Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 26, 2001, 465–482

THE SECOND MOMENT OF THE SYMMETRIC SQUARE L-FUNCTIONS

H. Iwaniec and P. Michel

Rutgers University, Department Mathematics New Brunswick, NJ-08903, U.S.A.; iwaniec@math.rutgers.edu Current address: Institute for Advanced Study, Olden Lane, Princeton, NJ-08540, U.S.A. Université Montpellier II CC 051, Mathématiques FR-34095 Montpellier Cedex 05, France; michel@darboux.math.univ-montp2.fr Current address: Institute for Advanced Study, Olden Lane, Princeton, NJ-08540, U.S.A.

Abstract. In this paper we investigate the second power moment of symmetric square Lfunctions on the critical line, which are associated with primitive cusp forms. We establish an upper bound which is sharp with respect to the level.

1. Introduction

Estimates for L-functions on the critical line appear in many applications in number theory. The Riemann hypothesis, if true, provides essentially the best possible results. Quite often estimates on average over suitable families of Lfunctions are sufficient for practical purpose and the results are as good as the Riemann hypothesis can do.

In this paper we are interested in estimating the second power moment of L-functions associated with the symmetric square representations of GL_{2} automorphic forms in the conductor aspect. According to the Gelbart–Jacquet lift [GJ] these correspond to automorphic forms on GL_{3} . Our main result is

Theorem 1.1. Let k, N be positive integers, k even, N squarefree. Let ℬ_{k}^{∗}(N) be the set of primitive cusp forms of weight k with respect to the group Γ_{0}(N). For each f ∈ ℬ_{k}^{∗}(N) let L(sym^{2} f, s) be the corresponding symmetric square L-function. Let Re s = \frac{1}{2} . We have

(1.1)  ∑_{f∈ℬ_{k}^{∗}(N)} |L(sym^{2} f, s)|^{2} ≪ |s|^{8}N^{1+ε}

for any ε > 0, the implied constant depending only on ε and k.

2000 Mathematics Subject Classification: Primary 11M99.

The first author is supported by the Ambrose Monell Foundation, the Hansmann Membership (by grants to the Institute for Advanced Study) and by NSF Grant DMS-98-01642, and the second author by NSF Grant DMS-97-29992 and the Ellentuck Fund (by grants to the Institute for Advanced Study) and by the Institut Universitaire de France.

Note that the number of primitive forms satisfies (see (2.72) of [ILS])

(1.2)  |ℬ_{k}^{∗}(N)| ∼ \frac{k − 1}{12}φ(N)

so that (1.1) implies

(1.3)  L(sym^{2} f, s) ≪ N^{ε}

for almost all f (the implied constant depending on ε, k and s), while the Riemann hypothesis would imply this for every f. Ignoring all but one form in (1.1) we get

(1.4)  L(sym^{2} f, s) ≪ N^{(1/2)+ε}

for every f ∈ ℬ_{k}^{∗}(N). This last estimate is nearly as good as the convexity bound which follows from the functional equation (2.23).

Inspite of three gamma factors in the functional equation for L(sym^{2} f, s) (which fact is consistent with sym^{2} f being a GL_{3} form) in the N aspect (the conductor) L(sym^{2} f, s) behaves like L^{2}(s, χ_{N}) for the real primitive Dirichlet character χ_{N} of conductor N. In that sense Theorem 1.1 can be compared with the estimate of Heath-Brown [HB] for the fourth power moment of Dirichlet L functions over the real characters on the critical line

(1.5)  \begin{matrix}∑ & ♭ \\ q≤Q \end{matrix} |L(s, χ_{q})|^{4} ≪ |s|^{A}Q^{1+ε}

where (and throughout this paper) ^{∑♭} indicates restriction to positive odd squarefree integers. Here A is an absolute positive constant, ε is any positive number, and the implied constant depends only on ε. This analogy is justified because both L(sym^{2} f, s) and L^{2}(s, χ_{N}) can be well approximated by the corresponding partial sums of comparable length relative to the conductor N. However the case of the symmetric square is somewhat harder. The point of distinction is that we are having the Fourier coefficients λ_{f}(n^{2}) in place of the characters χ_{N}(n)τ(n) for n ≪ N, and the squares are sparse while the divisor function is quite regular by comparison. Heath-Brown derives (1.5) (among some other results) directly from his more general

Theorem 1.2. For any complex numbers a_{n} we have

(1.6)  \begin{matrix}∑ & ♭ \\ m≤M \end{matrix}|\begin{matrix}∑ & ♭ \\ n≤N \end{matrix} a_{n}(\frac{n}{m})|^{2} ≪ (MN)^{ε}(M + N) \begin{matrix}∑ & ♭ \\ n≤N \end{matrix} |a_{n}|^{2}

with any ε > 0, the implied constant depending only on ε.

The second moment of the symmetric square L-functions 467

This is a deep and indeed powerful estimate. In the proof of our Theorem 1.1 we shall also use (1.6), but not as directly. Our method does not yield an asymptotic formula for the second power moment of L(sym^{2} f, s) because the estimate (1.6) is not precise enough.

Since our result (1.1) just yields the convexity bound (1.4) one is encouraged to use the amplification method to get an improvement for individual L(sym^{2} f, s). This would be possible if we could prove that

(1.7)  ∑_{f∈ℬ_{k}^{∗}(N)} λ_{f}(l^{2})|L(sym^{2} f, s)|^{2} ≪ l^{−β}N^{1+ε}

for every l ≤ N^{α} with some small α, β > 0. Investigating this question, some preliminary analysis led us to character sums of type

(1.8)  \begin{matrix}∑ & ♭ \\ c≤C \end{matrix} ∑ ∑_{{a≤A, b≤B \atop ab≠□}} (\frac{ab}{cN} )χ(a)ψ(b)

with any χ, ψ (mod l). One needs this triple sum (and other slightly more general sums) for A, B, C ≪ N to be bounded by O(N^{2−η}) for some absolute η > 0, which seems to be plausible due to extra cancellation from summation in c, but the current technology fails.

Acknowledgement. We began working on this paper in May 1999 during the visit of the first author to Université d’Orsay, and continued in the Institute for Advanced Study during the year 1999–2000. We thank both institutions for invitations and support. We would also like to thank Matti Jutila for his interest in our work and the referee for careful reading and corrections.

2. A brief account of automorphic L-functions

To fix notation and normalizations we recall standard facts about automorphic forms and their L-functions. The linear space S_{k}(N) of cusp forms of weight k and level N is a finite-dimensional Hilbert space with respect to the Petersson inner product

⟨f, g⟩ = ∫_{Γ_{0}(N)\H} f(z)\overline{g}(z)y^{k−2} dx dy.

We have

(2.1)  dim S_{k}(N) = ν(N)\frac{k − 1}{12} + O((kN)^{2/3})

where

(2.2)  ν(N) = [Γ_{0}(1) : Γ_{0}(N)] = N ∏_{p|N}(1 + \frac{1}{p}).

Expand f ∈ S_{k}(N) into Fourier series

(2.3)  f(z) = ∑_{n≥1} ψ_{f}(n)n^{(k−1)/2}e(nz).

Concerning the size of the Fourier coefficients ψ_{f}(n) on can show that

(2.4)  ∑_{n≤X} |ψ_{f}(n)|^{2} ≤ 3ω_{f}(1 + \frac{80πX}{kN} )

where

(2.5)  ω_{f} = \frac{(4π)^{k−1}}{Γ(k − 1)}⟨f, f⟩.

Although the proof is elementary (we do not use (2.4) here) the result is quite good (only the constants in the bound (2.4) can be improved). The ω_{f} given by (2.5) appears to be a natural normalizing factor in the spectral trace

(2.6)  Δ(m, n) = ∑_{f∈ℬ_{k}(N)} ω_{f}^{−1}\overline{ψ}_{f}(m)ψ_{f}(n).

Here ℬ_{k}(N)is any orthogonal basis of S_{k}(N) and the sum (2.6) is basis independent. We have (cf. Theorem 3.6 of [I]) the following formula of Petersson

Proposition 2.1. For any m, n ≥ 1

(2.7)  Δ(m, n) = δ(m, n) + 2πi^{−k} ∑_{c≡0(N)} c^{−1}S(m, n; c)J_{k−1}(\frac{4π\sqrt{mn}}{c})

where δ(m, n) is the diagonal symbol, J_{k−1}(x) is the Bessel function and

(2.8)  S(m, n; c) = ∑_{ad≡1(c)} e(\frac{am + dn}{c})

is the classical Kloosterman sum.

The Hecke operators T_{n}: S_{k}(N) → S_{k}(N) are defined by

(2.9)  (T_{n}f)(z) = \frac{1}{\sqrt{n}} ∑_{{ad=n \atop (a,N)=1}}(\frac{a}{d})^{k/2} ∑_{b (mod d)} f(\frac{az + b}{d}).

For any m, n ≥ 1 we have

(2.10)  T_{m}T_{n} = ∑_{{d|(m,n) \atop (d,N)=1}} T_{mn/d^{2}}.

Hence the Hecke operators commute. Moreover the operators T_{n} with (n, N) = 1 are self-adjoint, i.e., ⟨T_{n}f, g⟩ = ⟨f, T_{n}g⟩ for all f, g in S_{k}(N) if (n, N) = 1. Therefore we can choose the orthogonal basis ℬ_{k}(N) which consists of common eigenfunctions of all the Hecke operators with (n, N) = 1, i.e., for any f ∈ ℬ_{k}(N) we require

(2.11)  T_{n}f = λ_{f}(n)f,  if (n, N) = 1.

We shall call any f which satisfies (2.11) a Hecke cusp form.

By (2.9), (2.10) and (2.11) one derives the formula

(2.12)  ψ_{f}(m)λ_{f}(n) = ∑_{d|(m,n)} ψ_{f}(mn/d^{2})

for any m, n ≥ 1 with (n, N) = 1. In particular we obtain ψ_{f}(n) = ψ_{f}(1)λ_{f}(n) if (n, N) = 1. Moreover, it follows from (2.10) and (2.11) that

(2.13)  λ_{f}(m)λ_{f}(n) = ∑_{d|(m,n)} λ_{f}(mn/d^{2}),  if (mn, N) = 1.

We assume that the Hecke eigenbasis contains the subset ℬ_{k}^{∗}(N) of primitive forms (in the sense of Atkin–Lehner theory, see [I]). A primitive form f has many useful properties. First of all f is an eigenfunction of all the Hecke operators and the formulas (2.10), (2.12), (2.13) hold for all m, n ≥ 1. Moreover f can be normalized so that

(2.14)  ψ_{f}(1) = 1,

and the Fourier expansion becomes

(2.15)  f(z) = ∑_{n≥1} λ_{f}(n)n^{(k−1)/2}e(nz).

The associated Hecke L-function

(2.16)  L(f, s) = ∑_{n≥1} λ_{f}(n)n^{−s}

has Euler product L(f, s) = ∏_{p} L_{p}(f, s) with the local factors

(2.17)  L_{p}(f, s) = (1 − λ_{f}(p)p^{−s} + χ_{0}(p)p^{−2s})^{−1}

where χ_{0} denotes the principal character to modulus N. Define the local factor at p = ∞ by

(2.18)  L_{∞}(f, s) = π^{−s}Γ(\frac{s}{2} + \frac{k − 1}{4})Γ(\frac{s}{2} + \frac{k + 1}{4}).

Then the complete product Λ(f, s) = N^{s/2}L_{∞}(f, s)L(f, s) is entire and it satisfies the functional equation

(2.19)  Λ(f, s) = ε_{f}Λ(f, 1 − s)

with the root number ε_{f} = i^{k}μ(N)λ_{f}(N)N^{1/2} = ±1.

The symmetric square L-function is defined by

(2.20)  L(sym^{2} f, s) = ζ^{(N)}(2s) ∑_{n≥1} λ_{f}(n^{2})n^{−s},

here and hereafter ζ^{(N)}(2s) stands for the partial zeta function with local factors at primes of N removed. This has the Euler product L(sym^{2} f, s) = ∏_{p} L_{p}(sym^{2} f, s) with the local factors

(2.21)  L_{p}(sym^{2} f, s) = (1 − λ_{f}(p^{2})p^{−s} + λ_{f}(p^{2})p^{−2s} − p^{−3s})^{−1}

if p ∤ N and

L_{p}(sym^{2} f, s) = (1 − p^{−s−1})^{−1}

if p|N.

Remark. For p ∤ N the L_{p}(f, s) factors into (1−α_{f}(p)p^{−s})^{−1}(1−β_{f}(p)p^{−s})^{−1} with α_{f}(p) + β_{f}(p) = λ_{f}(p) and α_{f}(p)β_{f}(p) = 1, and then

L_{p}(sym^{2} f, s) = (1 − α_{f}^{2}(p)p^{−s})^{−1}(1 − α_{f}(p)β_{f}(p)p^{−s})^{−1}(1 − β_{f}^{2}(p)p^{−s})^{−1}.

Define the local factor at p = ∞ by

(2.22)  L_{∞}(sym^{2} f, s) = π^{−3s/2}Γ(\frac{s + 1}{2})Γ(\frac{s + k − 1}{2})Γ(\frac{s + k}{2} ).

Then the complete product Λ(sym^{2} f, s) = N^{s}L_{∞}(sym^{2} f, s)L(sym^{2} f, s) is entire and it satisfies the functional equation

(2.23)  Λ(sym^{2}f, s) = Λ(sym^{2}f, 1 − s).

For a primitive form one has (see Lemma 2.5 of [ILS])

(2.24)  ω_{f} = \frac{k − 1}{2π^{2}}NL(sym^{2} f, 1).

By the Ramanujan conjecture (proved by P. Deligne [D])

(2.25)  |λ_{f}(n)| ≤ τ(N)

and the functional equation (2.23) we derive by convexity arguments that

(2.26)  L(sym^{2} f, 1) ≪ (log kN)^{3}

where the implied constant is absolute. Hence

(2.27)  ω_{f} ≪ kN(log kN)^{3}.

3. Representation of L(sym^{2} f, s) by partial sums

The main Theorem 1.1 will be reduced to corresponding estimates for partial sums of (2.20). To this end we use the following formula which is obtained by standard contour integration of the functional equation (2.23).

Lemma 3.1. Let A be a positive integer and G(t) = cos(πt/4A)^{−3A}. For any s with 0 ≤ Re s ≤ 1 we have

L(sym^{2} f, s) = ∑_{n≥1} \frac{λ_{f}(n^{2})}{n^{s}}V_{s}(\frac{n}{N} ) + ε(s) ∑_{n≥1} \frac{λ_{f}(n^{2})}{n^{1−s}}V_{1−s}(\frac{n}{N} )

where

V_{s}(y) = ∫_{(2)} G(t)\frac{L_{∞}(sym^{2} f, s + t)}{L_{∞}(sym^{2} f, s)}ζ^{(N)}(2s + 2t)y^{−t} \frac{dt}{t}

and ε(s) = N^{1−2s}L_{∞}(sym^{2} f, 1 − s)/L_{∞}(sym^{2} f, s).

Of course, the function V_{s}(y) is independent of f ∈ ℬ_{k}^{∗}(N) so our notation is justified. We suppose from now on that Re s = \frac{1}{2} so |ε(s)| = 1. If y is large we shift the contour of integration in V_{s}(y) to Re t = A, if y is small we shift the contour to Re t = −\frac{1}{2} , meeting poles at t = 0 and t = \frac{1}{2} − s (eventually a double pole if s = \frac{1}{2}). Then by Stirling’s formula we derive

(3.1)  V_{s}(y) ≪ τ(N)(1 + \frac{y}{|s|^{3/2}} )^{−A} log(2 + y^{−1}).

Moreover, taking the derivatives we derive

(3.2)  V_{s}^{(j)}(y) ≪ τ(N)y^{−j}(1 + \frac{y}{|s|^{3/2}} )^{−A} log(2 + y^{−1}).

Now we apply a smooth partition of unity

1 = ∑_{α=−∞}^{∞} h(\frac{x}{2^{α/2}} ), for x > 0,

and obtain from Lemma 3.1, (3.1) and (3.2) that for f ∈ ℬ_{k}^{∗}(N)

(3.3)  L(sym^{2} f, s) ≪ τ(N) ∑_{X} |S_{f}(X)|\frac{log 4NX}{\sqrt{X}} (1 + \frac{X}{N|s|^{3/2}} )^{−A}

where X = 2^{α/2} with α ≥ −1, and S_{f}(X) is a sum of type

(3.4)  S_{f}(X) = ∑_{n} ψ_{f}(n^{2})g(\frac{n}{X} )

where g is a smooth function supported in the interval [1, 2] which satisfies

(3.5)  g^{(j)}(x) ≪ |s|^{j},  for j ≥ 0

and the implied constant depends on j and k only. By Theorem 5.1 for the particular sums (3.4) we have

∑_{f∈ℬ_{k}(N)} ω_{f}^{−1}|S_{f}(X)|^{2} ≪ |s|^{5}(NX)^{ε}(N^{−1}X^{2} + X).

We need this only for the primitive forms. Dropping the other forms we derive by (3.3)

∑_{f∈ℬ_{k}^{∗}(N)} ω_{f}^{−1}|L(sym^{2} f, s)|^{2} ≪ |s|^{8}N^{ε}.

Finally applying the upper bound (2.27) for the weight ω_{f} we conclude Theorem 1.1. It remains to prove Theorem 5.1 for which we spend the rest of this paper.

4. Some properties of Bessel functions

In the sequel the following expression for the Bessel function will be useful (see [W, p. 206])

(4.1)  J_{κ}(x) = e^{ix}W (x) + e^{−ix}\overline{W} (x)

where

W (x) = \frac{e^{i((πκ/2)−π/4)}}{Γ(κ + \frac{1}{2})} \sqrt{\frac{2}{πx}} ∫_{0}^{∞} e^{−y}(y(1 + \frac{iy}{2x}))^{κ−1/2} dy.

When κ is a positive integer, we derive (using the Taylor expansion for J_{κ}(x) if 0 < x ≤ 1, or the above integral expression for W (x) if x ≥ 1) the following bounds for the derivatives of W

(4.2)  x^{j}W^{(j)}(x) ≪ \frac{x}{(1 + x)^{3/2}}

for any j ≥ 0, the implied constant depending on j and κ.

5. The mean-square of a partial sum

We consider g a smooth function supported in the interval [1, 2] which satisfies

(5.1)  g^{(j)}(x) ≪ P^{j}

for some P ≥ 1 and any j ≥ 0, the implied constant depending on j only. For X ≥ 1 we define the partial sums

S_{f}(X) = ∑_{n} ψ_{f}(n^{2})g(\frac{n}{X} )

and their mean-square

S(X) = ∑_{f∈ℬ_{k}(N)} ω_{f}^{−1}|S_{f}(X)|^{2}.

Our goal is the following

Theorem 5.1. For X ≥ 1

S(X) ≪ P^{3}(NP X)^{ε}(N^{−1}X^{2} + X)

for any ε > 0, the implied constant depending on ε only.

6. Application of Petersson’s formula

By Petersson’s formula (2.6) we transform S(X) into sums of Kloosterman sums

(6.1)  S(X) = R_{0}(X) + 2πi^{k} ∑_{c≡0(N)} R_{c}(X)

with

(6.2)  R_{0}(X) = ∑_{n}|g(\frac{n}{X} )|^{2} ≪ X

and

R_{c}(X) = \frac{1}{c} ∑_{m} ∑_{n} S(m^{2}, n^{2}; c)J_{k−1}(\frac{4πmn}{c})\overline{g}(\frac{m}{X} )g(\frac{n}{X} ).

Our next step will be to apply the Poisson summation formula for R_{c}(X) to the n variable. However, this operation would produce poor results for c very large, so first we split the sum of R_{c}(X) into sums of the type

(6.3)  R(C; X) = ∑_{{C<c≤2C \atop c≡0(N)}} R_{c}(X)

for C ≤ N^{2000} and estimate the tail (the sum over the c > 2N^{2000} ) by Weil’s bound for individual Kloosterman sums getting O(X^{4}N^{−1000}). From now on we deal with each remaining R(C; X) separately with c ∼ C ≤ N^{2000} .

Applying Poisson’s formula on the n variable in R_{c}(X), we get

(6.4) R_{c}(X) = ∑_{m} ∑_{h} G(m, h; c)\overline{g}(\frac{m}{X} ) ∫_{0}^{∞} J_{k−1}(4πmx)g(\frac{cx}{X})e(−hx) dx

where

G(m, h; c) = \frac{1}{c} ∑_{a(c)} e(\frac{ah}{c})S(m^{2}, a^{2}; c).

The complete sum G(m, h; c) is computed in Section 8; it is shown there that G(m, h; c) = G(D; c), where D = h^{2} − 4m^{2} and G(D; c) is defined in (8.3). In particular we see that G(m, h; c) depends on D = h^{2} − 4m^{2} rather than on h and m separately. Moreover an important feature is the factorization D = (h − 2m)(h + 2m) = ab, say, with a ≡ b (mod 4). Making the change of variables h = \frac{1}{2}(a + b), m = \frac{1}{4}(b − a), we arrive at

R_{c}(X) = ∑_{D≡0,1(4)} G(D; c) ∑_{{ab=D \atop a≡b(4)}} I_{c}(a, b)

where

I_{c}(a, b) = \overline{g}(\frac{b − a}{4X}) ∫ J_{k−1}(π(b − a)x)g(\frac{cx}{X})e(−\frac{(a + b)x}{2}) dx.

7. Separation of the variables a, b

In order to separate the variables a and b we proceed by Fourier transform, and a very clean way to do it is to make use of the properties of the Bessel function which we have recalled in Section 4. From (4.1) we have a decomposition

(7.1)  \begin{aligned}I_{c}(a, b) & = \overline{g}(\frac{b − a}{4X}) ∫ e(−ax)W (π(b − a)x)g(\frac{cx}{X}) dx \\ & + \overline{g}(\frac{b − a}{4X}) ∫ e(bx)\overline{W} (π(b − a)x)g(\frac{cx}{X}) dx \\ & = Ic^{+}(a, b) + Ic^{−}(a, b), \end{aligned}

say. According to (7.1) we decompose R(C; X) = R^{+}(C; X) + R^{−}(C; X). We treat R^{+}(C; X), the other term being similar.

First of all we need to localize the range of a, b. By repeated integration by parts using (4.2) and (5.1) we obtain

Proposition 7.1. We have,

(7.2)  I_{c}(a, b) ≪ |g(\frac{b − a}{4X})|\frac{X}{C} (1 + \frac{|a|X}{CP} )^{−α}

for any α ≥ 0, the implied constants depending on α only.

Using this estimate we see that, for any ε > 0, the contribution to R^{+}(C; X) from the a, b such that |a| ≥ Y (NX)^{ε} (where Y = CP X^{−1} ) is negligible, in fact ≪ N^{−A} for all A > 0 (the implied constant depending on ε and A). Note that the range of b is controlled by a since |b − a| ≪ X. For the remaining portion we split the a, b sum into dyadic ranges and assume from now on that

(7.3) |a| ∼ A, |b| ∼ B,  for some A ≤ Y (NX)^{ε}, B ≤ (NX)^{ε}(X + Y).

Accordingly we denote by R^{+}(A, B, C; X) the corresponding portion of R^{+}(C; X). We have for x ∼ X/C,

\overline{g}(\frac{z}{4X} )W (πzx) = ∫_{R} h(t, x)e(tz) dt

with (using twice partial integration and (4.2))

h(t, x) = ∫_{R} \overline{g}(\frac{z}{4X} )W (πzx)e(−tz) dz ≪ X(1 + \frac{|t|X}{P} )^{−2} \frac{xX}{(1 + xX)^{3/2}}

so that

∫_{R} |h(t, x)| dt ≪ \frac{P xX}{(1 + xX)^{3/2}} ≪ \frac{P X^{2}/C}{(1 + X^{2}/C)^{3/2}} .

Hence we have

(7.4) \begin{aligned}R^{+} & (A, B, C; X) = ∫_{X/2C}^{2X/C} ∫_{−∞}^{+∞} h(t, x) \\ & × ∑_{{c∼C \atop c≡0(N)}} g(\frac{cx}{X}) ∑_{D} G(D; c) ∑ ∑_{{a∼A, b∼B \atop ab=D, a≡b(4)}} e(−(t + x)a)e(tb) dt dx \\ & ≪ \frac{X}{C} \frac{P X^{2}/C}{(1 + X^{2}/C)^{3/2}} ∑_{{c∼C \atop c≡0(N)}} | ∑_{D≡0,1(4)} G(D; c) ∑ ∑_{{a∼A, b∼B \atop ab=D, a≡b(4)}} x_{a}y_{b}| \end{aligned}

where x_{a}, y_{b} are complex numbers of modulus bounded by 1. Finally we detect the congruence a ≡ b(4) by means of additive characters, and at the expense of changing the x_{a} and y_{b} we may remove this congruence condition. This finishes the process of separation of the variables a, b.

Now we may apply Proposition 10.1 to (7.4) which gives us

(7.5)  \begin{aligned}R^{+}(A, B, C; X) & ≪ (P NX)^{ε} \frac{P X^{3}/C^{2}}{(1 + X^{3}/C^{2})^{3/2}} \\ & × [Y (X + Y)(\frac{C}{N} + Y )(\frac{C}{N} + X + Y )]^{1/2} \\ & ≪ (P NX)^{ε}P^{3}(N^{−1}X^{2} + X). \end{aligned}

Summing over relevant A, B we get R^{+}(C; X) ≪ (P NX)^{ε}P^{3}(N^{−1}X^{2} +X). The same bound can be proved for R^{−}(C; X). Finally summing these bounds over relevant C ≤ N^{2000} by (6.1) and (6.2) we conclude the proof of Theorem 5.1. It remains to prove Proposition 10.1 for which we spend the rest of this paper.

8. Evaluation of an exponential sum

Our objective in this section is to compute the following exponential sum

(8.1)  G(m, h; c) = \frac{1}{c} ∑_{a(c)} e(\frac{ah}{c})S(m^{2}, a^{2}; c).

Our computations will be completed in the next section. In Section 10 we will use the results to provide bounds for certain sums of bilinear forms involving G(m, h; c).

Opening the Kloosterman sum and changing the variables we get

G(m, h; c) = \frac{1}{c} ∑_{a (mod c)} S(0, a^{2} + ah + m^{2}; c)

where S(0, w; c) is the Ramanujan sum. This sum has a simple formula

S(0, w; c) = ∑_{{bd=c \atop b|w}} μ(d)b.

Hence we find that

G(m, h; c) = ∑_{bd=c} μ(d)ν_{b}(h^{2} − 4m^{2})

where ν_{b}(h^{2} − 4m^{2}) is the number of solutions to the congruence

a^{2} + ah + m^{2} ≡ 0 (mod b)

in a (mod b). Note that by completing square this is also equal to half the number of solutions to the congruence

(8.2)  x^{2} ≡ D (mod 4b)

in x (mod 4b) with D = h^{2} − 4m^{2} . Therefore G(m, h; c) = G(D; c) for D = h^{2} − 4m^{2} , where

(8.3)  G(D; c) := ∑_{bd=c} μ(d)ν_{b}(D)

with ν_{b}(D) = \frac{1}{2}|\{x (mod 4b), x^{2} ≡ D (mod 4b)\}|. From now on we consider the sum (8.3) for any integer D ≡ 0, 1 (mod 4). Note that G(D; c) is multiplicative in c so

(8.4)  G(D; c) = ∏_{p^{γ}∥c} G(D; p^{γ}) = ∏_{p^{γ}∥c}(ν_{p^{γ}} (D) − ν_{p^{γ−1}}(D)).

(In particular note that G(D; 1) = 1 since D ≡ 0, 1(4).) Let γ ≥ 1. If p ≠ 2 then ν_{p^{γ}} (D) is the number of solutions to x^{2} ≡ D (mod p^{γ}) while ν_{2^{γ}} (D) is half the number of solutions to x^{2} ≡ D (mod 2^{γ+2}). Suppose p ≠ 2 and p^{α} ∥ D. If γ ≤ α, then ν_{p^{γ}} (D) = p^{[γ/2]} . If γ > α then

ν_{p^{γ}} (D) = (1 + (\frac{Dp^{−α}}{p} ))p^{α/2}

when α is even, and ν_{p^{γ}} (D) = 0 when α is odd. Hence we conclude from (8.4) that G(D; p^{γ}) is given by

(8.5)

0,  if γ > α + 1,
(\frac{Dp^{−α}}{p} )p^{(γ−1)/2},  if γ = α + 1 odd,
−p^{(γ−2)/2},  if γ = α + 1 even,
(1 − \frac{1}{p})p^{γ/2},  if α ≥ γ even,
0,  if α ≥ γ odd.

Now let p = 2. If γ > α even then

ν_{2^{γ}} (D) = \frac{1}{2}(1 + χ_{4}(D2^{−α}))(1 + χ_{8}(D2^{−α}))2^{α/2},

if γ > α odd then ν_{2^{γ}} (D) = 0, if α = γ even then

ν_{2^{γ}} (D) = \frac{1}{2}(1 + χ_{4}(D2^{−α}))2^{α/2},

if α = γ odd then ν_{2^{γ}} (D) = 0, if α = γ + 1 even then

ν_{2^{γ}} (D) = \frac{1}{2}2^{α/2},

if α = γ + 1 odd then ν_{2^{γ}} (D) = 0, and finally if α ≥ γ + 2

ν_{2^{γ}} (D) = 2^{[γ/2]}.

Here χ_{4} , χ_{8} are primitive characters of moduli 4 and 8 respectively. From the above results we conclude that G(D; 2^{γ}) is given by

(8.6)

(1 + χ_{4}(D2^{−α}))χ_{8}(D2^{−α})2^{(γ−3)/2},  if γ = α + 1 odd,
χ_{4}(D2^{−α})2^{(γ/2)−1},  if γ = α even,
−2^{(γ/2)−1},  if γ = α − 1 even,
2^{(γ/2)−1},  if γ < α − 1 even,

and G(D; 2^{γ}) = 0 in the remaining cases. From the above computations we deduce the following

Proposition 8.1. Let D ≡ 0, 1 (mod 4) and c ≥ 1. Let G(D; c) be the sum defined in (8.3). It is a multiplicative function of c and for c = p^{γ} a prime power it is given by (8.5) for p ≠ 2 and by (8.6) for p = 2. Factoring uniquely c into qr, where q is squarefree, 4r is squareful and (q, 2r) = 1, we have

G(D; c) = G(D; q)G(D; r) = (\frac{D}{q} )G(D; r).

9. Computation of G(D; r)

We now give a synthetic formula for G(D; r) when 4r is squareful. For that we consider the (unique) factorization of D into D = D_{1}D_{2} , with D_{1} squarefree odd, 4D_{2} squareful and (D_{1}, D_{2}) = 1. Next we factor r accordingly into r = r′_{1}r_{2} with r′_{1}|D_{1}^{∞}, (r_{2}, D_{1}) = 1 (again this decomposition is unique), so that

G(D; r) = G(D_{1}D_{2}; r′_{1}r_{2}) = G(D_{1}D_{2}; r′_{1})G(D_{1}D_{2}; r_{2}).

Using (8.5) one sees that G(D; r) = 0, unless r′_{1} = r_{1}^{2} with r_{1}|D_{1} (so r_{1} is squarefree) in which case we have

(9.1)  G(D; r) = μ(r_{1})G(D_{1}D_{2}; r_{2}).

Finally we see from (8.5) and (8.6) that G(D_{1}D_{2}; r_{2}) = 0, unless D_{2} and r_{2} admit simultaneously factorization of the form

(9.2)  r_{2} = 2^{γ}b_{1}^{∗}b_{1}^{2}b_{2}^{2}b_{3}^{2},  D_{2} = 2^{α}b_{1}^{2}(b_{2}^{2}/b_{2}^{∗})b_{3}^{2}b′_{3} = 2^{α}d_{1}d_{2}d_{3}

with 2, b_{1}, b_{2}, b_{3} pairwise coprime, γ ≤ α + 1, b′_{3}|b_{3}^{∞} and b_{1}^{∗} being the product of distinct prime divisors of b_{1} (if this simultaneous factorization exists, it is unique). Then we have

(9.3)  \begin{aligned}G(D; r_{2}) & = G(D_{1}D_{2}; 2^{γ})G(D_{1}D_{2}; b_{1}^{∗}b_{1}^{2})G(D_{1}D_{2}; b_{2}^{2})G(D_{1}D_{2}; b_{3}^{2}) \\ & = G(D_{1}D_{2}; 2^{γ})(\frac{2^{α}D_{1}d_{2}d_{3}}{b_{1}^{∗}} )b_{1}μ(b_{2}^{∗})\frac{b_{2}}{b_{2}^{∗}} φ(b_{3}) \\ & = (\frac{2^{α}D_{1}}{b_{1}^{∗}} )φ(D_{1}, D_{2}, r_{2}), \end{aligned}

say, with

(9.4)  φ(D_{1}, D_{2}, r_{2}) = G(D_{1}D_{2}; 2^{γ})(\frac{2^{α}d_{2}d_{3}}{b_{1}^{∗}} )b_{1}μ(b_{2}^{∗})\frac{b_{2}}{b_{2}^{∗}} φ(b_{3}).

If the factorization (9.2) does not hold we define φ(D_{1}, D_{2}, r_{2}) to be zero. Summarizing what we have found so far we obtain

Proposition 9.1. Let D ≡ 0, 1 (mod 4) and r be such that 4r is squareful. Let D = D_{1}D_{2} and r = r′_{1}r_{2} be the factorizations such that D_{1} is squarefree, 4D_{2} is squareful, (D_{1}, 2D_{2}) = 1, r′_{1}|D_{1}^{∞} and (D_{1}, r_{2}) = 1. Then G(D; r) = 0, unless r′_{1} = r_{1}^{2} with r_{1}|D_{1} and r_{2}, D_{2} admit the factorizations (9.2). In this case one has

G(D; r) = μ(r_{1})(\frac{D_{1}}{b_{1}^{∗}} )φ(D_{1}, D_{2}, r_{2})

where b_{1}^{∗} is defined by (9.2) and φ(D_{1}, D_{2}, r_{2}) is given by (9.4).

Remark. It will be useful to know that for D_{2}, r_{2} fixed φ(D_{1}, D_{2}, r_{2}) depends only on the congruence class of D_{1} mod 8, see (8.6).

10. An upper bound for a sum of quadratic forms

Given A, B ≥ 1, C ≥ N, and two sequences {x_{a}}_{a≥1} , {y_{b}}_{b≥1} of complex numbers of modulus bounded by one, our goal is to obtain an upper bound for the average of the quadratic forms

ℬ(A, B, C) = ∑_{{c≤C \atop c≡0(N)}} | ∑_{D≡0,1(4)} G(D; c) ∑ ∑_{{a≤A,b≤B \atop ab=D}} x_{a}y_{b}|.

To this end we use the computations of G(D; c) from the previous sections.

Proposition 10.1. For x_{a}, y_{b} any complex numbers of modulus bounded by one,

ℬ(A, B, C) ≪ (ABC)^{ε}(AB)^{1/2}(\frac{C}{N} + A)^{1/2}(\frac{C}{N} + B)^{1/2}

for all ε > 0, the implied constant depending on ε.

Proof. In view of Propositions 8.1 and 9.1, we have

\begin{aligned}ℬ(A, B, C) & = ∑ ∑ ∑_{{q, r_{1}, r_{2} \atop qr_{1}^{2}r_{2}∼C} ; qr_{1}^{2}r_{2}≡0(N)}| ∑ ∑_{{D_{1},D_{2},(D_{1},D_{2})=1 \atop D_{1}D_{2}≡0,1(4), r_{1}|D_{1}}} (\frac{D_{1}}{b_{1}^{∗}} )φ(D_{1}, D_{2}, r_{2}) \\ & × ∑ ∑_{{a≤A, b≤B \atop ab=D_{1}D_{2}}} x_{a}y_{b}(\frac{ab}{q})| \end{aligned}

where q and D_{1} are squarefree and odd, 4r_{2} and 4D_{2} are squareful and have the same prime factors, and b_{1}^{∗} is a certain positive number depending on the pair {D_{2}, r_{2}}. We write a = a_{1}a_{2} , b = b_{1}b_{2} with a_{1} = (a, D_{1}), b_{1} = (b, D_{1}) (recall that D_{1} is squarefree) so that a_{1}b_{1} = D_{1} . We decompose the sum according to the congruence classes of a_{1}, b_{1} modulo 8 (see the remark at the end of Section 9) and so our sum is bounded by 16 sums each one of the type

∑_{D_{2}} ∑ ∑_{{r_{1},r_{2} \atop (r_{1},D_{2})=1}} |φ(uv, D_{2}, r_{2})| ∑_{a_{2}b_{2}=D_{2}} \begin{matrix}∑ & ♭ \\ q≡0(N/(N,r_{1}r_{2})) \end{matrix}| ^{∑♭}^{∑♭} ; {a,b, r_{1}|ab \atop (a,b)=(ab,2D_{2})=1} x_{aa_{2}}y_{bb_{2}}(\frac{ab}{q})|

where u, v are odd congruence classes modulo 8 and the last three sums are restricted by qr_{1}^{2}r_{2} ≤ C, aa_{2} ≤ A, bb_{2} ≤ B, respectively. Here the x_{a}, y_{b} may differ from the original ones by multiplication by complex numbers of modulus 0 or 1 (in particular we transfer the congruence conditions a ≡ u(8), b ≡ v(8) into the coefficients x_{a}, y_{b}). Next we estimate the last three sums by τ(r_{1}) sums of the form

\begin{matrix}∑ & ♭ \\ q≤Q \end{matrix}|^{∑♭}^{∑♭} ; {a,b \atop (a,b)=1} x_{a}y_{b}(\frac{ab}{q})|

with Q = C(N, r_{1}r_{2})/Nr_{1}^{2}r_{2} , a ≤ A/c, b ≤ B/d, cd = r_{1}D_{2} . Then we use the following result which is an immediate consequence of Theorem 1.6.

Proposition 10.2. For any complex numbers x_{a}, y_{b} we have

\begin{matrix}∑ & ♭ \\ q≤Q \end{matrix}|^{∑♭}^{∑♭} ; {a≤A, b≤B \atop (a,b)=1} x_{a}y_{b}(\frac{ab}{q})| ≪ (ABQ)^{ε}(Q + A)^{1/2}(Q + B)^{1/2}∥x∥ ∥y∥

for all ε > 0, the implied constant depending on ε only.

Using Proposition 10.2 we infer that ℬ(A, B, C) satisfies

\begin{aligned}ℬ(A, B, C) & ≪_{ε} (ABC)^{ε}(AB)^{1/2} ∑_{D_{2}≤AB} ∑ ∑_{r_{1}^{2}r_{2}≤C} \frac{|φ(u, D_{2}, r_{2})|}{D_{2}^{1/2}r_{1}^{1/2}} \\ & × [\frac{C(N, r_{1}r_{2})}{Nr_{1}^{2}r_{2}} + \frac{C^{1/2}(N, r_{1}r_{2})^{1/2}}{N^{1/2}r_{1}r_{2}^{1/2}} (A^{1/2} + B^{1/2}) + \frac{(AB)^{1/2}}{D_{2}r_{1}} ] \end{aligned}

where u is an odd congruence class modulo 8. First we sum over r_{1} , using the following three estimates

∑_{r_{1}≤C^{1/2}} \frac{(N, r_{1})}{r_{1}^{5/2}},  ∑_{r_{1}≤C^{1/2}} \frac{(N, r_{1})^{1/2}}{r_{1}^{3/2}} ,  ∑_{r_{1}≤C^{1/2}} \frac{1}{r_{1}^{3/2}} ≪ τ(N),

we arrive at

∑ ∑_{D_{2}≤AB, r_{2}≤C} |φ(u, D_{2}, r_{2})|(\frac{(N, r_{2})}{D_{2}^{1/2}r_{2}} + \frac{(N, r_{2})^{1/2}}{D_{2}^{1/2}r_{2}^{1/2}} + \frac{1}{D_{2}^{3/2}} ).

Now appealing to the definition of φ(u, D_{2}, r_{2}) (see (9.4)) we find that this is bounded by O_{ε}((ABCN)^{ε}) for all ε > 0 (to see this we recall that the variables D_{2} , r_{2} run over lacunary sets of integers, essentially squarefull numbers). Combining the above estimates we complete the proof of Proposition 10.1.

482 H. Iwaniec and P. Michel

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Received 20 March 2000