Eisenstein series, Jacobi theta functions, modular forms, sum of inte- ger squares, Hankel determinant evaluation. The second author was supported by Grant ...
A SHORT PROOF OF MILNE’S FORMULAE FOR SUMS OF INTEGER SQUARES LING LONG AND YIFAN YANG Abstract. We give a short proof of Milne’s formulae for sums of 4n2 and 4n2 + 4n integer squares using the theory of modular forms. Other identities of Milne are also discussed.
1. Introduction Representing a natural number n as the sums of integer squares is a long standing (and difficult) problem dating back to Diophantus (c.f. [2, VI-IX]). Let rk (n) be the number of presentations of n as the sum of k integer squares. In the early 17th century, Bachet conjectured that r4 (n) > 0 for all natural number n. Jacobi generalized Euler’s idea of generating functions and introduced elliptic and theta functions to obtained a precise formula for r4 (n) and also those for r2 (n), r6 (n), and r8 (n) [3, Section 40]. All of his formulae involve the Lambert series. Later, Ramanujan studied r10 (n), r16 (n) etc. by using eta products [2]. Interested readers may consult [5] for an exhaustive list of references. In general, it is hard to find formula for rk (n) when k is large. Kac and Wakimoto, using root systems in Lie superalgebras, have conjectured a surprising formula that leads to various known classical identities as well as some interesting new ones. Among them are conjectured formulae for representing integers as sums of 4n2 and 4n2 + 4n triangular numbers [4]. These conjectured formulae were later established by Zagier [8]. His method was mostly elementary, and used only basic properties of modular forms. Later on, Ono [6] converted these formulae into those for sums of squares by considering a suitable modular transformation. (This particular modular transformation also appears in the present article. See Section 4 for more details.) In his monumental work [5], Milne, combining observations from elliptic functions, continued factions, Lie algebras, hypergeometric functions, etc, obtained new families of exact formulae for 4n2 and 4n2 + 4n squares, along with another proof of Kac-Wakimoto formulae and many other interesting formulae for elliptic functions. One main tool Milne used is Hankel determinant evaluation and hence most of his identities involve determinants of Generalized Lambert series. Date: January 19, 2005. 2000 Mathematics Subject Classification. Primary, secondary. Key words and phrases. Eisenstein series, Jacobi theta functions, modular forms, sum of integer squares, Hankel determinant evaluation. The second author was supported by Grant 93-2115-M-009-014 from the National Science Council (NSC) of Taiwan and by the National Center for Theoretical Sciences (NCTS) of Taiwan. 1
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LING LONG AND YIFAN YANG
More recently, Chan and Chua [1], motivated by Milne’s work, gave a new formula for representing integers as the sums of 32 squares by using sums of products of two Lambert series. They also made a few conjectures along the same line. The main purpose of this article is to give a short proof of Milne’s formulae for sums of 4n2 and 4n2 + 4n integer squares. Theorem 1 (Milne). Let θ3 (q) =
X
2
qn
/2
.
n∈Z
Then, for any positive integer n we have ) ( 2n−1 Y 1 4n2 2n2 +n θ3 (q) = 2 · det(G− 2a+2b−2 )1≤a,b≤n , i! i=1 where ∞
G− k (q) = and
(2k − 1)Bk X rk−1 q r/2 + , 2k 1 + (−1)r q r/2 r=1 (
4n(n+1)
θ3 (q)
=
2n2 +3n
2
2n Y 1 i! i=1
) · det(G+ 2a+2b )1≤a,b≤n ,
where ∞
G+ k (q) = −
(2k − 1)Bk X rk−1 q r/2 . + 2k 1 − (−1)r q r/2 r=1
Here Bk denotes the kth Bernoulli number. (Note that the rk (n) is the coefficient of q n/2 in θ3 (τ )k .) Our method uses only the theory of modular forms. We show that, with the setting of q = e2πiτ , the two sides of either identity above transform in the same way under the action of a certain congruence subgroup of level 2 and of index 3 in SL2 (Z). Then by showing that the two sides have the same zeroes and that the leading Fourier coefficients are identical, we conclude that they are equal. Furthermore, our approach can also be used to prove other identities appearing in Milne’s work. As an example, we give a new proof of Theorems 5.7 and 5.9 of [5] in Section 5. In fact, it is expected that most identities in [5] involving the theta functions can be proved using our approach. The rest of the paper is organized as follows. In Section 2 we discuss the modular properties of the theta functions. In Section 3 we compute the Fourier expansions of the Eisenstein series relevant to our consideration. That is, we show that the functions G± k in Theorem 1 are in fact constant multiples of the Eisenstein series associated with the cusp ∞ and with characters on a certain congruence subgroup of SL2 (Z). In Section 4 we describe our main idea (Proposition 1), and complete the proof of Theorem 1. In the last section, we show that our approach works more generally by giving a new proof of Theorems 5.7 and 5.9 of [5]. Consequently, some Hankel determinant evaluations of Milne which involving Bernoulli numbers and Euler numbers are easily derived. More explicitly, they are obtained by comparing the leading coefficients of two different representations of the same modular forms. See Corollaries 1 and 2.
A SHORT PROOF OF MILNE’S FORMULAE FOR SUMS OF INTEGER SQUARES
3
2. Basic properties of theta functions In this section we shall describe basic properties of the Jacobi theta functions relevant to our consideration. Let q = e2πiτ . The Jacobi theta functions are defined by X η(2τ )2 θ2 (τ ) := q 1/8 q n(n+1)/2 = 2 , η(τ ) n∈Z
θ3 (τ ) :=
X
2
qn
/2
η(τ )5 , η(τ /2)2 η(2τ )2
=
n∈Z
and θ4 (τ ) :=
X
2
(−1)n q n
/2
n∈Z
=
η(τ /2)2 . η(τ )
They are modular forms of weight 1/2 on the congruence subgroups ½ µ ¶ µ ¶ ¾ 1 0 1 1 Γ2 = γ ∈ SL2 (Z) : γ ≡ or mod 2 , 0 1 0 1 ½ µ ¶ µ ¶ ¾ 1 0 0 1 Γ3 = γ ∈ SL2 (Z) : γ ≡ or mod 2 , 0 1 1 0 and ¶ ¾ ¶ µ ½ µ 1 0 1 0 mod 2 or Γ4 = γ ∈ SL2 (Z) : γ ≡ 1 1 0 1 of level 2, respectively. These are all index 3 subgroups of SL2 (Z). Here we remark that in the standard theory of modular forms of half-integral weights, the theta functions are often considered as modular on Γ0 (4) by setting, P 2 for example, θ3 (τ ) = n∈Z q 2πin τ , instead of the definitions given above. Since Γ0 (4) is conjugate to Γ(2) by a matrix in SL2 (Q), this difference is more or less superficial. However, our choice of definitions make the discussion much easier because the largest congruence subgroups that theta functions are modular on now have simple descriptions, as we have seen above. We now describe the action of Γ3 on θ3 . We first recall the transformation formula for the Dedekind eta function. (See, for example, [7].) Lemma 1. For
µ a c the transformation formula for η(τ ) is γ=
¶ b ∈ SL2 (Z), d given by, for c = 0,
η(τ + b) = eπib/12 η(τ ), and, for c 6= 0,
r η(γτ ) = ²(a, b, c, d)
cτ + d η(τ ) i
with
µ ¶ 2 d i(1−c)/2 eπi(bd(1−c )+c(a+d))/12 , (1) ²(a, b, c, d) = ³ c´ 2 c eπi(ac(1−d )+d(b−c+3))/12 , d µ ¶ d is the Legendre-Jacobi symbol. where c
if c is odd, if d is odd,
4
LING LONG AND YIFAN YANG
µ ¶ a b Lemma 2. The action of γ = ∈ Γ3 on the theta function θ3 is as follows. c d µ ¶ 1 0 mod 2, then If γ ≡ 0 1 ¯ θ3 (τ )2 ¯γ = (−1)(d−1)/2 θ3 (τ )2 = χ−1 (d)θ3 (τ )2 . µ ¶ 0 1 If γ ≡ mod 2, then 1 0 ¯ θ3 (τ )2 ¯γ = i−c θ3 (τ )2 = −iχ−1 (c)θ3 (τ )2 . ¡ ¢ is the odd Dirichlet character modulo 4. Here χ−1 (n) = −4 n µ ¶ µ ¶ a b 1 0 Proof. Suppose that γ = ≡ ∈ Γ3 . We have, by the transformation c d 0 1 formula for the Dedekind η-function (Lemma 1), µ ¶ a(τ /2) + (b/2) η(γτ /2) = η 2c(τ /2) + d r µ ¶ 2c πi(2ac(1−d2 )+d(b/2−2c+3))/12 cτ + d = e η(τ /2), d i r ³c´ cτ + d πi(ac(1−d2 )+d(b−c+3))/12 η(γτ ) = e η(τ /2), d i ¶ µ a(2τ ) + (2b) η(2γτ ) = η (c/2)(2τ ) + d r ³ c ´ cτ + d πi(ac(1−d2 )/2+d(2b−c/2+3))/12 = e η(2τ ). 2d i Thus, using the fact that θ3 (τ ) = η(τ )5 /(η(τ /2)2 η(2τ )2 ), we obtain ¯ ¯ θ3 (τ )2 ¯ = eπi(d−1)/2 θ3 (τ )2 = χ−1 (d)θ3 (τ )2 . γ
µ ¶ 1 0 mod 2. 0 1 The proof of the other formula is similar. We write ¶ µ ¶ µ (a/2)(2τ ) + b (2a)(τ /2) + b , η(2γτ ) = η , η(γτ /2) = η c(2τ ) + 2d c(τ /2) + d/2
This is the claimed formula for the case γ ≡
and then apply Lemma 1 to obtain the claimed result.
¤
3. Eisenstein series on Γ3 In this section we will study the properties of the Eisenstein series on Γ3 . In particular, we will derive their Fourier expansions and their behavior under the action of elements of SL2 (Z). Here we first recall the definition of Eisenstein series. Let Γ be a discrete subgroup of SL2 (R) commensurable with SL2 (Z), and χ be a character of µ Γ. Let ¶ Γ∞ denote the stabilizer subgroup of the cusp ∞. For an a b element γ = ∈ SL2 (R) and τ ∈ H, the Poincar´e upper half plane, we write c d
A SHORT PROOF OF MILNE’S FORMULAE FOR SUMS OF INTEGER SQUARES
5
j(γ, τ ) = (cτ + d). Then the Eisenstein series Ek,χ (τ ) of weight k on Γ associated with the cusp ∞ and the character χ is defined by X (2) Ek,χ (τ ) = χ(γ)j(γ, τ )−k . γ∈Γ∞ \Γ
(The Eisenstein series associated with other cusps are not needed in this article. We therefore do not define them here.) It is easy to check that when k is an integer greater than 2, the series converges absolutely, and that it is a modular form with character χ of weight k on Γ. In order to deduce the Fourier expansion of Ek,χ on Γi the following two lemmas are needed. Lemma 3. Let k be an integer greater than 1. For τ with Im τ > 0 we have X n∈Z
∞ 1 (−2πi)k X k−1 2πirτ = r e . (τ + n)k Γ(k) r=1
Proof. By the Poisson summation formula we have X X Z ∞ e−2πirx 1 = dx. k (τ + n)k −∞ (x + τ ) n∈Z
r∈Z
When r < 0, we move the line of integration to the horizontal line Im x = T and let T tend to infinity. We find that the integral is equal to zero since the integrand is analytic on the upper half-plane and decays exponentially as T → ∞. When r = 0, it is easy to see that the integral is zero. When r > 0, we move the line of integration to the line Im x = −T and let T go to infinity. By doing so, we cross the pole of the integrand at x = −τ . Thus, we have µ −2πirx ¶ Z ∞ −2πirx e e (−2πi)k k−1 2πirτ dx = −2πi Res ; x = −τ = r e , k (x + τ )k (k − 1)! −∞ (x + τ ) and the lemma follows.
¤
Lemma P 4. Let k be an even integer greater than 2. Let Sk (x, y), x, y = 0, 1, denote the sum m,n (mτ + n)−k over all integers m, n with the restriction m ≡ x mod 2, n ≡ y mod 2, and gcd(m, n) = 1. Then we have, for τ ∈ H, ∞
X 1 (2πi)k rk−1 q r Sk (0, 1) = 1 + k (−1)r , 2 2 Γ(k)L(k, χ0 ) r=1 1 − qr ∞
X rk−1 q r/2 1 (2πi)k Sk (1, 0) = k , 2 2 Γ(k)L(k, χ0 ) r=1 1 − q r and ∞
X 1 (2πi)k rk−1 q r/2 Sk (1, 1) = k . (−1)r 2 2 Γ(k)L(k, χ0 ) r=1 1 − qr
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LING LONG AND YIFAN YANG
Proof. We have, by the inclusion-exclusion principle, X X 1 1 Sk (0, 1) = 1 + 2 (2mτ + n)k m∈N n∈Z,(2m,n)=1
=1+
X
X
X
µ(d)
m∈N d|m,(d,2)=1
1 X =1+ k 2
1 (2mτ + d(2n + 1))k
n∈Z
X
m∈N d|m,(d,2)=1
1 µ(d) X . dk (mτ /d + 1/2 + n)k n∈Z
We now apply Lemma 3 to the innermost sum. We obtain 1 (−2πi)k X Sk (0, 1) = 1 + k 2 2 Γ(k)
∞
X
µ(d) X k−1 2πir(mτ /d+1/2) r e dk r=1
m∈N d|m,(d,2)=1
X
k
=1+
(2πi) 2k Γ(k)
d∈N,(d,2)=1
=1+
(2πi) 2k Γ(k)
k
=1+
dk
d∈N,(d,2)=1
(2πi) 2k Γ(k)L(k, χ0 )
∞ X
∞ X (−1)r rk−1 q rm/d
X
m∈N,d|m r=1
∞ µ(d) X
X
k
µ(d) dk
(−1)r
r=1
(−1)r
r=1
rk−1 q r 1 − qr
rk−1 q r 1 − qr
This proves the case Sk (0, 1). The proof of the case Sk (1, 0) is similar. We have 1 Sk (1, 0) = 2
X
(2πi) 2k Γ(k) k
=
µ(d)
(2πi) 2k Γ(k)
X n∈Z
m∈N,(m,2)=1 d|m k
=
X
1 (mτ + 2nd)k
∞ X µ(d) X
X
m∈N,(m,2)=1 d|m
dk
rk−1 eπimr/d
r=1
∞ µ(d) X rk−1 q r/2
X
dk
d∈N,(d,2)=1
r=1
1 − qr
∞
=
X rk−1 q r/2 (2πi)k . 2k Γ(k)L(k, χ0 ) r=1 1 − q r
For the sum S(1, 1), we can either proceed as above, or use the fact that (S(0, 1)+ S(1, 0) + S(1, 1))/2 is equal to the normalized Eisenstein series ∞
1+
(2πi)k X rk−1 q r Γ(k)ζ(k) r=1 1 − q r
of weight k on SL2 (Z) to obtain the claimed Fourier expansion. We omit the details here. ¤ The next lemma will be used to determine the action of SL2 (Z) on the Eisenstein series.
A SHORT PROOF OF MILNE’S FORMULAE FOR SUMS OF INTEGER SQUARES
Lemma 5. Let S(x, y) be defined as in the previous lemma. Let γ = ¯ SL2 (Z). Then we have S(x, y)¯γ = S(ax + cy mod 2, bx + dy mod 2). Proof. We have ¯ S(x, y)¯γ = (cτ + d)−k
X m≡x mod 2,n≡y mod 2,(x,y)=1
X
=
m≡x mod 2,n≡y mod 2,(x,y)=1
7
µ ¶ a b ∈ c d
1 (m(aτ + b)/(cτ + d) + n)k
1 . ((ma + nc)τ + (mb + nd))k
Thus, we are required to show that there is a one-to-one correspondence between the terms in the last sum and those in S(ax + cy mod 2, bx + dy mod 2). This, however, follows from the assumptions that the discriminant of γ is 1 and that m and n in the sum are relatively primes. This gives the result. ¤ In the last lemmas of this section we give the Fourier expansions of Eisenstein series on Γ3 , one with the trivial character and the other with a quadratic character. This lemma shows that the functions G± k in Theorem 1 are scalar multiples of Eisenstein series on Γ3 . We remark that the case k = 2 is treated separately because the series defining the Eisenstein series of weight 2 does not converge absolutely. Lemma 6. Let k be an even integer greater than 2. The normalized Eisenstein series Ek+ (τ ) associated with the cusp ∞ of weight k on Γ3 with the trivial character has the Fourier expansion ∞ X (2πi)k rk−1 q r/2 Ek+ (τ ) = 1 + k , 2 Γ(k)L(k, χ0 ) r=1 1 − (−1)r q r/2 and the Eisenstein series Ek− (τ ) with the character λ on Γ3 given by µµ ¶¶ ( 1, if d is odd, a b λ = c d −1, if c is odd has the Fourier expansion ∞
(3)
Ek− (τ ) = 1 −
X rk−1 q r/2 (2πi)k . 2k Γ(k)L(k, χ0 ) r=1 1 + (−1)r q r/2
Proof. By definition, we have Ek+ (τ ) =
1 1 S(0, 1) + S(1, 0). 2 2
Thus, by Lemma 4, we have ∞
Ek+ (τ ) = 1 +
r r r/2 X (2πi)k k−1 (−1) q + q . r 2k Γ(k)L(k, χ0 ) r=1 1 − qr
Noticing that the summand, for odd r, is equal to rk−1
rk−1 q r/2 q r/2 − q r = 1 − qr 1 + q r/2
rk−1
rk−1 q r/2 q r/2 + q r = , 1 − qr 1 − q r/2
and, for even r, is equal to
8
LING LONG AND YIFAN YANG
we obtain the first part of the lemma. For the second part, we have 1 1 Ek− (τ ) = S(0, 1) − S(1, 0), 2 2 and the rest of proof is similar to that of the first part. We therefore skip the details here. ¤ Finally, the lemma below shows that if we set k = 2 in (3), then we will again get a modular form of weight 2 with character λ on Γ3 . Lemma 7. We have θ3 (q)4 = E2− (τ ) = 1 + 8
∞ X r=1
rq r/2 . 1 + (−1)r q r/2
Proof. See [3].
¤ 4. A new proof of Theorem 1
Throughout the section we let γ0 denote the matrix µ ¶ 1 −1 γ0 = . 1 0
¯ ¯ Notice that γ0−1 Γ3 γ0 = Γ2 and hence if f is a modular form of Γ3 , then f ¯ is a γ0 ¶ µ a b ∈ SL2 (R), the action of γ on modular form of Γ2 . Here for a matrix γ = c d a modular form f of weight k is defined by ¶ µ ¯ aτ + b ¯ −k f ¯ = (cτ + d) f . cτ + d γ Lemma 8. The action of the matrix γ0 on θ32k , Ek,λk , and Ek± is ¯ ¯ θ32k ¯ = (−i)k θ22k , γ0
and
¯ ¯ Ek± ¯
∞
γ0
=
for even k ≥ 4, and ¯ ¯ θ34 ¯
X (2πi)k k 2 Γ(k)L(k, χ0 ) r=1
γ0
¯ ¯ = E2− (τ )¯
γ0
½
rk−1 q r/2 rk−1 q r/2 ± (−1)r r 1−q 1 − qr
= −θ24 = 16
∞ X (2r − 1)q r−1/2 r=1
1 − q 2r−1
¾
.
Proof. By Lemma 1 we have
√ √ η(γ0 τ ) = −i²(1, −1, 1, 0) τ η(τ ) = eπi/6 τ η(τ ), µµ ¶ ¶ √ 1 0 τ −1 η(γ0 τ /2) = η = e−πi/3 τ η((τ − 1)/2), 2 1 2
and
µµ η(2γ0 τ ) = η
2 1
r ¶ ¶ τ τ −πi/12 =e η(τ /2). 2 2
−1 0
A SHORT PROOF OF MILNE’S FORMULAE FOR SUMS OF INTEGER SQUARES
9
Noticing that µ ¶ ∞ Y τ −1 η(τ )3 η = e−πi/24 q 1/48 (1 − (−1)n q n/2 ) = e−πi/24 , 2 η(τ /2)η(2τ ) n=1 we see that
¯ ¯ θ3 ¯
γ0
= e−πi/4 θ2 ,
and the assertion about θ3 follows. Moreover, by Lemma 5, we have ¯ ¯ 1 1 ¯ ¯ Ek± ¯ = (S(0, 1) ± S(1, 0)) ¯ = (S(1, 0) ± S(1, 1)) , 2 2 γ0 γ0 and an application of Lemma 4 yields the claimed result for the case k ≥ 4. The case k = 2 follows from Jacobi’s work [3]. This completes the proof of the lemma. ¤ The following proposition constitutes our main idea of proving identities involving the theta functions. Proposition 1. Let k be a positive integer. Let λ be the character on Γ3 with ¯ θ32k ¯γ = λ(γ)θ32k for γ ∈ Γ3 . Suppose that fj are modular forms of weight k on Γ3 ¯ P with character λ and ak are complex numbers such that j aj fj ¯γ0 has a Fourier expansion (−4i)k q k/4 + · · · . Then one has X θ32k = aj fj . j
Proof. By Lemma 8, it suffices to show that ¯ X ¯ (−i)k θ22k = aj fj ¯ . j
γ0
The fundamental domain D of Γ2 = Γ0 (2) has two cusps ∞ and 0 with widths 1 and 2, respectively, and it has an elliptic point (1 + i)/2 of order 2. By the standard complex analysis argument, a modular form f of weight k on Γ2 satisfies 1 v∞ (f ) + v0 (f ) + v(1+i)/2 (f ) + 2
X P ∈D−{∞,1,(1+i)/2}
vP (f ) =
k , 4
where vP (f ) denotes the this formula and the ¯ order of zero of f at P . From ¯ P ¯ , we see that this sum of fj ¯ has zeroes only at ∞. assumption on a f j j j γ0 γ0 ¯ P On the other hand, θ22k is also such a function. Therefore, θ22k = c j aj fj ¯γ0 for some constant coefficients, we conclude that P c ∈ C.¯ Comparing the leading Fourier P (−i)k θ22k = j aj fj ¯γ0 , or equivalently, θ32k = j aj kj . This completes the proof of the proposition. ¤ Lastly, we collect identities about the Hankel determinants in the following lemma. Here we evaluate the Hankel determinants by elementary methods.
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LING LONG AND YIFAN YANG
Lemma 9. One has X
det(rb2a+2b−1 )1≤a,b≤n
2−n
=
r1 ,...,rn
X
2n Y
det((2rb − 1)2a+2b−3 )1≤a,b≤n
2
22n
=
−2n
r1 ,...,rn
X
2n−1 Y
(r!)
r=1
det((2rb − 1)2a+2b−2 )1≤a,b≤n
2
2n
=
−n
r1 ,...,rn
X
(r!),
r=1
n Y
((2r − 1)!)2 ,
r=1
det((2rb − 1)2a+2b−4 )1≤a,b≤n
2
2n
=
−n
r1 ,...,rn
n−1 Y
((2r)!)2 ,
r=1
where the summation index (r1 , . . . , rn ) is over all possible permutations of (1, 2, . . . , n). Q Q Proof. Let A(n) = 1≤i