to [22] S. Ramanujan, Collected Papers of Srinivasa Ramanu- be continued. jan (G. H. Hardy, P. V. Seshu Aiyar, and B. M. Wilson, eds.), AMS Chelsea Publishing, Third Printing 1999. [23] , The Lost Notebook and Other Unpublished Pa- References pers, Narosa, New Delhi, and Springer-Verlag, Berlin, [1] L. Alaoglu and P. Erd˝os, On highly composite and 1988. similar numbers, Trans. Amer. Math. Soc. 56 (1944) [24] , Highly composite numbers, annotated and 448–469. with a foreword by J-L Nicolas and G. Robin, [2] R. Arratia, A. D. Barbour, and S. Tavaré, Random Ramanujan J. 1 (1997) 119–153. combinatorial structures and prime factorizations, [25] L. G. Sathe, On a problem of Hardy on the distribution Notices Amer. Math. Soc. 44 (1997), 903–910. of integers having a given number of prime factors. [3] Y. Buttkewitz, C. Elsholtz, K. Ford, and J-C. I–IV, J. Indian Math. Soc. 17 (1953) 63–141; and 18 Schlage-Puchta, A problem of Ramanujan, Erd˝os (1954), 27–81. and Kátai on the iterated divisor function, Int. Math. [26] A. Selberg, Note on a paper of L. G. Sathe, J. Indian Res. Notices, to appear. Math. Soc. 18 (1954) 83–87. [4] J. M. De Koninck, A. Granville, and F. Luca (eds.), [27] G. Tenenbaum, Introduction to Analytic and Proba- Anatomy of Integers, CRM Proceedings and Lecture bilistic Number Theory, Cambridge Stud. Adv. Math., Notes, Vol. 46, Amer. Math. Soc., Providence, RI, 2008. vol. 46, Cambridge Univ. Press, 1995. Third Edition, in [5] P. Diaconis, J. Fulman, and R. Guralnick, On fixed French, Échelles Belin, (2008). points of permutations, J. Algebr. Comb. 28 (2008) [28] , Sur la probabilité qu’un entier possède un 189–218. diviseur dans un intervalle donné, Compositio Math. [6] P. Diaconis and K. Soundararajan, Unpublished 51 (1984), 243–263. notes on fixed sets of random permutations (2012). [7] P. D. T. A. Elliott, Probabilistic Number Theory (2 vol- umes), Grund. Math. Wiss., vol. 239 and 240, Springer R. C. Vaughan Verlag, 1979 and 1980. [8] P. Erd˝os, Integers having exactly k prime factors, Ann. The Hardy–Littlewood–Ramanujan Method Math. 49 (1948), 53–66. [9] P. Erd˝os and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. In modern analytic number theory there are three J. Math. 62 (1940), 738–742. very powerful standard techniques that are central [10] K. Ford, The distribution of integers with a divisor in to many investigations. One is the use of the theory a given interval, Annals of Math. 168 (2008), 367–433. [11] A. Granville, Smooth numbers: computational num- of Dirichlet series, including the Riemann zeta ber theory and beyond, in Algorithmic Number Theory: function and Dirichlet L-functions; a second is the Lattices, Number Fields, Curves and Cryptography, application of sieve theory in all its manifesta- MSRI Publications, Vol. 44, Cambridge Univ. Press, tions; and the third is the subject of this survey. 2008 pp. 267–323. It has its genesis in two papers of Ramanujan. [12] A. Granville and J. Granville, MSI: Anatomy of inte- The most famous of these is the celebrated pa- gers and permutations (graphic novel, art by Robert J. Lewis, cover by Spain Rodriguez), Princeton University per [Hardy, G. H. & Ramanujan, S. 1918] (there are Press, forthcoming (2013). also announcements in [Hardy, G. H. & Ramanu- [13] R. R. Hall and G. Tenenbaum, Divisors, Cambridge jan, S. 1917a] and [Hardy, G. H. & Ramanujan, Tracts in Math., vol. 90, Cambridge Univ. Press, 1988. S. 1917b]), which is concerned mostly with the par- [14] A. Hildebrand, Integers free of large prime factors tition function, but the paper [Ramanujan, S. 1918] and the Riemann hypothesis, Mathematika 31 (1984), 258–271. also describes the same fundamental idea. This [15] A. Hildebrand and G. Tenenbaum, On the number is that interesting number theoretic functions of prime factors of an integer, Duke Math. J. 56 (1988), have representations, or close approximations, as 471–501. infinite series indexed by the positive rational [16] , Integers without large prime factors, J. Théorie numbers—that is, the a/q with 1 ≤ a ≤ q and des Nombres Bordeaux 5 (1993), 411–484. (a, q) = 1—and moreover that such expressions [17] J. Kubilius, Probabilistic methods in the theory of numbers, Trans. Math. Monographs, vol. 11, Amer. can be obtained by consideration of generating Math. Soc., 1964. functions that have the unit circle as a natural [18] J. Lagarias, An elementary problem equivalent to boundary and the terms in the aforementioned the Riemann Hypothesis, Amer. Math. Monthly 109 series arise from the neighborhood of singularities (2002), 534–543. [19] T. Luczak and L. Pyber, On random generation of the symmetric group, Comb. Probab. Comput. 2 (1993), R. C. Vaughan is professor of mathematics at Pennsylva- 505–512. nia State University. His email address is rvaughan@math. [20] S. S. Pillai, On the number of numbers which contain psu.edu. a fixed number of prime factors, Math. Student (1929), Research supported by NSA grant number H98230-12-1- 250–251. 0276.

January 2013 Notices of the AMS 15 at e(a/q). A very simple example of this is the In the last twenty-five years there has been interesting formula a huge explosion of activity. This note is too ∞ short to do more than mention a few highlights, π 2 X (1) σ (n) = n q−2c (n), with the hope that the reader may be pointed 6 q q=1 in an interesting direction. On Waring’s prob- which first appears in the second of these papers. lem the development of efficient differencing, building on the earlier work, especially that of Here cq(n) is the eponymous sum q Davenport, led to significant further progress X on the upper bound for G(k), and there is a (2) cq(n) = e(an/q). a=1 survey of this and some other connected results (a,q)=1 in [Vaughan, R. C. & Wooley, T. D. 2002]. Very In addition to the partition function, Hardy and recently, the sensational work of [Wooley, T. D. Ramanujan also observe that the same ideas can to appear] now means that the asymptotic formula be applied to the number of representations of a for the number of representations of a natural number as a sum of a fixed number of squares. number as the sum of s kth powers is known to One might speculate how this could have devel- hold when s ≥ 2k2 − 2k − 8. On questions involv- oped if Ramanujan had not been struck down by ing forms there has been impressive progress illness. Fortunately, Hardy and Littlewood realized on applications of the method on a number the very great importance and generality of the of fronts, especially by [Schmidt, W. M. 1979] underlying conceptions and were able to develop and [Wooley, T. D. 1997] on small zeros; them so as to apply to the Goldbach and Waring, [Heath–Brown, D. R. 1983], [Hooley, C., 1988, and cognate, problems. There was also the realiza- 1991, 1994], and [Hooley, C., 1988, 1991, 1994] tion of the predictive possibilities of the method on cubic forms; [Heath-Brown, D. R. & Browning, in those cases where the arguments could not be T. D. 2009] on cubic polynomials; and [Brüdern, otherwise pursued to a satisfactory conclusion. J & Wooley, T. D. 1998] and [Brüdern, J & Wooley, T. D. Later this led, through further important ideas of 2003] on a variety of topics. I. M. Vinogradov and Davenport, in the 1930s to In a different direction, [Heath-Brown, D. R. the essential resolution of the ternary Goldbach 1981] has ingeniously combined the circle method problem (namely, that every large odd integer is with sieve theory, and this was taken up by the sum of three primes) and substantial progress [Brüdern, J. 1988] in his thesis and has been in the known upper bounds for G(k) (the smallest exploited extensively by [Brüdern, J. 1995], [Brü- s such that every large natural number is the sum dern, J. & Kawada, K. 2002], [Kawada, K. 1997], and of at most s kth powers) in Waring’s problem [Kawada, K. 2005]. This has spawned a whole new (see [Vaughan, R. C. 1997]). Davenport, along with industry of ongoing work on sums of powers of Birch and Lewis, embarked on the question of primes and almost primes. applying the method to nondiagonal questions, In the original series of papers “On some for example, the nontrivial representation of problems of partitio numerorum” by Hardy and zero by cubic forms (see [Davenport, H. 1963] Littlewood there was an unpublished manuscript and [Davenport, H. 1962, 2005]). [Davenport, H. & on small differences between consecutive primes, Lewis, D. J. 1963] had also established the and it was this manuscript which, when combined Artin conjecture concerning the local-to-global with developments in the large sieve, led to principle for diagonal forms for all but a finite the work of [Bombieri, E. & Davenport, H. 1966] set of degrees. When I was a postgraduate and the later refinements of [Pil’tja˘ı,G. Z. 1972], student in the late 1960s, I became interested [Huxley, M. N. 1977], and [Maier, H. 1985] on in the method. However, I was warned by a p − p (3) δ = lim inf n+1 n . number of distinguished mathematicians that n→∞ log pn the method was already played out and it would The later proof by [Goldston, D. A., Pintz, J. & be better to work on other things! Nevertheless, Yıldırım 2009] that δ = 0 rests mostly on sieve in [Vaughan, R. C. 1977] I did manage to use the theory, yet the motivation for the shape of some of method to deal with the cases not dealt with by the main terms comes from heuristics arising from Davenport and Lewis, and in a different direction the Hardy-Littlewood method. In many ways the [Montgomery, H. L. & Vaughan, R. C. 1975] we motivation for the large sieve also comes from the were able to establish that there is a positive Hardy–Littlewood method, and perhaps this can be number δ such that the number E(x) of even seen most clearly in [Gallagher, P. X. 1967]’s proof numbers not exceeding x which are not the sum of of the large sieve inequality. two primes satisfies An unexpected development was the adaptation E(x) x1−δ. of the method to deal with questions involving

16 Notices of the AMS Volume 60, Number 1 arithmetic progressions in general sets. This be- [Brüdern, J. & Kawada, K. 2002] Ternary problems in addi- gan with the seminal paper of [Roth, K. F. 1953] tive prime number theory, Analytic Number Theory dealing with sets having no three elements in arith- (Chaohua Jia and Kohji Matsumoto, eds.), Kluwer, metic progression and led to [Gowers, W. T. 1998]’s 2002, pp. 39–91. [Brüdern, J & Wooley, T. D. 1998] The addition of binary proof of Szemerédi’s theorem and the theorem of cubic forms, Philos. Trans. Roy. Soc. London Ser. A 356 [Green, B. J. & Tao, T. 2008] that the primes con- (1998), 701–737. tain arbitrarily long arithmetic progressions. This [Brüdern, J & Wooley, T. D. 2003] The paucity problem for is another very active area; see, for example, certain pairs of diagonal equations, Quart. J. Math. 54 [Green, B. J. 2007] or [Tao, T. & Vu, Van, 2006]. A (2003), 41–48. cognate area concerns the mean square distribu- [Davenport, H. 1962, 2005] Analytic Methods for Diophan- tion of sets of integers in arithmetic progressions. tine Equations and Diophantine Inequalities, First edition, Ann Arbor Publishers, 1962; Second edition, The prototype is the more precise version of the Cambridge University Press, 2005. Barban-Davenport-Halberstam theorem concerning [Davenport, H. 1963] Cubic forms in sixteen variables, the set of primes due to [Montgomery, H. L. 1970] Proc. Royal Soc. London 272A (1963), 285–303. and refined by [Hooley, C. 1975]. Montgomery [Davenport, H. & Lewis, D. J. 1963] Homogeneous addi- used a variant of the Hardy-Littlewood method, tive equations, Proc. Royal Soc. London 274A (1963), and more recently this was revived by [Gold- 443–460. [Gallagher, P. X. 1967] The large sieve, Mathematika 14 ston, D. A. & Vaughan, R. C. 1996] for the set (1967), 14–20. of primes and by [Vaughan, R. C. 1998] to treat [Goldston, D. A., Pintz, J. & Yıldırım 2009] Primes in tu- general dense sets. This has been taken a step ples. I, Annals of Mathematics 170 (2009), 819–862. further by [Brüdern, J. 2009], which extends the [Goldston, D. A. & Vaughan, R. C. 1996] On the Mont- ideas of the latter paper to treat general binary gomery-Hooley asymptotic formula, Sieve Methods, additive problems. Exponential Sums, and Their Application in Number Another aspect of the method is that the idea Theory, Greaves, Harman, Huxley (eds.), Cambridge University Press, 1996, pp. 117-142. of large peaks near “rational points” e(a/q) with q [Green, B. J. 2007] Generalizing the Hardy-Littlewood relatively small (the major arcs) and lesser peaks method for primes, Proc. ICM (Madrid, 2006), vol. 2, near e(a/q) with q relatively large (the minor European Math. Soc., 2007, pp. 373–399. arcs) has suggested divisions of arguments in, [Green, B. J. & Tao, T. 2008] The primes contain arbitrarily apparently, otherwise quite unrelated topics. An long arithmetic progressions, Ann. Math. 167 (2008), important example of this is the work of Bombieri, 481–547. Iwaniec, Huxley, and Watt on exponential sums [Gowers, W. T. 1998] A new proof of Szemerédi’s theorem for progressions, Geom. Funct. Anal. 8 (1998), 529– which led to the currently best-known bounds 1 551. for the Riemann zeta function on the 2 -line, and [Hardy, G. H. & Ramanujan, S. 1917a] Une formule asymp- the error terms in the Dirichlet divisor problem totique pour le nombre des partitions de n, Comptes and Gauss’s estimation of the number of lattice Rendus (January 2, 1917). points in a large disc centered at the origin. For an [Hardy, G. H. & Ramanujan, S. 1917b] Asymptotic for- exposition of this, see [Huxley, M. N. 1996]. mulæ in combinatory analysis, Proc. London Math. Soc. 2 (1917), records for March 1, 1917. For standard accounts of the Hardy-Littlewood [Hardy, G. H. & Ramanujan, S. 1918] Asymptotic formulæ method in its more classical forms, see in combinatory analysis, Proc. London Math. Soc. 2 [Davenport, H. 1962, 2005] and [Vaughan, R. C. (1918), 75–115. 1997]. The foreword to the second edition of [Heath–Brown, D. R. 1981] Three primes and an almost [Davenport, H. 1962, 2005] gives a good survey prime in arithmetic progression, J. London Math. Soc. of a number of the applications as they stood in (2) 23 (1981), 396–414. 2005. [Heath–Brown, D. R. 1983] Cubic forms in ten variables, Proc. London Math. Soc. (3) 47 (1983), 225–257. [Heath–Brown, D. R. 2007] Cubic forms in 14 variables, References Inventiones Mathematicae 170 (2007), 199–230. [Bombieri, E. & Davenport, H. 1966] Small difference be- [Heath–Brown, D. R. & Browning, T. D. 2009] Integral tween prime numbers, Proc. Royal Soc. London 293A points on cubic hypersurfaces, Analytic Number (1966), 1–18. Theory, Essays in Honour of Klaus Roth, Cambridge [Brüdern, J. 1988] Iterationsmethoden in der additiven University Press, 2009, 75–90. Zahlentheoerie, Thesis, Göttingen, 1988. [Hooley, C. 1975] On the Barban-Davenport–Halberstam [Brüdern, J. 1995] A sieve approach to the Waring- theorem. I, J. Reine Angew. Math. 274/275 (1975), Goldbach problem. I: Sums of four cubes, Ann. Scient. 206–223. École. Norm. Sup. (4) 28 (1995), 461–476. [Hooley, C., 1988, 1991, 1994] On nonary cubic forms. I, [Brüdern, J. 2009] Binary additive problems and the circle II, III, J. Reine Angew. Math. 386 (1988), 32–98; 415 method, multiplicative sequences and convergent sets, (1991), 95–165; 456 (1994), 53–63. Analytic Number Theory, Essays in Honour of Klaus [Huxley, M. N. 1977] Small differences between consecu- Roth, 2009, Cambridge University Press, pp. 91–132. tive primes. II, Mathematika 24 (1977), 142–152.

January 2013 Notices of the AMS 17 [Huxley, M. N. 1996] Area, Lattice Points, and Exponen- S. Ole Warnaar tial Sums, London Math. Soc. Monographs, Oxford University Press, 1996. Ramanujan’s ψ Summation [Kawada, K. 1997] Note on the sum of cubes of primes and 1 1 an almost prime, Arch. Math. 69 (1997), 13–19. Notation. It is impossible to give an account of the [Kawada, K. 2005] On sums of seven cubes of almost primes, Acta Arith. 117 (2005), 213–245. 1ψ1 summation without introducing some q-series [Maier, H. 1985] Primes in short intervals, Michigan Math. notation. To keep the presentation as simple as J. 32 (1985), 221–225. possible, we assume that 0 < q < 1. Suppressing [Montgomery, H. L. 1970] Primes in arithmetic progres- q-dependence, we define two q-shifted factorials: sions, Michigan Math. J. 17 (1970), 33–39. Q∞ k z (a)∞ := k=0(1 − aq ) and (a)z := (a)∞/(aq )∞ [Montgomery, H. L. & Vaughan, R. C. 1975] The exception- for z ∈ C. Note that 1/(q)n = 0 if n is a nega- al set in Goldbach’s problem, Acta Arithmetica 27 tive integer. For x ∈ C − {0, −1,... }, the q-gamma (1975), 353–370. x−1 [Pil’tja˘ı,G. Z. 1972] On the size of the difference between function is defined as Γq(x) := (q)x−1/(1 − q) . consecutive primes, Izdat. Saratov Univ., 1972, pp. Ramanujan recorded his now famous ψ sum- 73–79. 1 1 [Ramanujan, S. 1918] On certain trigonometrical sums and mation as item 17 of Chapter 16 in the second their applications in the theory of numbers, Trans. of his three notebooks [13, p. 32], [46]. It was Cambridge Phil. Soc. 13 (1918), 259–276. brought to the attention of the wider mathematical [Roth, K. F. 1953] On certain sets of integers, J. London community in 1940 by Hardy, who included it in Math. Soc. 28 (1953), 245–252. his twelfth and final lecture on Ramanujan’s work [Schmidt, W. M. 1979] Small zeros of additive forms in [31]. Hardy remarked that the result constituted many variables. I, II, Trans. Amer. Math. Soc. 248 “a remarkable formula with many parameters.” (1979), 121–133; Acta Math. 143 (1979), 219–232. [Tao, T. & Vu, Van, 2006] Additive Combinatorics, Cam- Instead of presenting the 1ψ1 sum as given by bridge University Press, 2006. Ramanujan and Hardy, we will state its modern [Vaughan, R. C. 1977] Homogeneous additive equations form: and Waring’s problem, Acta Arith. 33 (1977), 231–253. ∞ [Vaughan, R. C. 1997] The Hardy-Littlewood Method, Cam- X (a)n (az)∞(q/az)∞(b/a)∞(q)∞ (1) zn = , bridge Tracts in Mathematics, vol. 125, Cambridge n=−∞ (b)n (z)∞(b/az)∞(q/a)∞(b)∞ University Press, 1997. |b/a| < |z| < 1, [Vaughan, R. C. 1998] On a variance associated with the distribution of general sequences in arithmetic pro- where it is understood that a, q/b 6∈ {q, q2,... }. gressions. I & II, Philos. Trans. Royal Soc. Lond. Ser. A Characteristically, Ramanujan did not provide a Math. Phys. Eng. Sci., 356 (1998), 781–791, 793–809. proof of (1). Neither did Hardy, who, however, [Vaughan, R. C. & Wooley, T. D. 2002] Waring’s Problem: A remarked that it could be “deduced from one survey, Number Theory for the Millennium. III, A K which is familiar and probably goes back to Peters, 2002, pp. 301–340 (see also Surveys in Number Theory, A K Peters, 2003, pp. 285–324 (M. A. Bennett, B. Euler.” The result to which Hardy was referring C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand, is another famous identity—known as the q- and W. Philipp, eds.). binomial theorem—corresponding to (1) with b = P∞ n [Wooley, T. D. 1997] Forms in many variables, Analytic q: n=0 z (a)n/(q)n = (az)∞/(z)∞ and valid for Number Theory: Proceedings of the 39th Taniguchi |z| < 1. Although not actually due to Euler, the International Symposium, Kyoto, May 1996 (Y. q-binomial theorem is certainly classic. It seems Motohashi, ed.), London Mathematical Society Lec- to have appeared first and without proof (for ture Notes, vol. 247, Cambridge University Press, −N Cambridge, 1997, pp. 361–376. a = q ) in Rothe’s 1811 book Systematisches [Wooley, T. D.] Vinogradov’s mean value theorem via Lehrbuch der Arithmetik, and in the 1840s many efficient congruencing, II, Duke Math. J. (to appear). mathematicians of note, such as Cauchy (1843), Eisenstein (1846), Heine (1847), and Jacobi (1847), published proofs. The first proof of the 1ψ1 sum is due to Hahn in 1949 [30] and, as hinted by Hardy, uses the q-binomial theorem. After Hahn, a large number of alternative proofs of (1) were found, including one probabilistic and three combinatorial proofs [2], [3], [5], [16], [17], [19],

S. Ole Warnaar is professor of mathematics at The University of Queensland. His email address is [email protected]. The author thanks Dick Askey, Bruce Berndt, Susanna Fishel, Jeﬀ Lagarias, and Michael Schlosser for their helpful corre- spondence. This work was supported by the Australian Research Council.

18 Notices of the AMS Volume 60, Number 1 [20], [18], [23], [24], [32], [34], [35], [44], [48], [53], which may also be found in Kronecker’s 1881 [50]. The proof from the book, which again relies paper “Zur Theorie der elliptischen Functionen”. on the q-binomial theorem, was discovered by For a, b → −1 the right side gives R4(q), whereas Ismail [32] and is short enough to include here. the left side becomes Assuming |z| < 1 and |b| < min{1, |az|}, both ∞ ∞ X X sides are analytic functions of b. Moreover, they 1 − 8 qm n(−1)n+k coincide when b = qk+1 with k = 0, 1, 2,... by the m=1 n,k=1 −k nk=m q-binomial theorem with a , aq . Since 0 is the ∞ X X accumulation point of this sequence of b’s, the = 1 + 8 (−q)m d. proof is complete. m=1 d≥1 Apart from the q-binomial theorem, the 4öd|m 1ψ1 sum generalizes another classic identity, P Hence r4(n) = 8 d≥1; 4öd|n d. This result of Jacobi known as the Jacobi triple-product identity: implies Lagrange’s theorem that every positive P∞ n n (n) n=−∞(−1) z q 2 = (z)∞(q/z)∞(q)∞ =: θ(z). integer is a sum of four squares. By taking This result plays a central role in the theory of a, b2 → −1 in (3) the reader will have little trouble theta and elliptic functions. showing that r2(n) = 4(d1(n) − d3(n)), with dk(n) the number of divisors of n of the form 4m + k. The 1ψ1 Sum as Discrete Beta Integral This is a result of Gauss and Lagrange which As pointed out by Askey [8], [9], the 1ψ1 summation implies Fermat’s two-square theorem. may be viewed as a discrete analogue of Euler’s Other simple but important applications of the beta integral. First, deﬁne the Jackson or q-integral 1ψ1 sum concern orthogonal polynomials. In [11] R c·∞ P∞ n n 0 f (t)dqt := (1−q) n=−∞ f (cq )cq . Replacing it was employed by Askey and Wilson to compute α+β α (a, b, z) , (−c, −cq , q ) in (1) then gives a special case—corresponding to the continuous q- Z c·∞ tα−1 θ(−cqα) (α) (β) Jacobi polynomials—of the Askey-Wilson integral, = α Γq Γq (2) dqt c , and in [10] Askey gave an elementary proof of 0 (−t)α+β θ(−c) Γq(α + β) the full Askey-Wilson integral using the ψ sum. where Re(α), Re(β) > 0. For real, positive c the 1 1 The sum also implies the norm evaluation of the limit q → 1 can be taken, resulting in the beta weight functions of the q-Laguerre polynomials [45]. integral modulo the substitution t , t/(1 − t). These form a family of orthogonal polynomials Askey further noted in [8] that the specialization with discrete measure µ on [0, c · ∞) given by (α, β) , (x, 1 − x) in (2) (so that 0 < Re(x) < 1) R d µ(t) = tα/(−t) d t. The normalization d µ(t) may be viewed as a q-analogue of Euler’s reﬂection q ∞ q q formula. thus follows from the q-beta integral (2) in the limit of large β.

Simple Applications of the 1ψ1 Sum Generalizations in One Dimension There are numerous easy applications of the There exist several generalizations of Ramanujan’s 1ψ1 sum. For example, Jacobi’s well-known four- and six-square theorems as well as a number sum containing one additional parameter. In his of similar results readily follow from (1); see work on partial theta functions, Andrews [4] e.g., [1], [14], [15], [21], [22], [25]. To give a obtained a generalization in which each product of four infinite products on the right-hand side is flavor of how the 1ψ1 implies these types of results, we shall sketch a proof of the four- replaced by six such products. Another example is the curious identity of Guo and Schlosser, which is square theorem. Let rs (n) be the number of representations of n as the sum of s squares. no longer hypergeometric in nature [27]: P n ∞ The generating function Rs (q) := n≥0 rs (n)(−q) k 2 X (a)k(1 − ackq )(ckq)∞(b/ack)∞ k P∞ − m m s c is given by m=−∞( 1) q . By the triple- (b) (1 − azqk)(ac (q/ac k s k=−∞ k k ∞ k ∞ product identity this is also (q)∞/(−q)∞ . Any 1 (q) (b/a) identity that allows the extraction of the coefficient = ∞ ∞ , n of (−q) results in an explicit formula for rs (n). (1 − z) (q/a)∞(b)∞ Back to (1): replace (b, z) , (aq, b) and multiply k k where ck := z(1 − aczq )/(1 − azq ) and |b/ac| < both sides by (1 − b)/(1 − ab). By the geometric |z| < 1. For c = 1 this is (1). series this yields As discovered by Schlosser [49], a quite different ∞ (1 − a)(1 − b) X extension of the 1ψ1 sum arises by considering (3) 1 + qkn(akbn − a−kb−n) 1 − ab noncommutative variables. Let R be a unital Banach = k,n 1 algebra with identity 1, central elements b and q, 2 (abq)∞(q/ab)∞(q) −1 = ∞ , and norm k · k. Write a for the inverse of an (aq) (q/a) (bq) (q/b) Qn ∞ ∞ ∞ ∞ invertible element a ∈ R. Let i=m ai stand for 1

January 2013 Notices of the AMS 19 −1 −1 if n = m − 1, am ··· an if n ≥ m and am−1 ··· an+1 multivariable extension of the product formula for if n < m − 1, and define the Poincaré polynomial of a Coxeter group: ± (4) a1, . . . , ar w(α) ht(α) ; z X Y 1 − tα e Y 1 − tαt b , . . . , b W(t) := = . 1 r k 1 − ew(α) 1 − tht(α) r w∈W α∈R+ α∈R+ Y Y i−1 i−1 −1 := z (1 − as q )(1 − bs q ) , Here R is a reduced, irreducible finite root system i s=1 in a Euclidean space V , R+ the set of positive roots, Q + where k ∈ Z ∪ {∞}, a1, . . . , ar , b1, . . . , br ∈ R, i = W the Weyl group, and tα for α ∈ R a set of formal Qk Q Q1 i=1 in the + case and i = i=k in the − variables constant along Weyl orbits. The symbol 2 case. Subject to max{kqk, kzk, kba−1z−1k} < 1, ht(α) Q (β,α)/kαk t stands for β∈R+ tβ with (·, ·) the the following noncommutative 1ψ1 sum holds: W -invariant positive definite bilinear form on V . If ht(α) ht(α) all tα are set to t, then t = t with ht(α) ∞ X a + the usual height function on R, in which case W(t) ; z b k k=−∞ reduces to the classical Poincaré polynomial W (t). za − qa−1z−1 + bza−1z−1, q − Now let S be a reduced, irreducible affine root = ; 1 − − ; 1 − − ; 1 . z ∞ qza 1z 1 ∞ ba 1z 1, b ∞ system of type S = S(R) [38]. In analogy with the finite case, assume that ta for a ∈ S is constant along orbits of the affine Weyl group W of S. Then Higher-Dimensional Generalizations Macdonald generalized (4) to [40] Various authors have generalized (1) to multiple X Y 1 − t ew(a) (5) a 1ψ1 sums. Below we state a generalization due to w(a) + 1 − e Gustafson and Milne [28], [41] which is labelled w∈W a∈S ht(α) ht(α) χ(α∈B) Y (tαt )∞(t q /tα)∞ by the A-type root system. Similar such 1ψ1 = , (tht(α))2 sums are given in [6], [7], [29], [43], [47]. More α∈R+ ∞ involved multiple ψ sums with a Schur or 1 1 where B is a base for R. The parameter q on Macdonald polynomial argument can be found the right is fixed by q = Q exp(n a), where in [12], [36], [42], [52]. For r = (r , . . . , r ) ∈ Zn a∈B(S) a 1 n B(S) is a basis for S and the n are the labels denote |r| := r + · · · + r . Then a 1 n of the extended Dynkin diagrams given in [38]. n ri − rj If R is simply-laced, then t = t. In the case X |r| Y xiq xj q Y (aj xij )ri a z (1) x − x (b x ) of S(R) = A1 , q = exp(a0 + a1) so that after r∈Zn 1≤i

1 1 courtesy of Krishnaswami Alladi. type integrals. The most important such integrals are due to Aomoto [6], [7] and Ito [33], and are Photo closely related to (5). Further examples may be The museum at SASTRA University in found in [37], [51]. Kumbakonam, where several manuscripts, letters, papers, documents, photographs, etc., References relating to Ramanujan are displayed. [1] C. Adiga, On the representations of an integer as a sum of two or four triangular numbers, Nihonkai Math. J. 3 (1992), 125–131. [18] W. Y. C. Chen, and E. X. W. Xia, The q-WZ method for [2] C. Adiga, B. C. Berndt, S. Bhargava, and G. N. Wat- inﬁnite series, J. Symbolic Comput. 44 (2009), 960–971. son, Chapter 16 of Ramanujan’s second notebook: [19] X. Chen, Q. Liu and X. Ma, A short proof of Ramanu- theta-functions and q-series, Mem. Amer. Math. Soc. jan’s 1ψ1 formula, J. Math. Res. Exposition 28 (2008), 53 (1985), no. 315. 46–48. [3] G. E. Andrews, On Ramanujan’s 1ψ1(a; b; z), Proc. [20] W. Chu, Abel’s lemma on summation by parts and Amer. Math. Soc. 22 (1969), 552–553. Ramanujan’s 1ψ1-series identity, Aequationes Math. [4] , Ramanujan’s “lost” notebook. I. Partial 72 (2006), 172–176. θ-functions, Adv. in Math. 41 (1981), 137–172. [21] S. Cooper, On sums of an even number of squares, [5] G. E. Andrews and R. Askey, A simple proof of and an even number of triangular numbers: an elemen- Ramanujan’s 1ψ1, Aequationes Math. 18 (1978), tary approach based on Ramanujan’s 1ψ1 summation 333–337. formula, Contemp. Math., vol. 291, Amer. Math. Soc., [6] K. Aomoto, On a theta product formula for the sym- Providence, RI, 2001, pp. 115–137. metric A-type connection function, Osaka J. Math. 32 [22] S. Cooper and H. Y. Lam, Sums of two, four, six and (1995), 35–39. eight squares and triangular numbers: an elementary [7] , On elliptic product formulas for Jackson approach, Indian J. Math. 44 (2002), 21–40. integrals associated with reduced root systems, J. [23] S. Corteel, Particle seas and basic hypergeometric Algebraic Combin. 8 (1998), 115–126. series, Adv. Appl. Math. 31 (2003), 199–214. [8] R. Askey, The q-gamma and q-beta functions, [24] S. Corteel and J. Lovejoy, Frobenius partitions and Applicable Anal. 8 (1978/79), 125–141. the combinatorics of Ramanujan’s 1ψ1 summation, [9] , Ramanujan’s extensions of the gamma and J. Combin. Theory Ser. A 97 (2002), 177–183. beta functions, Amer. Math. Monthly 87 (1980), 346– [25] N. J. Fine, Basic Hypergeometric Series and Applica- 359. tions, Mathematical Surveys and Monographs, vol. 27, [10] , An elementary evaluation of a beta type Amer. Math. Soc., Providence, RI, 1988. integral, Indian J. Pure Appl. Math. 14 (1983), 892–895. [26] S. Fishel, I. Grojnowski, and C. Teleman, The strong [11] R. Askey and J. Wilson, Some basic hypergeomet- Macdonald conjecture and Hodge theory on the loop ric orthogonal polynomials that generalize Jacobi Grassmannian, Ann. of Math. (2) 168 (2008), 175–220. polynomials, Mem. Amer. Math. Soc. 54 (1985), [27] V. J. Guo and M. J. Schlosser, Curious extensions of no. 319. Ramanujan’s 1ψ1 summation formula, J. Math. Anal. [12] T. H. Baker and P. J. Forrester, Transformation for- Appl. 334 (2007), 393–403. mulas for multivariable basic hypergeometric series, [28] R. A. Gustafson, Multilateral summation theorems Methods Appl. Anal. 6 (1999), 147–164. for ordinary and basic hypergeometric series in U(n), [13] B. C. Berndt, Ramanujan’s Notebooks. Part III, SIAM J. Math. Anal. 18 (1987), 1576–1596. Springer-Verlag, New York, 1991. [29] R. A. Gustafson and C. Krattenthaler, Heine trans- [14] S. Bhargava and C. Adiga, Simple proofs of Jacobi’s formations for a new kind of basic hypergeometric two and four square theorems, Int. J. Math. Ed. Sci. series in U(n), J. Comput. Appl. Math. 68 (1996), Tech. 19 (1988), 779–782. 151–158. [15] S. H. Chan, An elementary proof of Jacobi’s six squares [30] W. Hahn, Beiträge zur Theorie der Heineschen Reihen, theorem, Amer. Math. Monthly 111 (2004), 806–811. Math. Nachr. 2 (1949), 340–379. [16] , A short proof of Ramanujan’s famous 1ψ1 [31] G. H. Hardy, Ramanujan. Twelve Lectures on Subjects summation formula, J. Approx. Theory 132 (2005), Suggested by His Life and Work, Cambridge University 149–153. Press, Cambridge, UK; Macmillan Company, New York, [17] S. H. L. Chen, W. Y. C. Chen, A. M. Fu, and W. J. T. Zang, 1940. The Algorithm Z and Ramanujan’s 1ψ1 summation, [32] M. E. H. Ismail, A simple proof of Ramanujan’s 1ψ1 Ramanujan J. 25 (2011), 37–47. sum, Proc. Amer. Math. Soc. 63 (1977), 185–186.

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22 Notices of the AMS Volume 60, Number 1