PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 86(100) (2009), 97–105 DOI: 10.2298/PIM0900097E

A RELATION BETWEEN FOURIER COEFFICIENTS OF HOLOMORPHIC CUSP FORMS AND EXPONENTIAL SUMS

Anne-Maria Ernvall-Hytönen

Communicated by Aleksandar Ivić

Abstract. We consider certain specific exponential sums related to holomorphic cusp forms, give a reformulation for the Lehmer conjecture and prove that certain exponential sums of Fourier coefficients of holomorphic cusp forms contain information on other similar non-overlapping exponential sums. Also, we prove an Omega result for short sums of Fourier coefficients.

1. Introduction

Holomorphic cusp forms can be represented as Fourier series

F (z) = ∑_{n=1}^{∞} a(n)n^{(κ−1)/2} e(nz),

where Imz > 0, e(x) = e^{2πix} , and the numbers a(n) are called normalized Fourier coefficients and κ is the weight of the form; see e.g. [1] or [13] for an account of the theory of holomorphic modular forms. For properties of exponential sums and related techniques, see [10].

It is of interest to consider exponential sums of the normalized Fourier coefficients:

A(M, Δ, α) = ∑_{M⩽n⩽M+Δ} a(n) e(nα)

with 0 < Δ ⩽ M and α ∈ ℝ. For similar exponential sums involving the divisor function d(n) = ∑ _{d|n} 1, the notation D(M, Δ, α) will be used. Wilton’s estimate [17]

∑_{n⩽M} a(n) e(nα) ≪ M^{1/2} logM

from the year 1929 is a classical result. This estimate is nearly sharp, only the logarithm can be removed and that was done by Jutila in 1987 [11]. Therefore, moving the focus to short sums was a logical next step. Karppinen and Ernvall-Hytönen [5] proved that, for 1 ⩽ Δ ≪ M^{3/4} ,

2000 Mathematics Subject Classification: Primary 11L07; Secondary 11F11, 11F30.

A(M, Δ, α) ≪ \{ \begin{matrix}ΔM^{ε}, & when 1 ⩽ Δ ≪ M^{2/5} \\ Δ^{1/6} M^{1/3+ε} , & when M^{2/5} ≪ Δ ≪ M^{5/8} \\ ΔM^{−9/48+ε} , & when M^{5/8} ≪ ΔM^{11/16} \\ M^{−1/4} Δ + M^{1/2−1/32+ε} , & when M^{11/16} ≪ ΔM^{3/4} . \end{matrix}

In this article, we will consider the sum

∑_{M⩽n⩽M+Δ} c(n)e( \frac{n\sqrt{k}}{\sqrt{M}} ) w(n),

where c(n) is either a(n) or d(n), k ∈ ℕ, and w is a smooth weight function. In particular, we will show a connection between this sum with c(n) = a(n) and the coefficient a(k). For k = 1, such a relation was established in [5] for c(n) = a(n) and in [4] for c(n) = d(n). We will also show that this sum contains information about similar shifted (not necessarily overlapping) sums.

Also, we will show the Ω-result

∑_{M⩽n⩽M+c\sqrt{M}} a(n) = Ω(M^{1/4} ),

where the Ω-symbol is to be understood in the following way: f = Ω (g) if f = o(g) does not hold. The question of good Ω-results has been earlier tackled by several mathematicians, Joris [9], Redmond [16], Corrádi and Katai [2], to mention a few. In 1989, Ivić and Hafner [6] proved the existence of a positive constant D such that

∑_{n⩽M} a(n)n^{(κ−1)/2} = Ω_{±} ( M^{κ/2−1/4} exp( D \frac{(loglogM)^{1/4}}{(logloglogM)^{3/4}} ) ) ,

where Ω_{±} means the following: f = Ω_{±} (g) if lim supf/g >0 and lim inf f/g <0. One year later appeared Ivić’s paper [8] in which he showed that there are A, B, T_{0} > 0 such that, for T ⩾ T_{0}, every interval [T, T + A\sqrt{T}] contains t_{1} and t_{2} for which A(1, t_{1} − 1, 0) > Bt_{1}^{1/4} and A(1, t_{2} − 1, 0) < −Bt_{2}^{1/4} . Very recently, Ivić [7] proved an Ω-result for short sums:

A(M, Δ, 1) = Ω(\sqrt{Δ} )

when M^{ε} ⩽ Δ ⩽ M^{1/2−ε} . The result in this article extends this result by treating the “missing" case Δ ≍ M^{1/2} .

The author would like to thank professors Jutila and Ivić for valuable insight and comments.

2. Preliminaries

Let us begin with

Definition 2.1. Given X, Y, Z ∈ ℝ we write

d(X, Y, Z) = {x ∈ ℂ : ∃y ∈ [X, Y] : |x − y| < Z} .

Now we may state a lemma [12, Lemma 6] which will be used repeatedly in this article:

Lemma 2.1. Let A be a function which is compactly supported in a finite interval [M_{1}, M_{2}] and at least P ⩾ 0 times differentiable. Assume also that there exist two natural numbers A_{0} and A_{1} such that for any non-negative integer ν ⩽ P and for any x ∈ [M_{1}, M_{2}],

A^{(ν)}(x) ≪ A_{0}A_{1}^{−ν} .

Also, let B be a function which is real-valued on [M_{1}, M_{2}], and regular throughout the complex domain d(M_{1}, M_{2}, ρ); and assume that there exists a quantity B_{1} such that

0 < B_{1} ≪ |B′(x)|

for any point x in the domain. Then we have

∫_{−∞}^{∞} A(x) e(B(x)) dx ≪ A_{0}(A_{1}B_{1})^{−P} ( 1 + \frac{A_{1}}{ϱ} )^{P}(M_{2} − M_{1}).

3. Connecting exponential sums and individual coefficients

The following theorem was proved in [5]:

Theorem 3.1. Let M^{1/2+δ} < Δ ⩽ λM^{3/4} , where 0 < λ < 1 is a constant. Let w be a smooth weight function on the interval [M, M + Δ] which equals 1 on the interval [a, b] ⊂ [M, M + Δ] where a − M = M + Δ − b = Δ^{1−δ} with δ a sufficiently small fixed positive real number. Assume further that α = M^{−1/2} . Then

| ∑_{M⩽n⩽M+Δ} a(n) w(n) e(αn)| ≍ ΔM^{−1/4} .

The symbol ≍ has to be understood in the following way: f ≍ g if f = O(g) and g = O(f).

However, the following more general theorem holds:

Theorem 3.2. Let M^{1/2+θ} ≪ Δ ⩽ λM^{3/4} and 0 ⩽ T ⩽ M^{3/4} , where 0 < λ ⩽ 1/\sqrt{k} is a constant, θ an arbitrarily small fixed positive number, k a positive integer, and let w be a smooth weight function on the interval [M, M + Δ] such that w is a constant function 1 on the interval [a, b] ⊂ [M, M + Δ] where a − M, M + Δ − b = Δ^{1−δ} with δ < \frac{2θ}{1+2θ} a sufficiently small fixed positive real number. Then

∑_{M+T ⩽n⩽M+T +Δ} c(n) w(n − T) e( \frac{\sqrt{k}n}{\sqrt{M}})

= Cc(k)k^{−1/4} ∫_{M+T}^{M+T +Δ} x^{−1/4} w(x − T) e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} x − 2\sqrt{kx}) dx + O(1),

where c(n) = a(n) or d(n) and C is a constant depending only whether c(n) equals d(n) or a(n) and on the weight of the form.

Notice that the size of the integral is ≍ M^{−1/4} Δ. This can be easily proved using the fact that the exponential part is stationary.

Proof of Theorem 3.2. The proof for c(n) = a(n) with k = 1 and T = 0 can be found in [5] and the proof for both c(n) = d(n) and c(n) = a(n) with k = 1 and T = 0 can be found in [4] and the proof of the above formula is similar. As the case with c(n) = a(n) is easier and similar to the case c(n) = d(n), we are only going to prove the latter case.

Let us first use a Voronoi type summation formula [10, Theorem 1.7]

D( M + T, Δ, \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} ) = ∫_{M+T}^{M+T +Δ} (logx + 2γ) w(x − T) e( \frac{\sqrt{k}x}{\sqrt{M}}) dx

+ ∑_{n=1}^{∞} d(n) ∫_{M+T}^{M+T +Δ} \{−2πY_{0}(4π\sqrt{nx} ) + 4K_{0}(4π\sqrt{nx} )\}w(x − T) e( \frac{\sqrt{k}x}{\sqrt{M}}) dx,

where Y_{0} and K_{0} are Bessel functions in the standard notation. The following estimate is well known (see formula (5.16.5) of [14])

K_{0}(z) ∼ \sqrt{\frac{π}{2z}} e^{−z} , when z → ∞.

Therefore, the integral corresponding to the K-function yields

∫_{M+T}^{M+T +Δ} 4K_{0}(4π\sqrt{nx} ) w(x − T) e( \frac{\sqrt{k}x}{\sqrt{M}}) dx

≪ \frac{1}{n^{1/4}} ∫_{M+T}^{M+T +Δ} x^{−1/4} e^{−4π\sqrt{nx}} dx ≪ n^{−3/2} .

Hence, the corresponding sums converges to O(1) (as a function of M). Let us now move to the Y -Bessel function. We write it first using Hankel functions [14, (5.6.1)]:

Y_{0}(z) = \frac{1}{2i}(H_{0}^{(1)}(z) − H_{0}^{(2)}(z)).

The asymptotic expansions for the Hankel functions [14, (5.11.5)] give

(3.1)  H_{0}^{(j)} (z) = ( \frac{2}{πz} )^{1/2} e^{i(−1)^{j−1} (z−π/4)} (1 + c_{1j} z^{−1} + O(|z|^{−2} )).

The first step to treat these terms is first to integrate and then sum over the O-term:

∑_{n=1}^{∞} d(n) ∫_{M+T}^{M+T +Δ} (nx)^{−5/4} dx ≪ ∑_{n=1}^{∞} d(n) n^{−5/4} ΔM^{−5/4} ≪ 1.

Use Lemma 2.1 to treat the integral over the second term in (3.1), except in the case of n = d, with the following choices: M_{2} − M_{1} = Δ, ϱ = \frac{1}{2}M, A_{1} = Δ^{1−δ} ,

B_{1} ≍ \begin{matrix}√ & \overline{n} \\ √ & \overline{M} \end{matrix} and A_{0} = n^{−3/4} M^{−3/4} . We obtain

∫_{M+T}^{M+T +Δ} (nx)^{−3/4} e( \frac{x\sqrt{k}}{\sqrt{M}} ± 2\sqrt{nx}) w(x − T) dx
≪ Δ^{−P (1−δ)+1} n^{−P/2−3/4} M^{P/2−3/4}
≪ n^{−P/2−3/4} M^{−P (1/2+θ)(1−δ)+P/2−1/4+θ} ≪ n^{−P/2−3/4} .

Therefore, the series converges and produces an error term of size O(1). When n = k, use integration over the absolute values to obtain the same estimate. Let us now treat the integral corresponding to the first term in the asymptotic Expan-Sion (3.1). When n ≠ k, we obtain by use of Lemma 2.1 the estimate

≪ Δ^{(1−δ)(1−P)} n^{−P/2−1/4} M^{P/2−1/4}
≪ M^{−P (1/2+θ) (1−δ)+P/2+1/4+θ} n^{−P/2−1/4} ≪ n^{−P/2−1/4} ,

when P is sufficiently large. When n = k, the first term in the asymptotic expansion for H_{0}^{(1)} also gives the same estimate. Hence, we have now derived

D( M + T, Δ, \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} ) = ∫_{M+T}^{M+T +Δ} ( 2γ + logx − cd(k)k^{−1/4} x^{−1/4} e(−2\sqrt{kx} ))

× w(x − T) e( \frac{x\sqrt{k}}{\sqrt{M}} ) dx + O(1),

where c is a constant. Write q(x) = w(x − T)(lnx + 2γ). Now q^{(P)}(x) ≪ Δ^{(1−δ)(ε−P)} . Using Lemma 2.1 we obtain

∫_{M+T}^{M+T +Δ} q(x) e( \frac{x\sqrt{k}}{\sqrt{M}} ) dx ≪ Δ^{(1−δ)(ε−P)+1} M^{P/2}
≪ M^{P/2+(1−δ)(ε−P)+1/2+θ} ≪ 1.

This proves the theorem. □

As a simple corollary, we obtain

Corollary 3.1. With the assumptions of the previous theorem and supposing, moreover, that a(k) = 0, we have

∑_{M⩽n⩽M+Δ} a(n) w(n) e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} n) = O(1).

On the other hand, if a(k) ≠ 0, then

∑_{M⩽n⩽M+Δ} a(n) w(n) e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} n) ≍ M^{−1/4} Δ.

In other words, the Lehmer conjecture for the eigenfunctions of the Hecke operators is equivalent to the corresponding sums being large.

Remark 3.1. Notice that if a(k) ≠ 0, then

| ∑_{M⩽n⩽M+Δ′} a(n) e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} n) | ≫ M^{−1/4} Δ

for some Δ′ ∈ (0, Δ]. Otherwise, it would follow from partial summation that the estimate for the smoothed sum would be o(M^{−1/4} Δ).

102 ERNVALL-HYTÖNEN

Theorem 3.3. With the assumptions of Theorem 3.2, the following holds:

∑_{M⩽n⩽M+Δ} c(n) w(n) e( \frac{\sqrt{k}n}{\sqrt{M}}) − ∑_{M+T ⩽n⩽M+T +Δ} c(n) w(n − T) e( \frac{\sqrt{k}n}{\sqrt{M}})

≪ \frac{T Δ(T + Δ)}{M^{7/4}} + \frac{Δ(Δ + T)}{M^{5/4}} + 1.

Proof. Using Theorem 3.2, we see that it is sufficient to consider the difference

∫_{M}^{M+Δ} x^{−1/4} w(x) e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} x − 2\sqrt{kx}) dx

− ∫_{M}^{M+Δ} (x + T)^{−1/4} w(x) e( \frac{\sqrt{k}(x + T)}{\sqrt{M}} x − 2\sqrt{k(x + T)}) dx.

We first use the Taylor expansion to treat the terms x^{−1/4} and (x + T)^{−1/4} :

x^{−1/4} = M^{−1/4} + O(M^{−5/4} |x − M|).

Hence,

∫_{M}^{M+Δ} ( \frac{w(x)}{x^{1/4}} e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} x − 2\sqrt{kx}) − \frac{w(x)}{(x+T)^{1/4}} e( \frac{\sqrt{k}(x+T)}{\sqrt{M}} x − 2\sqrt{k(x+T)})) dx

= M^{−1/4} ∫_{M}^{M+Δ} w(x)( e( \frac{\sqrt{k}x}{\sqrt{M}} − 2\sqrt{kx}) − e( \frac{\sqrt{k}(x + T)}{\sqrt{M}} − 2\sqrt{k(x + T)})) dx

+ O ( \frac{Δ(Δ + T)}{M^{5/4}} ) .

Let us now consider the difference

|e( \frac{\sqrt{k}x}{\sqrt{M}} − 2\sqrt{kx}) − e( \frac{\sqrt{k}(x + T)}{\sqrt{M}} − 2\sqrt{k(x + T)}) |

= |e( \frac{\sqrt{k}x}{\sqrt{M}} − 2\sqrt{kx} − \frac{\sqrt{k}(x + T)}{\sqrt{M}} + 2\sqrt{k(x + T)}) − 1|.

Since |e^{iy} − 1| ⩽ |y|, it is sufficient to consider the exponent to obtain an upper bound for the difference of the exponent functions, and thereby for the original integral expression:

| \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} x − 2\sqrt{kx} − \frac{\sqrt{k}(x + T)}{\sqrt{M}} + 2\sqrt{k(x + T)} | ≪ \frac{T (Δ + T)}{M^{3/2}} .

We obtain

∫_{M}^{M+Δ} ( \frac{w(x)}{x^{1/4}} e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} x−2\sqrt{kx}) − \frac{w(x)}{(x+T)^{1/4}} e( \frac{\sqrt{k}(x+T)}{\sqrt{M}} x−2\sqrt{k(x+T)})) dx

≪ \frac{T Δ(T + Δ)}{M^{7/4}} + \frac{Δ(Δ + T)}{M^{5/4}} . □

4. An Omega-result for short sums of Fourier coefficients

Theorem 4.1. Let c > 0 be an arbitrary real number. Then

∑_{M⩽n⩽M+c\sqrt{M}} a(n) = Ω(M^{1/4} ).

Before proving the theorem, let us prove a lemma:

Lemma 4.1. Write D = Kc^{4}, where K is a sufficiently large constant. Write ∥x∥ to denote the distance from x to the nearest integer. Let b be a sufficiently large constant. Then it is possible to choose an integer k ∈ [b^{−1} D, bD] such that the following two conditions are satisfied: (1) ∥c\sqrt{k}∥ > D^{−1/4} , (2) a(k) ≠ 0.

Proof. First, consider the difference

c\sqrt{k + 1} − c\sqrt{k} = \frac{c}{\sqrt{k} + \sqrt{k + 1}} ≍ D^{−1/2} .

Therefore, the values ∥c\sqrt{k}∥ are somewhat uniformly distributed on the interval [ 0, 1). It is now easy to conclude that only ≍ D^{3/4} of k ∈ [b^{−1} D, bD] satisfy the condition ∥c\sqrt{k}∥ ⩽ D^{−1/4} . Since a(k) ≪ k^{ε} by Deligne’s estimate [3], we obtain

∑_{\begin{matrix}b^{−1} D⩽k⩽bD, \\ ∥c\sqrt{k}∥<D^{−1/4} \end{matrix}} |a(k)|^{2} ≪ D^{3/4+ε} .

The Rankin–Selberg mean value theorem (see e.g. Rankin [15]) gives the estimate

∑_{\begin{matrix}b^{−1} D⩽k⩽bD, \\ ∥c\sqrt{k}∥>D^{−1/4} \end{matrix}} |a(k)|^{2} + O(D^{3/4+ε} ) ≍ D,

which proves the existence of a coefficient satisfying both conditions. □

We may now turn to the proof of the actual theorem.

Proof of Theorem 4.1. Take k as in Lemma 4.1. From the first condition we obtain

| ∑_{0⩽h⩽c\sqrt{M}} e( \frac{h\sqrt{k}}{\sqrt{M}}) | = | \frac{1 − e(⌊c\sqrt{M}⌋\sqrt{k}M^{−1/2} + \sqrt{k}M^{−1/2} )}{1 − e(\sqrt{k}M^{−1/2} )} | ≫ M^{1/2} ,

since the denominator is ≍ M^{−1/2} \sqrt{k} ≍ M^{−1/2} as k is a constant, and the nominator is ≍ 1 by condition (1) of Lemma 4.1. From Remark 3.1 we know that there exists Δ′ ⩽ λM^{3/4} , where λ ∈ (0, 1) is a constant, such that

| ∑_{M⩽n⩽M+Δ′} a(n) e( \begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} ) | ≫ M^{1/2} .

Multiplying these two sums together, we obtain

M ≪ |( ∑_{0⩽h⩽c\sqrt{M}} e( \frac{h\sqrt{k}}{\sqrt{M}}) ) ( ∑_{M⩽n⩽M+Δ′} a(n) e( \frac{n\sqrt{k}}{\sqrt{M}})) |.

Change the variable m = h + n and estimate further:

= |  ∑_{M⩽m⩽M+Δ′+c\sqrt{M}} e( m\begin{matrix}√ & \overline{k} \\ √ & \overline{M} \end{matrix} ) ∑_{n∈[M,M+Δ′]∩[m−c\sqrt{M} ,m]} a(n)|

⩽ |  ∑_{M+c\sqrt{M} ⩽m⩽M+Δ′} e( \frac{c\sqrt{k}}{\sqrt{M}}) ∑_{m−c\sqrt{M} ⩽n⩽m} a(n)|

+ |  ∑_{M⩽m<M+c\sqrt{M}} e( \frac{c\sqrt{k}}{\sqrt{M}}) ∑_{n∈[M,M+Δ′]∩[m−c\sqrt{M} ,m]} a(n)|

+ |  ∑_{M+Δ′<m⩽M+Δ′+c\sqrt{M}}  e( \frac{c\sqrt{k}}{\sqrt{M}}) ∑_{n∈[M,M+Δ′]∩[m−c\sqrt{M} ,m]} a(n)|

We may now use the well-known estimate (see [10]) ∑ _{n⩽M} a(n) ≪ M^{1/3+ε} to treat the second and third term and then use the triangle inequality to the first term to obtain

≪ ∑_{M+c\sqrt{M} ⩽m⩽M+λM^{3/4}}  |  ∑_{m−c\sqrt{M} ⩽n⩽m} a(n)| + M^{5/6} ,

Therefore the mean of the sums |∑ _{m−c\sqrt{M} ⩽n⩽m} a(n)| is ≫ M^{1/4} and hence, at least one of them has to be ≫ M^{1/4} . This proves the theorem. □

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Department of Mathematics (Received 08 12 2008) University of Turku (Revised 27 03 2009) Turku Finland

anmaer@utu.fi