pmc logo image
Logo of pnasPNAS Home page.Reference to the article.PNAS Info for AuthorsPNAS SubscriptionsPNAS About
Journal List > Proc Natl Acad Sci U S A > v.93(26); Dec 24, 1996

Formats:
Proc Natl Acad Sci U S A. 1996 December 24; 93(26): 15004–15008.
PMCID: PMC26345
Copyright © 1996, The National Academy of Sciences of the USA
Mathematics
New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function
Stephen C. Milne
Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210
Communicated by Walter Feit, Yale University, New Haven, CT
Received June 24, 1996; Accepted October 22, 1996.
Abstract
In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s C[ell] nonterminating 6[var phi]5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.
Keywords: Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, Schur functions
1. Introduction
In this paper, we announce two infinite families of explicit exact formulas that generalize Jacobi’s (1) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s (2) tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. (For this background material, see refs. 1 and 3–16.)
The problem of representing an integer as a sum of squares of integers has had a long and interesting history, which is surveyed in ref. 17 and chapters 6–9 of ref. 18. The review article (19) presents many questions connected with representations of integers as sums of squares. Direct applications of sums of squares to lattice point problems and crystallography can be found in ref. 20. One such example is the computation of the constant ZN, which occurs in the evaluation of a certain Epstein zeta function, needed in the study of the stability of rare gas crystals and in that of the so-called Madelung constants of ionic salts.
The s squares problem is to count the number rs(n) of integer solutions (x1, … , xs) of the Diophantine equation
equation M1
1
in which changing the sign or order of the xi’s gives distinct solutions.
Diophantus (325–409 A.D.) knew that no integer of the form 4n − 1 is a sum of two squares. Girard conjectured in 1632 that n is a sum of two squares if and only if all prime divisors q of n with q [equivalent] 3 (mod 4) occur in n to an even power. Fermat in 1641 gave an “irrefutable proof” of this conjecture. Euler gave the first known proof in 1749. Early explicit formulas for r2(n) were given by Legendre in 1798 and Gauss in 1801. It appears that Diophantus was aware that all positive integers are sums of four integral squares. Bachet conjectured this result in 1621, and Lagrange gave the first proof in 1770.
Jacobi, in his famous Fundamenta Nova (1) of 1829, introduced elliptic and theta functions, and used them as tools in the study of Eq. 1. Motivated by Euler’s work on 4 squares, Jacobi knew that the number rs(n) of integer solutions of Eq. 1 was also determined by
equation M2
2
where [theta]3(0, q) is the z = 0 case of the theta function [theta]3(z, q) in ref. 21 given by
equation M3
3
Jacobi then used his theory of elliptic and theta functions to derive remarkable identities for the s = 2, 4, 6, 8 cases of [theta]3(0, −q)s. He immediately obtained elegant explicit formulas for rs(n), where s = 2, 4, 6, 8. We recall Jacobi’s identities for s = 4 and 8 in the following theorem.
Theorem 1.1 (Jacobi).
equation M4
equation M5
4
and
equation M6
equation M7
5
Consequently, we have
equation M8
6
respectively.
In general it is true that
equation M9
7
where δ2s(n) is a divisor function and e2s(n) is a function of order substantially lower than that of δ2s(n). If 2s = 2, 4, 6, 8, then e2s(n) = 0, and Eq. 7 becomes Jacobi’s formulas for r2s(n), including Eq. 6. On the other hand, if 2s > 8 then e2s(n) is never 0. The function e2s(n) is the coefficient of qn in a suitable “cusp form.” The difficulties of computing Eq. 7, especially the nondominate term e2s(n), increase rapidly with 2s. The modular function approach to Eq. 7 and the cusp form e2s(n) is discussed in ref. 13. For 2s > 8, modular function methods such as those in refs. 22–27, or the more classical elliptic function approach of refs. 28–30, are used to determine general formulas for δ2s(n) and e2s(n) in Eq. 7. Explicit, exact examples of Eq. 7 have been worked out for 2 ≤ 2s ≤ 32. Similarly, explicit formulas for rs(n) have been found for (odd) s < 32. Alternate, elementary approaches to sums of squares formulas can be found in refs. 31–36.
We next consider classical analogs of Eqs. 4 and 5 corresponding to the s = 8 and 12 cases of Eq. 7.
Glaisher (37, 62–64) used elliptic function methods rather than modular functions to prove the following theorem.
Theorem 1.2 (Glaisher).
equation M10
8a
equation M11
8b
where we have
equation M12
9
Glaisher took the coefficient of qn to obtain r16(n). The same formula appears in ref. 13 (equation 7.4.32).
To find r24(n), Ramanujan (ref. 2, entry 7, table VI; see also ref. 13, equation 7.4.37) first proved Theorem 1.3.
Theorem 1.3 (Ramanujan). Let (q; q)∞ be defined by Eq. 9. Then
equation M13
10a
equation M14
10b
One of the main motivations for this paper was to generalize Theorem 1.1 to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms such as Eqs. 8b and 10b but still using just sums of products of at most n Lambert series similar to either Eq. 4 or Eq. 5, respectively. This is done in Theorems 2.1 and 2.2 below. Here, we state the n = 2 cases, which determine different formulas for 16 and 24 squares.
Theorem 1.4.
equation M15
equation M16
11
where
equation M17
equation M18
equation M19
12
Analogous to Theorem 1.3, we have Theorem 1.5.
Theorem 1.5.
equation M20
equation M21
13
where
equation M22
equation M23
equation M24
14
An analysis of Eq. 10b depends upon Ramanujan’s (2) tau function τ(n), defined by
equation M25
15
For example, τ(1) = 1, τ(2) = −24, τ(3) = 252, τ(4) = −1472, τ(5) = 4830, τ(6) = −6048, and τ(7) = −16744. Ramanujan (ref. 2, equation 103) conjectured, and Mordell (38) proved, that τ(n) is multiplicative.
In the case where n is an odd integer (in particular an odd prime), equating Eqs. 10a, 10b, and 13 yields two formulas for τ(n) that are different from Dyson’s (39) formula. We first obtain Theorem 1.6.
Theorem 1.6. Let τ(n) be defined by Eq. 15 and let n be odd. Then
equation M26
equation M27
equation M28
16
where
equation M29
17
Remark: We can use Eq. 16 to compute τ(n) in ≤6n ln n steps when n is an odd integer. This may also be done in n2+[var epsilon] steps by appealing to Euler’s infinite-product-representation algorithm (40) applied to (q; q)equation M30 in Eq. 15.
A different simplification involving a power series formulation of Eq. 13 leads to the following theorem.
Theorem 1.7. Let τ(n) be defined by Eq. 15 and let n ≥ 3 be odd. Then
equation M31
equation M32
18a
equation M33
equation M34
18b
Remark: The inner sum in Eq. 18b counts the number of solutions (y1, y2) of the classical linear Diophantine equation m1y1 + m2y2 = n. This relates Eqs. 18a and 18b to the combinatorics in sections 4.6 and 4.7 of ref. 15.
In the next section, we present the infinite families of explicit exact formulas that generalize Theorems 1.1, 1.4, and 1.5.
Our methods yield (elsewhere) many additional infinite families of identities analogous to the η-function identities in appendix I of Macdonald’s work (41). A special case of our analysis gives a proof (presented elsewhere) of the two identities conjectured by Kac and Wakimoto (42); these identities involve representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. The n = 1 case gives the classical identities of Legendre (ref. 43; see also ref. 3, equations ii and iii).
Theorems 1.4 and 1.5 were originally obtained via multiple basic hypergeometric series (44–51) and Gustafson’s* C[ell] nonterminating 6[var phi]5 summation theorem combined with Andrews’ (52) basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively.
Our sums of more than 8 squares identities are not the same as the formulas of Mathews (31), Glaisher (37, 62–64), Sierpinski (32), Uspensky (33–35), Bulygin (28, 53), Ramanujan (2), Mordell (26, 54), Hardy (23, 24), Bell (55), Estermann (56), Rankin (27, 57), Lomadze (25), Walton (58), Walfisz (59), Ananda-Rau (60), van der Pol (61), Krätzel (29, 30), Gundlach (22), and Kac and Wakimoto (42).
2. The 4n2 and 4n(n + 1) Squares Identities
To state our identities, we first need the Bernoulli numbers Bn defined by
equation M35
19
We also use the notation In := {1, 2, … , n}; ||S|| is the cardinality of the set S, and det(M) is the determinant of the n × n matrix M.
The determinant form of the 4n2 squares identity is Theorem 2.1.
Theorem 2.1. Let n = 1, 2, 3, … . Then
equation M36
equation M37
20
where [theta]3(0, −q) is determined by Eq. 3, and Mn,S is the n × n matrix whose ith row is
equation M38
21
equation M39
where U2i−1 is determined by Eq. 12, and ci is defined by
equation M40
22
with B2i the Bernoulli numbers defined by Eq. 19.
We next have Theorem 2.2.
Theorem 2.2. Let n = 1, 2, 3, … . Then
equation M41
equation M42
23
where [theta]3(0, −q) is determined by Eq. 3, and Mn,S is the n × n matrix whose ith row is
equation M43
24
equation M44
where G2i+1 and ai := ci+1 are determined by Eqs. 14 and 22, respectively.
We next use Schur functions sλ(x1, … , xp) to rewrite Theorems 2.1 and 2.2. Let λ = (λ1, λ2, … , λr, …) be a partition of nonnegative integers in decreasing order, λ1 ≥ λ2 ≥ … ≥ λr … , such that only finitely many of the λi are nonzero. The length [ell](λ) is the number of nonzero parts of λ.
Given a partition λ = (λ1, … , λp) of length ≤p,
equation M45
25
is the Schur function (12) corresponding to the partition λ. [Here, det(aij) denotes the determinant of a p × p matrix with (i, j)th entry aij]. The Schur function sλ(x) is a symmetric polynomial in x1, … , xp with nonnegative integer coefficients. We typically have 1 ≤ p ≤ n.
We use Schur functions in Eq. 25 corresponding to the partitions λ and ν, with
equation M46
equation M47
equation M48
26
where the [ell]r and jr are elements of the sets S and T, with
equation M49
equation M50
27
equation M51
equation M52
28
where Sc := In − S is the compliment of the set S. We also have
equation M53
equation M54
29
Keeping in mind Eqs. 25–29, symmetry and skew-symmetry arguments, row and column operations, and the Laplace expansion formula (9) for a determinant, we now rewrite Theorem 2.1 as Theorem 2.3.
Theorem 2.3. Let n = 1, 2, 3, … . Then
equation M55
equation M56
equation M57
equation M58
equation M59
30
where [theta]3(0, −q) is determined by Eq. 3; the sets S, Sc, T, and Tc are given by Eqs. 27 and 28; Σ(S) and Σ(T) are given by Eq. 29; the (n − p) × (n − p) matrix equation M60 := [equation M61]1≤r,s≤n−p, where the ci are determined by Eq. 22, with the B2i in Eq. 19; and sλ and sν are the Schur functions in Eq. 25, with the partitions λ and ν given by Eq. 26.
We next rewrite Theorem 2.2 as Theorem 2.4.
Theorem 2.4. Let n = 1, 2, 3, … . Then
equation M62
equation M63
equation M64
equation M65
equation M66
31
where the same assumptions hold as in Theorem 2.3, except that the (n − p) × (n − p) matrix equation M67 := [equation M68]1≤r,s≤n−p, where the ai := ci+1 are determined by Eq. 22.
We close this section with some comments about the above theorems. To prove Theorem 2.1, we first compare the Fourier and Taylor series expansions of the Jacobi elliptic function f1(u, k) := sc(u, k)dn(u, k), where k is the modulus. An analysis similar to that in refs. 3, 4, and 16 leads to the relation U2m−1(−q) = cm + dm, for m = 1, 2, 3, … , where U2m−1(−q) and cm are defined by Eqs. 12 and 22, respectively, and dm is given by dm = [(−1)mz2m/22m+1]·(sd/c)m(k2), where z := 2F1(1/2, 1/2; 1; k2) = 2K(k)/π [equivalent] 2K/π, with K(k) [equivalent] K the complete elliptic integral of the first kind in ref. 21, and (sd/c)m(k2) is the coefficient of u2m−1/(2m − 1)! in the Taylor series expansion of f1(u, k) about u = 0.
An inclusion/exclusion argument then reduces the q [not right arrow] −q case of Eq. 20 to finding suitable product formulas for the n × n Hankel determinants det(di+j−1) and det(ci+j−1). Row and column operations immediately imply that
equation M69
32
From theorem 7.9 of ref. 4, we have z = [theta]3(0, q)2, where q = exp[−πK(equation M70)/K(k)]. Setting z = [theta]3(0, q)2 in Eq. 32 and then taking q [not right arrow] −q produces the [theta]3(0, −q)4n2 in Eq. 20. The proof of Theorem 2.1 is complete once we show that
equation M71
and
equation M72
33
By a classical result of Heilermann (7, 8), more recently presented in ref. 10 (theorem 7.14), Hankel determinants whose entries are the coefficients in a formal power series L can be expressed as a certain product of the “numerator” coefficients of the associated Jacobi continued fraction J corresponding to L, provided that J exists. Modular transformations, followed by row and column operations, reduce the evaluation of det[(sd/c)i+j−1(k2)] in Eq. 33 to applying Heilermann’s formula to Rogers’ (14) J-fraction expansion of the Laplace transform of sd(u, k)cn(u, k). The evaluation of det(ci+j−1) can be done similarly, starting with sc(u, k) and the relation sc(u, 0) = tan(u).
The proof of Theorem 2.2 is similar to Theorem 2.1, except that we start with sc2(u, k)dn2(u, k).
Our proofs of the Kac and Wakimoto conjectures do not require inclusion/exclusion, and the analysis involving Schur functions is simpler than in those in Eqs. 30 and 31.
We have (elsewhere) written down the n = 3 cases of Theorems 2.3 and 2.4 which yield explicit formulas for 36 and 48 squares, respectively.
Acknowledgments
This work was partially supported by National Security Agency Grant MDA 904-93-H-3032.
Footnotes
*Gustafson, R. A., Ramanujan International Symposium on Analysis, Dec. 26–28, 1987, Pune, India, pp. 187–224.
References
1. Jacobi, C. G. J. (1829) Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti, Sumptibus fratrum Bornträger; reprinted in Jacobi, C. G. J. (1881–1891) Gesammelte Werke (Reimer, Berlin), Vol. 1, pp. 49–239 [reprinted (1969) by Chelsea, New York; now distributed by Am. Mathematical Soc., Providence, RI].
2. Ramanujan S. Trans Cambridge Philos Soc. 1916;22:159–184.
3. Berndt, B. C. (1991) Ramanujan’s Notebooks, Part III (Springer, New York), p. 139.
4. Berndt B C. In: CRM Proceedings & Lecture Notes. Murty M R, editor. Vol. 1. Providence, RI: Am. Mathematical Soc.; 1993. pp. 1–63.
5. Goulden I P, Jackson D M. Combinatorial Enumeration. New York: Wiley; 1983.
6. Gradshteyn I S, Ryzhik I M. Table of Integrals, Series, and Products. 4th Ed. San Diego: Academic; 1980. ; trans. Scripta TechnicaInc.JeffreyA.
7. Heilermann, J. B. H. (1845) De Transformatione Serierum in Fractiones Continuas, Dr. Phil. dissertation (Royal Academy of Münster).
8. Heilermann J B H. J Reine Angew Math. 1846;33:174–188.
9. Jacobson N. Basic Algebra I. San Francisco: Freeman; 1974. pp. 396–397.
10. Jones W B, Thron W J. In: Encyclopedia of Mathematics and Its Applications. Rota G-C, editor. Vol. 11. London: Addison–Wesley; 1980. (now distributed by Cambridge Univ. PressCambridgeU.K.)244246.
11. Lawden D F. Applied Mathematical Sciences. Vol. 80. New York: Springer; 1989.
12. Macdonald I G. Symmetric Functions and Hall Polynomials. 2nd Ed. Oxford: Oxford Univ. Press; 1995.
13. Rankin R A. Modular Forms and Functions. Cambridge, U.K.: Cambridge Univ. Press; 1977. pp. 241–244.
14. Rogers, L. J. (1907) Proc. London Math. Soc. (Ser. 2) 4, 72–89.
15. Stanley R P. Enumerative Combinatorics. Vol. 1. Belmont, CA: Wadsworth & Brooks Cole; 1986.
16. Zucker I J. SIAM J Math Anal. 1979;10:192–206.
17. Grosswald E. Representations of Integers as Sums of Squares. New York: Springer; 1985.
18. Dickson L E. History of the Theory of Numbers. Vol. 2. New York: Chelsea; 1966.
19. Taussky O. Am Math Monthly. 1970;77:805–830.
20. Glasser M L, Zucker I J. In: Theoretical Chemistry: Advances and Perspectives. Eyring H, Henderson D, editors. Vol. 5. New York: Academic; 1980. pp. 67–139.
21. Whittaker E T, Watson G N. A Course of Modern Analysis. 4th Ed. Cambridge, U.K.: Cambridge Univ. Press; 1927. p. 464. 498.
22. Gundlach K-B. Glasgow Math J. 1978;19:173–197.
23. Hardy G H. Proc Natl Acad Sci USA. 1918;4:189–193. [PubMed]
24. Hardy G H. Trans Am Math Soc. 1920;21:255–284.
25. Lomadze G A. Akad Nauk Gruzin SSR Trudy Tbiliss Mat Inst Razmadze. 1948;16:231–275. (in Russian; Georgian summary).
26. Mordell L J. Q J Pure Appl Math Oxford. 1917;48:93–104.
27. Rankin R A. Acta Arith. 1962;7:399–407.
28. Bulygin V. Bull Acad Imp Sci St Petersbourg Ser VI. 1914;8:389–404. ; announced by Boulyguine, B. (1914) Comptes Rendus Paris 158, 328–330.
29. Krätzel E. Wiss Z Friedrich–Schiller–Univ Jena Math Naturwiss Reihe. 1961;10:33–37.
30. Krätzel E. Wiss Z Friedrich–Schiller–Univ Jena Math Naturwiss Reihe. 1962;11:115–120.
31. Mathews G B. Proc London Math Soc. 1895;27:55–60.
32. Sierpinski W. Wiad Mat Warszawa. 1907;11:225–231. (in Polish).
33. Uspensky J V. Commun. Soc. Math. Kharkow. Ser. 2; 1913. 14 31–64 (in Russian).
34. Uspensky J V. Bull. Acad. Sci. USSR Leningrad. Ser. 6; 1925. 19 647–662 (in French).
35. Uspensky J V. Trans Am Math Soc. 1928;30:385–404.
36. Uspensky J V, Heaslet M A. Elementary Number Theory. New York: McGraw–Hill; 1939.
37. Glaisher J W L. Q J Pure Appl Math Oxford. 1907;38:1–62.
38. Mordell L J. Proc Cambridge Philos Soc. 1917;19:117–124.
39. Dyson F J. Bull Am Math Soc. 1972;78:635–653.
40. Andrews, G. E. (1986) q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, National Science Foundation Conference Board of the Mathematical Sciences Regional Conference Series, Vol. 66, p. 104.
41. Macdonald I G. Invent Math. 1972;15:91–143.
42. Kac V G, Wakimoto M. In: Progress in Mathematics. Brylinski J-L, Brylinski R, Guillemin V, Kac V, editors. Vol. 123. Boston, MA: Birkhäuser Boston; 1994. pp. 415–456.
43. Legendre A M. Traité des Fonctions Elliptiques et des Intégrales Euleriennes. III. Paris: Huzard-Courcier; 1828. pp. 133–134.
44. Lilly G M, Milne S C. Constr Approx. 1993;9:473–500.
45. Milne S C. Adv Math. 1985;57:34–70.
46. Milne S C. J Math Anal Appl. 1987;122:223–256.
47. Milne S C. Discrete Math. 1992;99:199–246.
48. Milne S C. SIAM J Math Anal. 1994;25:571–595.
49. Milne, S. C. (1995) Adv. Math., in press.
50. Milne S C, Lilly G M. Bull Am Math Soc (NS). 1992;26:258–263.
51. Milne S C, Lilly G M. Discrete Math. 1995;139:319–346.
52. Andrews G E. SIAM Rev. 1974;16:441–484.
53. Bulygin, V. (B. Boulyguine) (1915) Comptes Rendus Paris 161, 28–30 (in French).
54. Mordell L J. Trans Cambridge Philos Soc. 1919;22:361–372.
55. Bell E T. Bull Am Math Soc. 1919;26:19–25.
56. Estermann T. Acta Arith. 1936;2:47–79.
57. Rankin R A. Proc Cambridge Philos Soc. 1945;41:1–11.
58. Walton J B. Duke Math J. 1949;16:479–491.
59. Walfisz A Z. Uspehi Mat. Nauk (N.S.). Ser. 6; 1952. 52 91–178 (in Russian); English transl.(1956) Am. Math. Soc. Transl. (Ser. 2) 3 163–248.
60. Ananda-Rau K. J Madras Univ Sect B. 1954;24:61–89.
61. van der Pol B. Ned Akad Wetensch Indag Math. 1954;16:349–361.
62. Glaisher J W L. Q J Pure Appl Math Oxford. 1907;38:178–236.
63. Glaisher J W L. Q J Pure Appl Math Oxford. 1907;38:210.
64. Glaisher J W L. Q J Pure Appl Math Oxford. 1907;38:289–351.
Articles from Proceedings of the National Academy of Sciences of the United States of America are provided here courtesy of
National Academy of Sciences

PubMed articles by these authors

Links
See more articles cited in this paragraph