EXPLICIT BOUNDS FOR SUMS OF EXPLICIT BOUNDS FOR SUMS OF SQUARES JEREMY ROUSE Abstract. For an even...

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Transcript of EXPLICIT BOUNDS FOR SUMS OF EXPLICIT BOUNDS FOR SUMS OF SQUARES JEREMY ROUSE Abstract. For an even...

  • EXPLICIT BOUNDS FOR SUMS OF SQUARES

    JEREMY ROUSE

    Abstract. For an even integer k, let r2k(n) be the number of representations of n as a sum of 2k squares. The quantity r2k(n) is approximated by the classical singular series ρ2k(n) � nk−1. Deligne’s bound on the Fourier coefficients of Hecke eigenforms gives that r2k(n) = ρ2k(n) +

    O(d(n)n k−1 2 ). We determine the optimal implied constant in this estimate provided that either

    k/2 or n is odd. The proof requires a delicate positivity argument involving Petersson inner products.

    1. Introduction and Statement of Results

    In Hardy’s book on Ramanujan [11], he states the following (Chapter 9, pg. 132).

    The problem of the representations of an integer n as the sum of a given number k of integral squares is one of the most celebrated in the theory of numbers. Its history may be traced back to Diophantus, but begins effectively with Girard’s (or Fermat’s) theorem that a prime 4m + 1 is the sum of two squares. Almost every arithmetician of note since Fermat has contributed to the solution of the problem, and it has its puzzles for us still.

    If n is a non-negative integer, let

    rs(n) = #{(x1, x2, . . . , xs) ∈ Zs : x21 + x22 + · · ·+ x2s = n}

    be the number of representations of n as a sum of s squares.

    The classical work that Hardy refers to includes the formulas of Jacobi giving the following exact formulas. Let n be a positive integer and write n = 2αm, where m is odd. Then

    r4(n) =

    { 8σ1(m) if α = 0

    24σ1(m) if α ≥ 1, r8(n) =

    { 16σ3(m) if α = 0

    16 · 23α+3−15 7

    σ3(m) if α ≥ 1.

    The search for higher exact formulas (each involving more complicated arithmetic functions) for was carried out by many mathematicians. Glaisher [9] and Rankin [20] were interested in these formulas where the arithmetic functions involved were multiplicative.

    2010 Mathematics Subject Classification. Primary 11E25; Secondary 11F30. The author was supported by NSF grant DMS-0901090.

    1

  • 2 JEREMY ROUSE

    In a different direction, Hardy [10] and Mordell [18] applied the circle method to give an approximation

    rs(n) = ρs(n) +Rs(n)

    where ρs(n) is the “singular series” and Rs(n) is an error term. Here ρs(n) can be expressed as a divisor sum if s is even, and ρs(n) � n

    s 2 −1 provided s > 4. The contribution Rs(n) is more

    mysterious, and Deligne’s proof of the Weil conjectures (see [7]) implies an estimate of the form

    (1) Rs(n) = O(d(n)n s 4 − 1

    2 )

    provided s is even. The phenomena of exact formulas for rs(n) of the form rs(n) = ρs(n) only occurs for small s. In [21], Rankin shows that Rs(n) is identically zero if and only if s ≤ 8. Exact formulas of a different nature were given by Milne in [17] when s = 4n2 and s = 4n(n+1).

    The problem we study is the implied constant in equation (1) above. This is a natural ques- tion, and in [22, 14, 13], the author has studied the corresponding problem for powers of the ∆ function, p-core partitions (joint work with Byungchan Kim), and arbitrary level 1 cusp forms (joint work with Paul Jenkins), respectively. To prove their now famous “290-theorem” Bhargava and Hanke [4] compute this implied constant for about 6000 quadratic forms in four variables and use this to determine precisely which integers these forms represent.

    Returning to our problem, if s = 2k and k is even, we have that

    ρ2k(n) = 2k(−1)k/2+1

    (2k − 1)Bk ( σk−1(n) + (−1 + (−1)k/2+1)σk−1(n/2) + (−1)k/22kσk−1(n/4)

    ) ,

    where Bk is the kth Bernoulli number and σk−1(n) is the sum of the k − 1st powers of the positive integer divisors of n (and is hence zero if n is not an integer). Our main result is the following.

    Theorem 1. Suppose that k is even. If either k/2 is odd or n is odd, then we have

    |r2k(n)− ρ2k(n)| ≤ (

    4k + 2k(−1)k/2

    (2k − 1)Bk

    ) d(n)n

    k−1 2 .

    Remark. If 2k = 4 or 2k = 8, the right hand side is zero, and we recover the exact formulas of

    Jacobi. For arbitrary even k, we have r2k(1) = 4k and ρ2k(1) = 2k(−1)k/2+1

    (2k−1)Bk . Thus, the inequality

    above becomes an equality when n = 1. This shows that the implied constant

    4k + 2k(−1)k/2

    (2k − 1)Bk in (1) is best possible. The error term is smaller than the main term provided n� k2.

    Our approach to proving Theorem 1 is as follows. If

    θ(z) = 1 + 2 ∞∑ n=1

    qn 2

    , q = e2πiz

  • EXPLICIT BOUNDS FOR SUMS OF SQUARES 3

    is the classical Jacobi theta function, then

    θ2k(z) = ∞∑ n=0

    r2k(n)q n

    is a modular form of weight k on Γ0(4). If k is even, we can decompose

    (2) θ2k(z) = a1Ek(z) + a2Ek(2z) + a3Ek(4z) + ∑ i

    cigi(z) + ∑ i

    digi(2z) + ∑ i

    eigi(4z)

    where

    Ek(z) = 1− 2k

    Bk

    ∞∑ n=1

    σk−1(n)q n

    is the classical level 1 Eisenstein series, and the gi(z) are normalized newforms of level 1, 2, or 4. We prove the following.

    Theorem 2. Assume the notation above. Then for all i, ci ≥ 0.

    Theorem 2 allows us to read off∑ i

    |ci| = ∑ i

    ci = 4k + 2k(−1)k/2

    (2k − 1)Bk

    from the coefficient of q on both sides of (2), using that a1 = (−1)k/2

    (2k−1)Bk .

    To prove Theorem 2 we use properties of the Petersson inner product on Mk(Γ0(4)) (see Sec- tion 2 for precise definitions). If gi(z) is a newform of level 4, then gi(z) is orthogonal to every other term in the expansion (2). It follows that

    〈θ2k, gi〉 = ci〈gi, gi〉.

    It suffices to prove that 〈θ2k, gi〉 ≥ 0. This Petersson inner product consists of a contribution from each of the three cusps of Γ0(4): ∞, 0, and 1/2. The contribution from ∞ is

    2

    (4π)k

    ∞∑ n=1

    r2k(n)a(n)

    nk−1

    ∫ ∞ 4πn

    uk−2e−u du.

    Here gi(z) = ∑∞

    n=1 a(n)q n. Our approach is to show that the main term in the above sum

    comes from n = 1. If n is fixed, r2k(n) is a polynomial of degree 2k in n. We compute these polynomials explicitly, and use this to the bound the terms when 2 ≤ n ≤ 2500. Next, we use a simple induction bound on r2k(n) to show that the terms with 2500 ≤ n ≤ k2π log(k) are small enough. Finally, we use the exponential decay of

    ∫∞ 4πn

    uk−2e−u du when n ≥ k 2π

    log(k).

    The cusp at zero behaves in an essentially identical way to the cusp at∞, and the contribution from the cusp at 1/2 is very small, since θ(z) vanishes there.

  • 4 JEREMY ROUSE

    Remark. This result can be thought of as a refined form of the circle method. The Eisenstein series is the contribution of the major arcs, while the Deligne’s result, and the bounds we give on the constants ci can be thought of as explicit, uniform minor arc estimates. Further, it is plausible that the Fourier coefficients of distinct newforms are independent (an assertion that could be justified under the assumption of the holomorphy of certain Rankin-Selberg convolu- tions). Combining this with the recent proof of the Sato-Tate conjecture (see [3]) suggests that for any � > 0, there are infinitely many primes p so that

    |r2k(p)− ρ2k(p)| > (

    4k + 2k(−1)k/2

    (2k − 1)Bk − � ) d(p)p

    k−1 2 .

    Remark. The proof gives more detailed information about the constants ci in (2). In particular, if gi(z) is a newform of level 4 and k ≡ 2 (mod 4), then

    ci = 16k · (k − 2)!

    (4π)k〈gi, gi〉 (1 +O(αk))

    where α ≈ 0.918. If k ≡ 0 (mod 4), then ci = 0. Similar, but more complicated results are true for the constants ci associated with level 1 and level 2 newforms.

    An outline of the paper is as follows. In Section 2 we give precise definitions and review necessary background information. In Section 3 we prove a number of auxiliary results that will be used in the proof of Theorem 2. In Section 4, we prove Theorem 2 and use this to deduce Theorem 1. Finally, in Section 5, we address other values of k and n.

    Acknowledgements. The author used Magma [5] version 2.17 for computations.

    2. Background

    In this section we give definitions and review necessary background. For N ≥ 1, let Mk(Γ0(N))

    denote the C-vector space of modular forms of weight k on Γ0(N) := {(

    a b c d

    ) ∈ SL2(Z) : N |c

    } .

    Let Sk(Γ0(N)) denote the subspace of cusp forms.

    If f is a modular form of weight k, and α =

    [ a b c d

    ] ∈ GL2(Q) and has positive determinant,

    define the usual slash operator by

    f |α = (ad− bc)k/2(cz + d)−kf ( az + b

    cz + d

    ) .

    For a positive integer d, define the operator V (d) by f(z)|V (d) = f(dz). It is well-known (see [12], pg. 107 for a proof) that V (d) maps Mk(Γ0(N)) to Mk(Γ0(dN)) and Sk(Γ0(N)) to Sk(Γ0(dN)). For a positive integer d, define the operator U(d) by

    ∞∑ n=0

    a(n)qn|U(d) = ∞∑ n=0

    a(dn)qn.

  • EXPLICIT BOUNDS FOR SUMS OF SQUARES 5

    If d|N , then U(d) maps Mk(Γ0(N)) to itself and Sk(Γ0(N)) to itself. If p is a prime with p - N , define the usual Hecke operator T (p) by T (p) = U(p) + pk−1V (p).

    If f, g ∈Mk(Γ0(N)) and at least one of f or g is a cusp form, let

    〈f, g〉 = 3 π[SL2(Z) : Γ0(N)]

    ∫∫ H/Γ0(N)

    f(x+ iy)g(x+ iy)yk dx dy

    y2

    denote the usual Petersson inner product. If p - N , then the Hecke opera