REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES. Ken Ono. Dedicated to the memory of Robert Rankin.

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1 REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES Ken Ono Dedicated to the memory of Robert Rankin.. Introduction and Statement of Results. If s is a positive integer, then let rs; n denote the number of representations of a nonnegative integer n as a sum of s integer squares. If Θz := n= n q n q := e πiz throughout, then. n rs; nq n = Θz s. For small s, there are well known formulas such as Jacobi s four squares theorem: n r4; nq n = + 8 n dq n. The general problem of determining exact formulas for rs; n is classical in number theory. One may consult the popular book by E. Grosswald [G] for a thorough account as of the early 980 s of the subject complete with references. The series Θz s is a modular form, and so there are abstract formulas for rs; n as the Fourier coefficients of modular forms. Specifically, it is well known that Θz s = Es z+c s z, where Es z is an Eisenstein series with explicit coefficients and c s z is a cusp form. Using this fact, one may deduce asymptotic information for rs; n. Rankin proved [R] that c s z is non-trivial for every s > 8. Therefore, the problem of computing non-trivial formulas for rs; n remains since the coefficients of cusp forms, although small, rarely have simple descriptions. In a startling turnabout, Milne [M] announced formulas for r4s ; n and r4s + 4s; n for every s. His formulas were obtained by combining a variety of methods and observations The author thanks the Number Theory Foundation, the Alfred P. Sloan Foundation, the David and Lucile Packard Foundation, and the National Science Foundation for their generous support. d n, 4 d. Typeset by AMS-TEX

2 KEN ONO from the theory of elliptic functions, continued fractions, Lie algebras, Schur functions, and hypergeometric functions. The proofs of his formulas appear in [M]. Also in [M], he proves via similar methods conjectures of Kac and Wakimoto on the number of representations of positive integers as sums of triangular numbers. These conjectures were born out of observations arising in the theory of Lie algebras. In a recent paper [Z], Zagier also proves these conjectures. His method involves an elegant and surprisingly simple argument. Zagier notices that the generating functions in the Kac and Wakimoto Conjectures are modular forms on Γ 0 whose zeros are supported on the cusp at infinity. Two forms sharing this property with the same weight must be multiples of each other. Zagier then observes that the specializations of suitable polynomials with certain Eisenstein series yield such forms. Therefore, these specializations equal the relevant generating functions up to easily computable constants. For r4s ; n and r4s + 4s; n, it turns out that a similar analysis applies. The powers of Θz are modular forms on Γ 0 whose zeros are supported at the cusp inequivalent to infinity. Arguing as above with E ± k; z see.5 and.6 and the polynomials in Zagier s work, one easily obtains new formulas for r4s ; n and r4s +4s; n see Corollary. These formulas are sums of products of divisor functions, and are simpler than those of Milne. His formulas involve Schur functions and determinants of Lambert series. Instead of this approach, we use the fact that the map sending z to /z swaps see Proposition. Θz and the generating function for triangular numbers. Since the fundamental domain of Γ 0 has two cusps which are interchanged by this map, we obtain our formulas from Zagier s work on the Kac-Wakimoto conjectures. This is completely elementary. For every s, let A ± s λ denote the coefficients of the polynomials..3 s i= s i= X i X 3 i i<j s i<j s X i X j = X i X j = A + s λx a Xa s s, A s λx a Xa s s. As usual, let σ ν n := d n dν, and let {B k } denote the Bernoulli numbers defined by.4 B k t k /k! := t/e t. k=0 If k is an even integer, then define weight k modular forms E ± k; z by.5 E + k; z := k B k k + σ k nq 4n k B k k + n σ k nq n.6 E k; z := k B k k + σ k nq n B k k + σ k nq n.

3 Theorem. If s is a positive integer, then.7 Θz 4s = s 4 s s! s.8 Θz 4s +4s +3s = s s! s SUMS OF SQUARES 3 A + s λe + a + ; z E + a s + ; z, A s λe a + ; z E a s + ; z. Corollary. If t is an odd integer, then define divisor functions σ t ± n by { σ t + t t+ Bt+ t+ if n = 0, n := t+ σ t n/4 t n σ t n otherwise, { σt t+ Bt+ t+ if n = 0, n := t+ σ t n/ σ t n otherwise. If s is a positive integer, then for every non-negative integer n we have r4s ; n = s+n 4 s s! s A + s λ σ a + m σ a + s m s, m + +m s =n, m i 0. r4s + 4s; n = n s +3s s! s A s λ. Proofs m + +m s =n, m i 0. σ a m σ a s m s. If k is an even integer, then let G k z denote the weight k Eisenstein series. G k z = B k k + σ k nq n. If k 4, then G k is a weight k modular form on SL Z. As usual, let ηz. ηz := q /4 q n be Dedekind s eta-function. It is well known that.3 Θz = η z/ηz. Similarly, it is also well known that.4 T z := η z ηz = q /8 q n +n/ Up to the factor q /8, T z is the generating function for the triangular numbers..

4 4 KEN ONO Proposition.. If s is a positive integer and Imz > 0, then T /z 4s = s z s s Θz 4s. Proof. In view of.3 and.4, the proposition follows from the fact that [K, p. ]: η /z = z/i ηz. Proposition.. If k 4 is an even integer and Imz > 0, then G k /4z = 4z k G k 4z, G k 4z + = z k G k z +. 0 Proof. Since and SL 0 Z, the modularity of G k z implies.5 G k /z = z k Gz, z.6 G k = z k G k z. z Claim follows from.5, and claim follows by replacing z by z + Proposition.3. If Imz > 0, then G /4z = 4z G 4z + 4z πi. G 4z + = z G z + + 4z πi. 0 Proof. Let S := and T := 0 0 Claim follows from the fact that [K, p. 3].7 G Sz = G /z = z G z + 6z/πi. Since G T z = G z,.7 and in.6. be the standard generators of SL Z. = ST S T, implies z G = z G z + z /πi. z

5 Claim follows by replacing z by z +. SUMS OF SQUARES 5 Proof of Theorem. First we prove.7. If k is even, then define g + k; z by.8 g + k; z := z + G k z/ G k. Zagier [Z] proved that.9 T z 4s = 4 ss s! s A + s λg + a + ; z g + a s + ; z. By replacing z by /z, Proposition. implies s 4 s.0 Θz 4s = z s s! s A + s λg + a +; /z g + a s +; /z. By.8, Proposition. and Proposition.3, we find that g + k; /z = G k /4z G k 4z + = z k k G k 4z k G k z + = z k E + k; z. In view of.0, this implies.7. To prove.8, we begin with Zagier s formula [Z]. If g k; z = G k z G k z, then. T z 4ss+ s = s! s A s λg a + ; z g a s + ; z. By.5, it is easy to see that. g k; /z = k z k G k z z k G k z = z k E k; z. By Proposition.,. and. implies.8. References [G] E. Grosswald, Representations of integers as sums of squares, Springer-Verlag, 984. [K] N. Koblitz, Introduction to elliptic curves and modular forms, Springer-Verlag, 984. [M] S. Milne, New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan s tau function, Proc. Natl. Acad. Sci., USA , [M] S. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Ramanujan J. 6 00, [R] R. Rankin, Sums of squares and cusp forms, Amer. J. Math , [Z] D. Zagier, A proof of the Kac-Wakimoto affine denominator formula for the strange series, Math. Res. Letters 7 000, Department of Mathematics, University of Wisconsin, Madison, Wisconsin address: ono@math.wisc.edu