Proc. Natl. Acad. Sci. USA Vol. 93, pp. 15004–15008, December 1996 Mathematics New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function (Jacobi continued fractionsyHankel or Tura´nian determinantsyFourier seriesyLambert seriesySchur functions) 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, October 22, 1996 (received for review June 24, 1996) ABSTRACT In this paper, we give two infinite families of of the constant ZN, which occurs in the evaluation of a certain explicit exact formulas that generalize Jacobi’s (1829) 4 and Epstein zeta function, needed in the study of the stability of 8 squares identities to 4n2 or 4n(n 1 1) squares, respectively, rare gas crystals and in that of the so-called Madelung con- without using cusp forms. Our 24 squares identity leads to a stants of ionic salts. different formula for Ramanujan’s tau function t(n), when n The s squares problem is to count the number rs(n) of integer is odd. These results arise in the setting of Jacobi elliptic solutions (x1,...,xs) of the Diophantine equation functions, Jacobi continued fractions, Hankel or Tura´nian 2 2 determinants, Fourier series, Lambert series, inclusiony x1 1 zzz1 xs 5 n, [1] exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional in which changing the sign or order of the xi’s gives distinct infinite families of identities in this same setting that are solutions. analogous to the h-function identities in appendix I of Mac- Diophantus (325–409 A.D.) knew that no integer of the donald’s work [Macdonald, I. G. (1972) Invent. Math. 15, form 4n 2 1 is a sum of two squares. Girard conjectured in 91–143]. A special case of our methods yields a proof of the two 1632 that n is a sum of two squares if and only if all prime conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress divisors q of n with q [ 3 (mod 4) occur in n to an even power. in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, Fermat in 1641 gave an ‘‘irrefutable proof’’ of this conjecture. V. & Kac, V. (Birkha¨user Boston, Boston, MA), Vol. 123, pp. Euler gave the first known proof in 1749. Early explicit 415–456] identities involving representing a positive integer formulas for r2(n) were given by Legendre in 1798 and Gauss by sums of 4n2 or 4n(n 1 1) triangular numbers, respectively. in 1801. It appears that Diophantus was aware that all positive Our 16 and 24 squares identities were originally obtained via integers are sums of four integral squares. Bachet conjectured multiple basic hypergeometric series, Gustafson’s Cø nonter- this result in 1621, and Lagrange gave the first proof in 1770. minating 6f5 summation theorem, and Andrews’ basic hyper- Jacobi, in his famous Fundamenta Nova (1) of 1829, intro- geometric series proof of Jacobi’s 4 and 8 squares identities. duced elliptic and theta functions, and used them as tools in the We have (elsewhere) applied symmetry and Schur function study of Eq. 1. Motivated by Euler’s work on 4 squares, Jacobi techniques to this original approach to prove the existence of knew that the number rs(n) of integer solutions of Eq. 1 was similar infinite families of sums of squares identities for n2 or also determined by n(n 1 1) squares, respectively. Our sums of more than 8 ` squares identities are not the same as the formulas of Mathews s n n q3~0, 2q! :5 1 1 O~21! rs~n!q , [2] (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, n51 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others. where q3(0, q)isthez50 case of the theta function q3(z, q) in ref. 21 given by ` 1. Introduction j2 q3~0, q! :5 O q . [3] j52` In this paper, we announce two infinite families of explicit exact formulas that generalize Jacobi’s (1) 4 and 8 squares Jacobi then used his theory of elliptic and theta functions 2 to derive remarkable identities for the s 5 2, 4, 6, 8 cases of identities to 4n or 4n(n 1 1) squares, respectively, without s using cusp forms. Our 24 squares identity leads to a different q3(0, 2q) . He immediately obtained elegant explicit formulas formula for Ramanujan’s (2) tau function t(n), when n is odd. for rs(n), where s 5 2, 4, 6, 8. We recall Jacobi’s identities for These results arise in the setting of Jacobi elliptic functions, s 5 4 and 8 in the following theorem. Jacobi continued fractions, Hankel or Tura´nian determinants, THEOREM 1.1 (JACOBI). Fourier series, Lambert series, inclusionyexclusion, Laplace ` rqr 4 r21 expansion formula for determinants, and Schur functions. (For q3~0, 2q! 5 1 2 8 ~21! r this background material, see refs. 1 and 3–16.) rO51 1 1 q The problem of representing an integer as a sum of squares ` of integers has had a long and interesting history, which is 5 1 1 8 O~21!n O d qn, [4] surveyed in ref. 17 and chapters 6–9 of ref. 18. The review n51 F dun,d.0 G 4Bd article (19) presents many questions connected with represen- tations of integers as sums of squares. Direct applications of and sums of squares to lattice point problems and crystallography ` r3qr can be found in ref. 20. One such example is the computation q ~0, 2q!8 5 1 1 16 ~21!r 3 O r r51 1 2 q The publication costs of this article were defrayed in part by page charge ` payment. This article must therefore be hereby marked ‘‘advertisement’’ in 5 1 1 16 (21)dd3 qn. [5] accordance with 18 U.S.C. §1734 solely to indicate this fact. O O n51Fdun,d.0 G 15004 Downloaded by guest on September 28, 2021 Mathematics: Milne Proc. Natl. Acad. Sci. USA 93 (1996) 15005 Consequently, we have where r ~n! 5 8 d and r ~n! 5 16 ~21!n1dd3, [6] ` rsqr 4 O 8 O r21 dun,d.0 dun,d.0 U [ U (q) :5 (21) s s O r 4Bd r51 11q respectively. ` In general it is true that 5 O O (21)d1nydds qn n51Fdun,d.0 G r2s~n! 5 d2s~n! 1 e2s~n!, [7] 5 y11m1 s m1y1 [12] O (21) m1q . y1,m1$1 where d2s(n) is a divisor function and e2s(n) is a function of order substantially lower than that of d2s(n). If 2s 5 2, 4, 6, 8, Analogous to Theorem 1.3, we have Theorem 1.5. then e (n) 5 0, and Eq. 7 becomes Jacobi’s formulas for r (n), 2s 2s THEOREM 1.5. including Eq. 6. On the other hand, if 2s . 8 then e2s(n) is never n 0. The function e2s(n) is the coefficient of q in a suitable ‘‘cusp 16 form.’’ The difficulties of computing Eq. 7, especially the q ~0, 2q!24 5 1 1 ~17G 1 8G 1 2G ! 3 9 3 5 7 nondominate term e2s(n), increase rapidly with 2s. The mod- ular function approach to Eq. 7 and the cusp form e2s(n)is 512 2 discussed in ref. 13. For 2s . 8, modular function methods such 1 ~G3G7 2 G5!, [13] as those in refs. 22–27, or the more classical elliptic function 9 approach of refs. 28–30, are used to determine general for- where mulas for d2s(n) and e2s(n)inEq.7. Explicit, exact examples of Eq. 7 have been worked out for 2 # 2s # 32. Similarly, explicit ` rsqr r formulas for rs(n) have been found for (odd) s , 32. Alternate, Gs [ Gs(q) :5 (21) r elementary approaches to sums of squares formulas can be rO51 12q found in refs. 31–36. ` We next consider classical analogs of Eqs. 4 and 5 corre- 5 O O (21)dds qn sponding to the s 5 8 and 12 cases of Eq. 7. n51Fdun,d.0 G Glaisher (37, 62–64) used elliptic function methods rather 5 m1 s m1y1 [14] than modular functions to prove the following theorem. O (21) m1q . y ,m $1 THEOREM 1.2 (GLAISHER). 1 1 32 An analysis of Eq. 10b depends upon Ramanujan’s (2) tau q 16 5 1 m1 7 m1y1 [8a] function t(n), defined by 3~0, 2q! 1 O ~21! m1q 17 y1,m1$1 ` 512 q~q; q!24 :5 t(n)qn. [15] 8 2 2 8 ` O 2 q~q; q) (q ;q! , [8b] n 1 17 ` ` 5 For example, t(1) 5 1, t(2) 5224, t(3) 5 252, t(4) 521472, where we have t(5) 5 4830, t(6) 526048, and t(7) 5216744. Ramanujan r (ref. 2, equation 103) conjectured, and Mordell (38) proved, ~q; q!` :5 P(12q ). [9] r$1 that t(n) is multiplicative. In the case where n is an odd integer (in particular an odd n Glaisher took the coefficient of q to obtain r16(n). The same prime), equating Eqs. 10a, 10b, and 13 yields two formulas for formula appears in ref. 13 (equation 7.4.32). t(n) that are different from Dyson’s (39) formula. We first To find r24(n), Ramanujan (ref. 2, entry 7, table VI; see also obtain Theorem 1.6.