New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

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_ℓ nonterminating ₆φ₅ 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.

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 Z_N, 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 r_s(n) of integer solutions (x₁, … , x_s) of the Diophantine equation

1

in which changing the sign or order of the x_i’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 ≡ 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 r_s(n) of integer solutions of Eq. 1 was also determined by

2

where ϑ₃(0, q) is the z = 0 case of the theta function ϑ₃(z, q) in ref. 21 given by

3

Jacobi then used his theory of elliptic and theta functions to derive remarkable identities for the s = 2, 4, 6, 8 cases of ϑ₃(0, −q)^s. He immediately obtained elegant explicit formulas for r_s(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).

4

and

5

Consequently, we have

6

respectively.

In general it is true that

7

where δ_2s(n) is a divisor function and e_2s(n) is a function of order substantially lower than that of δ_2s(n). If 2s = 2, 4, 6, 8, then e_2s(n) = 0, and Eq. 7 becomes Jacobi’s formulas for r_2s(n), including Eq. 6. On the other hand, if 2s > 8 then e_2s(n) is never 0. The function e_2s(n) is the coefficient of qⁿ in a suitable “cusp form.” The difficulties of computing Eq. 7, especially the nondominate term e_2s(n), increase rapidly with 2s. The modular function approach to Eq. 7 and the cusp form e_2s(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 δ_2s(n) and e_2s(n) in Eq. 7. Explicit, exact examples of Eq. 7 have been worked out for 2 ≤ 2s ≤ 32. Similarly, explicit formulas for r_s(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).

8a

8b

where we have

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

10a

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.

11

where

12

Analogous to Theorem 1.3, we have Theorem 1.5.

Theorem 1.5.

13

where

14

An analysis of Eq. 10b depends upon Ramanujan’s (2) tau function τ(n), defined by

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

16

where

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 n^2+ɛ steps by appealing to Euler’s infinite-product-representation algorithm (40) applied to (q; q) 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

18a

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_ℓ nonterminating ₆φ₅ 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 B_n defined by

19

We also use the notation I_n := {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

20

where ϑ₃(0, −q) is determined by Eq. 3, and M_n,S is the n × n matrix whose ith row is

21

where U_2i−1 is determined by Eq. 12, and c_i is defined by

22

with B_2i the Bernoulli numbers defined by Eq. 19.

We next have Theorem 2.2.

Theorem 2.2. Let n = 1, 2, 3, … . Then

23

where ϑ₃(0, −q) is determined by Eq. 3, and M_n,S is the n × n matrix whose ith row is

24

where G_2i+1 and a_i := c_i+1 are determined by Eqs. 14 and 22, respectively.

We next use Schur functions s_λ(x₁, … , x_p) to rewrite Theorems 2.1 and 2.2. Let λ = (λ₁, λ₂, … , λ_r, …) be a partition of nonnegative integers in decreasing order, λ₁ ≥ λ₂ ≥ … ≥ λ_r … , such that only finitely many of the λ_i are nonzero. The length ℓ(λ) is the number of nonzero parts of λ.

Given a partition λ = (λ₁, … , λ_p) of length ≤p,

25

is the Schur function (12) corresponding to the partition λ. [Here, det(a_ij) denotes the determinant of a p × p matrix with (i, j)th entry a_ij]. The Schur function s_λ(x) is a symmetric polynomial in x₁, … , x_p 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

26

where the ℓ_r and j_r are elements of the sets S and T, with

27

28

where S^c := I_n − S is the compliment of the set S. We also have

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

30

where ϑ₃(0, −q) is determined by Eq. 3; the sets S, S^c, T, and T^c are given by Eqs. 27 and 28; Σ(S) and Σ(T) are given by Eq. 29; the (n − p) × (n − p) matrix := []_{1≤r,s≤n−p}, where the c_i are determined by Eq. 22, with the B_2i in Eq. 19; and s_λ and s_ν 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

31

where the same assumptions hold as in Theorem 2.3, except that the (n − p) × (n − p) matrix := []_{1≤r,s≤n−p}, where the a_i := c_i+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 U_2m−1(−q) = c_m + d_m, for m = 1, 2, 3, … , where U_2m−1(−q) and c_m are defined by Eqs. 12 and 22, respectively, and d_m is given by d_m = [(−1)^mz^2m/2^2m+1]·(sd/c)_m(k²), where z := ₂F₁(1/2, 1/2; 1; k²) = 2K(k)/π ≡ 2K/π, with K(k) ≡ K the complete elliptic integral of the first kind in ref. 21, and (sd/c)_m(k²) is the coefficient of u^2m−1/(2m − 1)! in the Taylor series expansion of f₁(u, k) about u = 0.

An inclusion/exclusion argument then reduces the q ↛ −q case of Eq. 20 to finding suitable product formulas for the n × n Hankel determinants det(d_i+j−1) and det(c_i+j−1). Row and column operations immediately imply that

32

From theorem 7.9 of ref. 4, we have z = ϑ₃(0, q)², where q = exp[−πK()/K(k)]. Setting z = ϑ₃(0, q)² in Eq. 32 and then taking q ↛ −q produces the ϑ₃(0, −q)⁴ⁿ² in Eq. 20. The proof of Theorem 2.1 is complete once we show that

and

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(c_i+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.