ON SUMS OF THREE SQUARES

Transcription

1 ON SUS OF THREE SQUARES S.K.K. CHOI, A.V. KUCHEV, AND R. OSBURN Abstract. Let r 3 (n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of Wagon concerning the asymptotic value of the mean square of r 3 (n). 1. Introduction Problems concerning sums of three squares have a rich history. It is a classical result of Gauss that n = x 1 + x + x 3 has a solution in integers if and only if n is not of the form 4 a (8k + 7) with a, k Z. Let r 3 (n) be the number of representations of n as a sum of three squares (counting signs and order). It was conjectured by Hardy and proved by Bateman [1] that (1) r 3 (n) = 4πn 1/ S 3 (n), the singular series S 3 (n) is given by (16) with Q =. While in principle this exact formula can be used to answer almost any question concerning r 3 (n), the ensuing calculations can be tricky because of the slow convergence of the singular series S 3 (n). Thus, one often sidesteps (1) and attacks problems involving r 3 (n) directly. For example, concerning the mean value of r 3 (n), one can adapt the method of solution of the circle problem to obtain the following r 3 (n) 4 3 πx3/. oreover, such a direct approach enables us to bound the error term in this asymptotic formula. An application of a result of Landau [9, pp ] yields r 3 (n) = 4 3 πx3/ + O ( x 3/4+ɛ) for all ɛ > 0, and subsequent improvements on the error term have been obtained by Vinogradov [19], Chamizo and Iwaniec [3], and Heath-Brown [6]. In this note we consider the mean square of r 3 (n). The following asymptotic formula was conjectured by Wagon and proved by Crandall (see [4] or []). Date: September 14, athematics Subject Classification. Primary: 11E5, 11P55 Secondary: 11E45. Research of Stephen Choi was supported by NSERC of Canada. 1

2 Theorem. Let r 3 (n) be the number of representations of a positive integer n as a sum of three squares of integers. Then () r 3 (n) 8π4 1ζ(3) x. Apparently, at the time they discussed this conjecture Crandall and Wagon were unaware of the earlier work of üller [11, 1]. He obtained a more general result which, in a special case, gives r 3 (n) = Bx + O ( x 14/9), B is a constant. However, since in üller s work B arises as a specialization of a more general (and more complicated) quantity, it is not immediately clear that B = 8 1 π4 /ζ(3). The purpose of this paper is to give two distinct proofs of this fact: one that evaluates B in the form given by üller and a direct proof using the Hardy Littlewood circle method.. A direct proof: the circle method Our first proof exploits the observation that the left side of () counts solutions of the equation m 1 + m + m 3 = m 4 + m 5 + m 6 in integers m 1,..., m 6 with m j x. This is exactly the kind of problem that the circle method was designed for. The additional constraint m 1 + m + m 3 x causes some technical difficulties, but those are minor. Set N = x and define f (α) = m N e ( αm ), e(z) = e πiz. Then for an integer n x, the number r (n) of representations of n as a sum of three squares of positive integers is r (n) = 1 0 f (α) 3 e( αn) dα. Since r 3 (n) = 8r (n) + O(r (n)), r (n) is the number of representations of n as a sum of two squares, we have (3) r 3 (n) = 64 r (n) + O ( x 3/+ɛ). Therefore, it suffices to evaluate the mean square of r (n). Let We introduce the sets and P = N/4 and Q = N 1/. (q, a) = { α [ Q 1, 1 + Q 1] : qα a PN } = q Q 1 a q (a,q)=1 (q, a), m = [ Q 1, 1 + Q 1] \.

3 We have (4) ( r (n) = + m ) f (α) 3 e( αn) dα = r (n, ) + r (n, m), say. We now proceed to approximate the mean square of r (n) by that of r (n, ). Cauchy s inequality, (5) r (n) = r (n, ) + O ( ) (Σ 1 Σ ) 1/ + Σ, Σ 1 = r (n, ), Σ = r (n, m). By (4) and By Bessel s inequality, (6) Σ = m f (α) 3 e( αn) dα m f (α) 6 dα. By Dirichlet s theorem of diophantine approximation, we can write any real α as α = a/q + β, 1 q N P 1, (a, q) = 1, β P/(qN ). When α m, we have q Q, and hence Weyl s inequality (see Vaughan [18, Lemma.4]) yields (7) f (α) N 1+ɛ( q 1 + N 1 + qn ) 1/ N 1+ɛ Q 1/. Furthermore, we have (8) 1 0 f (α) 4 dα N +ɛ, because the integral on the left equals the number of solutions of m 1 + m = m 3 + m 4 in integers m 1,..., m 4 N. For each choice of m 1 and m, this equation has N ɛ solutions. Combining (6) (8) and replacing ɛ by ɛ/3, we obtain (9) Σ N 4+ɛ Q 1. Furthermore, another appeal to Bessel s inequality and appeals to (8) and to the trivial estimate f (α) N yield 1 (10) Σ 1 f (α) 6 dα f (α) 6 dα N 4+ɛ. here We now define a function f on by setting f (α) = q 1 S (q, a)v(α a/q) for α (q, a) ; S (q, a) = e ( ah /q ), v(β) = 1 m 1/ e(βm). 1 h q 3 0 m x

4 Our next goal is to approximate the mean square of r (n, ) by the mean square of the integral R (n) = f (α) 3 e( αn) dα. Similarly to (5), (11) (1) Σ 3 = r (n, ) = R (n) + O ( Σ 3 + (Σ 1 Σ 3 ) 1/), [ f (α) 3 f (α) 3] e( αn) dα f (α) 3 f (α) 3 dα, after yet another appeal to Bessel s inequality. By [18, Theorem 4.1], when α (q, a), Thus, whence (q,a) f (α) = f (α) + O ( q 1/+ɛ). f (α) 3 f (α) 3 dα q 1+ɛ (q,a) ( f (α) 4 + q +4ɛ) dα, f (α) 3 f (α) 3 1 dα Q 1+ɛ f (α) 4 dα + PQ 4+6ɛ N. Bounding the last integral using (8) and substituting the ensuing estimate into (1), we obtain (13) Σ 3 QN +ɛ + PQ 4 N +3ɛ QN +ɛ. Combining (5), (9) (11), and (13), we deduce that (14) r (n) = R (n) + O ( N 4+ɛ Q 1/ + N 3+ɛ Q 1/). so We now proceed to evaluate the main term in (14). We have f (α) 3 e( αn) dα = q 3 S (q, a) 3 e( an/q) (q,a) R (n) = q Q A(q, n)i(q, n), 0 (q,0) v(β) 3 e( βn) dβ, Hence, (15) A(q, n) = 1 a q (a,q)=1 q 3 S (q, a) 3 e( an/q), I(q, n) = (q,0) R (n) = I(n) S 3 (n, Q) + O ( ) (Σ 4 Σ 5 ) 1/ + Σ 5, 4 v(β) 3 e( βn) dβ.

5 (16) S 3 (n, Q) = A(q, n), q Q I(n) = 1/ 1/ v(β) 3 e( βn) dβ, Σ 4 = ( ) I(n) A(q, n), Σ 5 = ( A(q, n)(i(n) I(q, n)) ). q Q q Q By [18, Theorem.3] and [18, Theorem 4.], (17) I(n) = Γ(3/) n + O(1) = π n + O(1), A(q, n) q 1/. 4 Furthermore, since A(q, n) is multiplicative in q, [18, Lemma 4.7] yields A(q, n) ( (18) 1 + A(p, n) + A(p, n) + ) q Q p Q ( 1 + c1 (p, n)p 3/ + 3c 1 p 1) (nq) ɛ, p Q c 1 > 0 is an absolute constant. In particular, we have (19) Σ 4 N 4+ɛ. We now turn to the estimation of Σ 5. By Cauchy s inequality and the second bound in (17), Σ 5 (log Q) I(n) I(n, q) Another application of Bessel s inequality gives q Q I(n) I(n, q) 1/ P/qN v(β) 6 dβ. Using [18, Lemma.8] to estimate the last integral, we deduce that Σ 5 log Q q Q ( q N 4 P + 1 ) N Q 3+ɛ. Substituting this inequality and (19) into (15), we conclude that (0) R (n) = I(n) S 3 (n, Q) + O ( N 3+ɛ Q 3/). We then use (17) and (18) to replace I(n) on the right side of (0) by π 4 n. We get I(n) S 3 (n, Q) = π 16 ns 3 (n, Q) + O ( N 3+ɛ). Together with (14) and (0), this leads to the asymptotic formula (1) r (n) = π 16 ns 3 (n, Q) + O ( N 4+ɛ Q 1/ + N 3+ɛ Q 3/). 5

6 Finally, we evaluate the sum on the right side of (1). On observing that S 3 (n, Q) is in fact a real number, we have S 3 (n, Q) = (q 1 q ) 3 S (q 1, a 1 ) 3 S (q, a ) 3 e ( (a 1 /q 1 a /q )n ). n t n t q 1,q Q 1 a 1 q 1 1 a q (a 1,q 1 )=1 (a,q )=1 As the sum over n equals t + O(1) when a 1 = a and q 1 = q and O(q 1 q ) otherwise, we get S 3 (n, Q) = t q 6 S (q, a) 6 + O ( Σ6), n t q Q We find that with Σ 6 = q Q 1 a q (a,q)=1 1 a q (a,q)=1 q S (q, a) 3 Q 3/. S 3 (n, Q) = B 1 t + O ( tq 1 + Q 3), n t B 1 = q=1 1 a q (a,q)=1 q 6 S (q, a) 6. Thus, by partial summation, ns 3 (n, Q) = (B 1 /)x + O ( x Q 1 + xq 3). Combining this asymptotic formula with (1), we deduce that r (n) = π 3 B 1x + O ( x 15/8+ɛ). Recalling (3), we see that () will follow if we show that B 1 = 8ζ() 7ζ(3). This, however, follows easily from the well-known formula (see [7, 7.5]) q if q 1 (mod ), () S (q, a) = q if q 0 (mod 4), 0 if q (mod 4). Indeed, () yields B 1 = 4 3 q odd q 3 φ(q) = 8ζ() 7ζ(3), the last step uses the Euler product of ζ(s). This completes the proof of our theorem. 6

7 3. Second Proof of Theorem Rankin [13] and Selberg [17] independently introduced an important method which allows one to study the analytic behavior of the Dirichlet series a(n) n s n=1 a(n) are Fourier coefficients of a holomorphic cusp form for some congruence subgroup of Γ = S L (Z). Originally the method was for holomorphic cusp forms. Zagier [0] extended the method to cover forms that are not cuspidal and may not decay rapidly at infinity. üller [11, 1] considered the case a(n) is the Fourier coefficient of non-holomorphic cusp or non-cusp form of real weight with respect to a Fuchsian group of the first kind. It is this last approach we wish to discuss. Note that if we apply a Tauberian theorem to the above Dirichlet series, we then gain information on the asymptotic behavior of the partial sum a(n). We now discuss üller s elegant work. For details regarding discontinuous groups and automorphic forms, see [8, 10, 11, 14, 15, 16]. Let H = {z C : I(z) > 0} denote the upper half plane and G = S L(, R) the special linear group of all matrices with determinant 1. G acts on H by z gz = az + b ( ) cz + d a b for g = G. We write y = y(z) = I(z). Thus we have c d y y(gz) = cz + d. Let dx dy denote the Lebesgue measure in the plane. Then the measure dµ = dx dy y is invariant under the action of G on H. A discrete subgroup Γ of G is called a Fuchsian group of the first kind if its fundamental domain Γ\H has finite volume. Let Γ be a Fuchsian group of the first kind containing ±I I is the identity matrix. Let F (Γ, χ, k, λ) denote the space of (non-holomorphic) automorphic forms of real weight k, eigenvalue λ = 1 4 ρ, R(ρ) 0, and multiplier system χ. For k R, g S L(, R) and f : H C, we define the stroke operator k by ( ) k cz + d ( f k g)(z) := f (gz) cz + d ( ) a b g = Γ. The transformation law for f F (Γ, χ, k, λ) is then c d ( f k g)(z) = χ(g) f (z) for all g Γ. Automorphic forms f F (Γ, χ, k, λ) have a Fourier expansion at every cusp κ of Γ, namely A κ,0 (y) + n 0 a κ,n W (sgn n) k,ρ(4π n + µ κ y)e((n + µ k )x), µ κ is the cusp parameter and a κ,n are the Fourier coefficients of f at κ. The functions W α,ρ are Whittaker functions (see [11, 3]), A κ,0 (y) = 0 if µ k 0 and 7

8 { aκ,0 y A κ,0 (y) = 1/+ρ + b κ,0 y 1/ ρ if µ κ = 0, ρ 0, a κ,0 y 1/ + b κ,0 y 1/ log y if µ κ = 0, ρ = 0. An automorphic form f is called a cusp form if a κ,0 = b κ,0 = 0 for all cusps κ of Γ. Now consider the Dirichlet series S κ ( f, s) = a κ,n (n + µ n>0 κ ). s This series is absolutely convergent for R(s) > R(ρ) and has been shown [1] to have meromorphic continuation in the entire complex plane. In what follows, we will only be interested in the case f is not a cusp form. If f is not a cusp form and R(ρ) > 0, then S κ ( f, s) has a simple pole at s = R(ρ) with residue (3) β κ ( f ) = res S κ ( f, s) = (4π) R(ρ) b + (k/, ρ) s=r(ρ) ι K ϕ κ,ι (1 + R(ρ)) a ι,0, K denotes a complete set of Γ-inequivalent cusps, ϕ κ,ι (1 + R(ρ)) > 0 and b + ( k, ρ) > 0 if ρ + 1 ± k is a non-negative integer. For the definition of the functions ϕ κ,ι and b +, see Lemma 3.6 and (69) in [1]. This result (3) and a Tauberian argument then provide the asymptotic behaviour of the summatory function a κ,n n + µ κ r. Precisely, we have (see [11, Theorem.1] or [1, Theorem 5.]) that (4) a κ,n n + µ κ r = res S κ( f, s) xr+s s=z r + s + O(xr+Rρ γ (log x) g ), z R R(ρ) + r 0, R = {±R(ρ), ±ii(ρ), 0, r}, γ = ( + 8R(ρ))(5 + 16R(ρ)) 1, and g =max(0, b 1); b denotes the order of the pole of S κ ( f, s)(r + s) 1 x r+s at s = R(ρ) (0 b 5). We now consider an application of (4). Let Q Z m m be a non-singular symmetric matrix with even diagonal entries and q(x) = 1 Q[x] = 1 xt Qx, x Z m, the associated quadratic form in m 3 variables. Here we assume that q(x) is positive definite. Let r(q, n) denote the number of representations of n by the quadratic form Q. Now consider the theta function θ Q (z) = x Z m e πizq[x]. By [11, Lemma 6.1], the Dirichlet series associated with the automorphic form θ Q is (4π) m/4 ζ Q ( m 4 + s) ζ Q (s) = n=1 r(q, n) n s = x Z m \{0} q(x) s for R(s) > m/. Using (4), üller proved the following (see [11, Theorem 6.1]) Theorem (üller). Let q(x) = 1 Q[x] = 1 xt Qx, x Z m be a primitive positive definite quadratic form in m 3 variables with integral coefficients. Then ( ) r(q, n) = Bx m 1 4m 5 (m 1) + O x 4m 3 8

9 and β (θ Q ) is given by (3). B = (4π) m/ β (θ Q ) m 1 We are now in a position to prove our theorem in page. Proof. We are interested in the case q(x) = x1 + x + x3 and so r(q, n) = r 3 (n) counts the number of representations of n as a sum of three squares. By üller s Theorem above, ) r 3 (n) = Bx + O (x 14/9 B is a computable constant. Specifically, we have by (3) (with k = 3/ and ρ = 1/4) B = 4π 3 1 b+ (3/4, 1/4) ι K ϕ,ι (3/) a ι,0, K denotes a complete set of Γ 0 (4)-inequivalent cusps and a ι,0 is the 0-th Fourier coefficient of θ Q (z) at a rational cusp ι. Choose K = {1, 1, 1 }. Then by p. 145 and (67) in [11], we have 4 a ι,0 = W 3 ι G(S ι ) ι = u/w, (u, w) = 1, w 1, W ι is width of the cusp ι, and G(S ι ) = 3 w 3 w e( u w x ) 6. As W 1/4 = W 1/ = 1, W 1 = 4, we have a 1,0 = 1, a 1/,0 = 0, and a 1/4,0 = 1. An explicit description of the functions ϕ,ι (s) in the case Γ 0 (4) is given by (see (1.17) and p. 47 in [5]) ϕ,1/4 (s) = 1 4s (1 s ) 1 1/ Γ(s 1/)ζ(s 1) π, Γ(s)ζ(s) ϕ,1/ (s) = ϕ,1 (s) = s (1 s ) 1 (1 1 s 1/ Γ(s 1/)ζ(s 1) )π. Γ(s)ζ(s) Thus for s = 3/, we have Now, from p. 65 in [1], we have By Lemma 3.3 and (16) in [1], x=1 ϕ,1/4 (3/) = 5 (1 3 ) 1 π ζ() Γ(3/)ζ(3), ϕ,1/ (3/) = ϕ,1 (3/) = 3 (1 3 ) 1 (1 )π ζ() Γ(3/)ζ(3). b + (3/4, 1/4) = G 1/4,1/4(3/). G 1/4,1/4(s) = Γ(s + 1/) 1 9

10 and so b + (3/4, 1/4) = Γ() 1. In total, Thus ( B = (4π) 1 3 (1 3 ) 1 (1 )π 1/ ζ() (3 1) Γ() Γ(3/)ζ(3) + 5 (1 3 ) 1 π 1/ ζ() Γ(3/)ζ(3) = 8π4 1ζ(3). r 3 (n) ) 8π4 1ζ(3) x. Remark. üller s Theorem can also be used to obtain the mean square value of sums of N > 3 squares. Precisely, if r N (n) is the number of representations of n by N > 3 squares, then a calculation similar to the second proof of our theorem yields (compare with Theorem 3.3 in []) ( ) r N (n) = W N x N 1 4N 5 (N 1) + O x 4N 3 W N = π N 1 ζ(n 1). (N 1)(1 N ) Γ(N/) ζ(n) Acknowledgments The authors would like to thank Wolfgang üller for his comments regarding the second proof of the theorem. The second author would like to take this opportunity to express his gratitude to the athematics Department at the University of Texas at Austin for the support during the past three years. The third author would like to thank the ax-planck-institut für athematik for their hospitality and support during the preparation of this paper. References [1] P.T. Bateman, On the representations of a number as the sum of three squares, Trans. Amer. ath. Soc. 71 (1951), [] J.. Borwein, S.K.K. Choi, On Dirichlet Series for sums of squares, Ramanujan J. 7 (003), [3] F. Chamizo, H. Iwaniec, On the sphere problem, Rev. at. Iberoamericana 11 (1995), [4] R. Crandall, S. Wagon, Sums of squares: computational aspects, manuscript (001). [5] J. Deshouillers, H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. ath. 70 (198), [6] D.R. Heath-Brown, Lattice points in the sphere, in Number Theory in Progress, vol., eds. K. Györy et al., 1999, pp [7] L.K. Hua, Introduction to Number Theory, Springer-Verlag, 198. [8] T. Kubota, Elementary Theory of Eisenstein Series, Wiley Halsted, New York, [9] E. Landau, Collected Works, vol. 6, Thales-Verlag, Essen, [10] H. aass, Lectures on odular Forms of One Complex Variable, Tata Institute, Bombay, [11] W. üller, The mean square of Dirichlet series associated with automorphic forms, onatsh. ath. 113 (199),

11 [1] W. üller, The Rankin-Selberg ethod for non-holomorphic automorphic forms, J. Number Theory 51 (1995), [13] R.A. Rankin, Contributions to the theory of Ramanujan s function τ(n) and similar functions. II. The order of the Fourier coefficients of integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939), [14] R. A. Rankin, odular forms and functions, Cambridge University Press, [15] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, ath. Ann. 167 (1966), [16] W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, II, ath. Ann. 168 (1967), [17] A. Selberg, Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der odulformen nahe verbunden ist, Archiv. ath. Natur. B 43 (1940), [18] R.C. Vaughan, The Hardy Littlewood ethod, nd ed., Cambridge University Press, [19] I.. Vinogradov, On the number of integer points in a sphere, Izv. Akad. Nauk SSSR Ser. ath. 7 (1963), , in Russian. [0] D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA ath. 8 (1981), Department of athematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 address: kkchoi@cecm.sfu.ca Department of athematics, 1 University Station C100, The University of Texas at Austin, Austin, TX 7871, U.S.A. address: kumchev@math.utexas.edu ax-planck Institut für athematik, Vivatsgasse 7, Bonn, Germany address: osburn@mpim-bonn.mpg.de 11