Youness Lamzouri
email: lamzouri@mathstat.yorku.ca

LARGE SUMS OF HECKE EIGENVALUES OF HOLOMORPHIC CUSP FORMS

Keywords:

Let f be a Hecke cusp form of weight k for the full modular group, and let {λ f (n)} n≥1 be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of λ f (n), we investigate the range of x (in terms of k) for which there are cancellations in the sum S f (x) = n≤x λ f (n). We first show that S f (x) = o(x log x) implies that λ f (n) < 0 for some n ≤ x. We also prove that S f (x) = o(x log x) in the range log x/ log log k → ∞ assuming the Riemann hypothesis for L(s, f ), and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which S f (x) A x log x, when x = (log k) A . Our results are GL 2 analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

Introduction

Let k be a positive even integer, and denote by H k the set of Hecke cusp forms of weight k for the full modular group Γ = SL(2, Z). Then, H k is an orthonormal basis for the space of holomorphic cusp forms of weight k for Γ and we have

|H k | = k 12 + O k 2/3 .
Given f ∈ H k , its Fourier expansion can be written in the form

f (z) = ∞ n=1 λ f (n)n (k-1)
/2 e(nz), for Im(z) > 0, where e(z) = e 2πiz . The λ f (n) are the normalized eigenvalues of the Hecke operators T n , and satisfy the well-known Hecke relations:

(1.1)

λ f (m)λ f (n) = d|(m,n) λ f mn d 2 ,
for all m, n ≥ 1. In particular, λ f is a real-valued multiplicative function of n. Moreover, it also satisfies the following deep bound due to Deligne

(1.2) |λ f (n)| ≤ τ (n),
where τ is the divisor function. These facts are standard and may be found for example in Chapter 14 of [START_REF] Iwaniec | Analytic number theory[END_REF].

In [START_REF] Kowalski | On modular signs[END_REF], Kowalski, Lau, Soundararajan and Wu studied the signs of the sequence λ f (n). Their results show a strong analogy between these signs and the values of quadratic Dirichlet characters, and especially between the first negative Fourier coefficient and the problem of the least quadratic non-residue, which has a long history in analytic number theory. Let n f be the smallest positive integer n such that λ f (n) < 0. The best known bound for n f is due to Matomäki [START_REF] Matomäki | On signs of Fourier coefficients of cusp forms[END_REF], who improved the authors of [START_REF] Kowalski | On modular signs[END_REF] by showing that

n f k 3/4 .
This is probably far from the truth, since it is known that n f (log k) 2 under the assumption of the generalized Riemann hypothesis (GRH). In the other direction, Theorem 3 of [START_REF] Kowalski | On modular signs[END_REF] shows that n f √ log k for many Hecke cusp forms f of weight k. A folklore conjecture asserts that the correct order of magnitude for the maximal values of n f should be (log k) 1+o (1) .

In this paper, we explore GL 2 analogues of certain classical problems concerning short character sums and the least quadratic non-residue. More precisely, we investigate the size of the short sum of Hecke eigenvalues

S f (x) := n≤x λ f (n),
and its relation to the first negative Fourier coefficient of f . Our results are inspired by the work of Granville and Soundararajan [START_REF] Granville | Large character sums[END_REF] on character sums. In particular, Corollaries 1.2 and 1.4 below can be regarded as GL 2 analogues of Corollary A of [START_REF] Granville | Large character sums[END_REF].

Using Deligne's bound (1.2), one obtains the "trivial" bound

S f (x) ≤ n≤x τ (n) = (1 + o(1))x log x.
Our first result shows that if S f (x) is substantially smaller than this bound, namely that

(1.3) S f (x) = o(x log x) (as x, k → ∞),
then we must have n f ≤ x. The proof relies on arguments of Kowalski, Lau, Soundararajan and Wu [START_REF] Kowalski | On modular signs[END_REF], together with ideas of Granville and Soundararajan [START_REF] Granville | The spectrum of multiplicative functions[END_REF] concerning mean values of multiplicative functions.

Theorem 1.1. Let f ∈ H k . There exist absolute positive constants b 0 , c 0 such that if x ≥ b 0 and λ f (n) ≥ 0 for all n ≤ x then n≤x λ f (n) ≥ c 0 x log x. Let f ∈ H k . The L-function attached to f is defined by (1.4) L(s, f ) = ∞ n=1 λ f (n) n s = p 1 - e iθ f (p) p -1 p 1 - e -iθ f (p) p -1 , for Re(s) > 1,
where

θ f (p) ∈ [0, π].
It is known that L(s, f ) extends analytically to the entire complex plane, and satisfies a functional equation that relates L(s, f ) to L(1 -s, f ) (see for example Section 5.11 of [START_REF] Iwaniec | Analytic number theory[END_REF]). A standard application of Perron's formula together with the convexity bound for L(s, f ) imply that

(1.5) S f (x) x 1/2+ε • k 1/2+ε ,
and hence one has S f (x) = o(x log x) in the range x ≥ k 1+ε . This range can be improved to x ≥ k 1-δ , for some δ > 0, by using subconvexity bounds for L(s, f ) (see for example [START_REF] Michel | The subconvexity problem for GL 2[END_REF]). Furthermore, assuming GRH for L(s, f ) one has the much stronger bound

(1.6) S f (x) x 1/2+ε exp c 1 log k log log k ,
for some absolute constant c 1 > 0. This shows that (1.3) is valid in the larger range

x ≥ exp c 2 log k log log k for some constant c 2 > 0, conditionally on the GRH. Exploiting an idea of Montgomery and Vaughan [START_REF] Montgomery | Exponential Sums with Multiplicative Functions[END_REF], we substantially improve this range under the assumption of GRH.

Corollary 1.2. Let f ∈ H k , and assume GRH for L(s, f ). In the range log x/ log log k → ∞, we have

n≤x λ f (n) = o(x log x).
We shall deduce this result from the following theorem, which shows that under GRH, we can approximate S f (x) by the corresponding sum of λ f (n) over friable (or smooth) numbers n, which are positive integers having only small prime factors. A positive integer n is said to be y-friable if P (n) ≤ y, where P (n) denotes the largest prime factor of n, with the standard convention P (1) = 1.

Theorem 1.3. Let f ∈ H k , and assume GRH for L(s, f ). Then, for all real numbers x, y such that

(log k) 2 (log log k) 8 ≤ y < x ≤ k we have n≤x λ f (n) = n≤x P (n)≤y λ f (n) + O (log k)(log y) 4 √ y x log x .
For an arithmetic function g, we define Ψ(x, y; g) :

= n≤x P (n)≤y g(n).
The asymptotic behaviour of Ψ(x, y; g) was investigated for a large class of multiplicative functions g by several authors, and notably by Tenenbaum and Wu [START_REF] Tenenbaum | Moyennes de certaines fonctions multiplicatives sur les entiers friables[END_REF]. When g is the divisor function τ , de Bruijn and van Lint [START_REF] De Bruijn | Incomplete sums of multiplicative functions, I, II[END_REF] proved that there exists a continuous function

ρ 2 : [0, ∞) → R such that (1.7)
Ψ(x, y; τ ) ∼ ρ 2 (u) • x log y, where u := log x log y , in the range u 1. The function ρ 2 is defined by the differential-difference equation

(1.8) uρ 2 (u) = ρ 2 (u) -2ρ 2 (u -1),
subject to the initial condition ρ 2 (u) = u for 0 ≤ u ≤ 1. It is known that ρ 2 (u) > 0 for any u > 0 and that ρ 2 (u) = u -u(1+o(1)) for large u (see for example [START_REF] Hensley | The convolution powers of the Dickman function[END_REF]). In fact, ρ 2 is the square convolution of the standard Dickman-de Bruijn function ρ, which appears in the asymptotic formula for the counting function of friable integers. The range of validity of the asymptotic formula (1.7) was improved to u ≤ exp (log y) 3/5-ε by Smida [START_REF] Smida | Valeur moyenne des fonctions de Piltz sur les entiers sans grand facteur premier[END_REF], and hence in this range we have Recall that n f √ log k for many Hecke cusp forms f of weight k by Theorem 3 of [START_REF] Kowalski | On modular signs[END_REF]. In view of Theorem 1.1, this shows that (1.10) is valid for such f with x = √ log k. Using probabilistic methods we improve this range, by showing that for any A > 1, there are many Hecke cusp forms f of weight k such that (1.10) holds for x = (log k) A . This implies that the range of validity of Corollary 1.2 is best possible, and that conditionally on GRH the converse of Theorem 1.1 does not hold.

Ψ(x, y; λ f ) ≤ Ψ(x, y; τ ) u -u(1+o(
Corollary 1.4. Let k be a large even integer. Let A > 1 be fixed, and x = (log k) A . There are at least k 1-1/ log log k Hecke cusp forms f ∈ H k such that

n≤x λ f (n) A x log x.
We shall deduce this result from the following theorem.

Theorem 1.5. Let k be a large even integer. Let A > 1 be fixed, y = log k/ log log k and x = (log k) A . There are at least k 1-1/ log log k Hecke cusp forms f ∈ H k such that

n≤x λ f (n) ≥ Ψ(x, y; τ ) 1 + O 1 √ log log k .
The key idea in the proof of Theorem 1.5 is to compare large moments of S f (x) (as f varies in H k ) with those of a corresponding probabilistic random model. This model was introduced by Cogdell and Michel in [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF] to study the complex moments of symmetric power L-functions at s = 1, and was subsequently used by various authors (see for example [START_REF] Lau | Twisted moments of automorphic L-functions[END_REF] and [START_REF] Liu | On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1[END_REF]) to explore similar problems. To describe this probabilistic model we consider the compact group G = SU (2) endowed with its natural Haar measure µ G ; we then let G be the set of conjugacy classes of G endowed with the Sato-Tate measure µ st (i.e. the direct image of µ G by the canonical projection). Let {g p } p prime be a sequence of independent random variables, with values in G and distributed according to the measure µ st . We construct the sequence of random variables {X(n)} n≥1 by first defining X(p a ) = tr Sym a g p for a prime p and a positive integer a, where Sym a is the symmetric a-th power representation of the standard representation of GL 2 . We then extend the X(p a ) multiplicatively by letting X(1) = 1 and

X(n) = X(p a 1 1 ) • • • X(p a ) if the prime factorization of n is n = p a 1 1 • • • p a .
We shall explore this probabilistic model and the motivation behind it in details in Section 3. Using the Petersson trace formula (see Lemma 3.1 below), we show that in a certain range of x, large (weighted) moments of S f (x) are very close to those of the sum of random variables n≤x X(n). We then estimate the moments of this sum by first restricting the random variables X(n) to those indexed by y-friable integers n, and then controlling these by restricting the range of the random variables X(p) for the primes p ≤ y.

Our approach is flexible and could be further generalized to obtain similar results for other families of automorphic forms. In particular, our results hold mutatis mutandis for primitive Hecke cusp forms of weight 2 and prime level q (in the level aspect), with the extra condition that x < q in Theorem 1.1. One should also obtain the analogues of Theorems 1.3 and 1.5 for Fourier coefficients of the symmetric square and other symmetric power L-functions attached to primitive Hecke cusp forms, assuming their automorphy.

2. The size of S f (x) and the first negative Hecke eigenvalue: Proof of Theorem 1.1

Let p be a prime number. It follows from (1.4) that λ f (p) = 2 cos θ f (p) and more generally we have

λ f (p b ) = sin((b + 1)θ f (p)) sin θ f (p) ,
for any integer b ≥ 0, by the Hecke relation

(1.1). Let α : [0, 1] → [-2, 2] be defined by α(0) = 2 and α(t) = 2 cos(π/(m + 1)) if 1/(m + 1) < t ≤ 1/m, for m ∈ N. For x ≥ 2,
let h x be the multiplicative function supported on squarefree numbers and defined on the primes by

h x (p) = α log p log x if p ≤ x, 0 otherwise.
By exploiting the Hecke relation (1.1), we obtain the following lemma which has essentially been proved in Remark 2.2 of [START_REF] Kowalski | On modular signs[END_REF].

Lemma 2.1. Let f ∈ H k . Let x ≥ 2 be such that λ f (n) ≥ 0 for all n ≤ x. Then, we have n≤x λ f (n) ≥ n≤x h x (n).
Proof. By our assumption we have

n≤x λ f (n) ≥ n≤x λ f (n),
where restricts the summation to squarefree integers. Finally, the fact that λ f (n) ≥ h x (n) for all squarefree n follows from Remark 2.2 of [START_REF] Kowalski | On modular signs[END_REF].

In order to bound n f , Kowalski, Lau, Soundararajan and Wu [START_REF] Kowalski | On modular signs[END_REF] suggested to use an extension h x of the function h x , defined as the multiplicative function supported on the squarefree integers and satisfying h x (p) = h x (p) for p ≤ x, and h x (p) = -2 for p > x.

They also remarked (see Remark 2.2 of [START_REF] Kowalski | On modular signs[END_REF]) that using the techniques of Granville and Soundararajan [START_REF] Granville | The spectrum of multiplicative functions[END_REF], one can obtain an asymptotic of the type (2.1)

n≤x u h x (n) ∼ Cβ(u)x u (log x u )
for u fixed, where C is a positive constant, and the function β is defined using the following inclusion-exclusion formula

uβ(u) = u + j≥1 (-1) j j! I j (u),
where

I j (u) = t 1 ,...,t j ≥0 t 1 +•••+t j ≤u (u -t 1 -• • • -t j ) j i=1 (2 -α(t j )) dt 1 • • • dt j t 1 • • • t j .
In particular, one has β(u) ≥ 1 -I 1 (u)/u, and hence (2.1) gives

(2.2) n≤x h x (n) ≥ C(1 -I 1 (1) + o(1))x log x x log x,
since I 1 (1) < 1 (see the computation at the end of page 9). However, the authors of [START_REF] Kowalski | On modular signs[END_REF] did not supply a proof of (2.1), so we cannot use it to deduce (2.2). Instead, we shall provide a proof of (2.2) using ideas of Granville-Soundararajan [START_REF] Granville | The spectrum of multiplicative functions[END_REF]. We also give an explicit description of the constant C.

Proposition 2.2. Let x be large. Then we have

n≤x h x (n) ≥ C(1 -I 1 (1))x log x + O(x log log x),
where

C := p 1 + 2 p 1 - 1 p 2 .
To prove this result, we first need the following lemma.

Lemma 2.3. For all squarefree integers n ≥ 1 we have

h x (n) ≥ τ (n)   1 - p|n 1 - h x (p) 2   .
Proof. We prove this result by induction on the number of prime factors of n. First, the inequality is trivial if

n = p is prime. Now, assume that n = p 1 • • • p k is squarefree, and let m = p 1 • • • p k-1
. By our induction hypothesis we have

h x (n) = h x (m)h x (p k ) ≥ τ (n) h x (p k ) 2   1 - p|m 1 - h x (p) 2   .
The result follows upon noting that

h x (p k ) 2   1 - p|m 1 - h x (p) 2   ≥ h x (p k ) 2 - p|m 1 - h x (p) 2 = 1 - p|n 1 - h x (p) 2 , since h x (p) ≤ 2 for all p.
Proof of Proposition 2.2. Let g(p) = 1-h x (p)/2. Then, it follows from Lemma 2.3 that

(2.3) n≤x h x (n) ≥ n≤x τ (n) - n≤x τ (n) p|n g(p),
where restricts the summation to squarefree integers. We now use the following standard mean value estimate (2.4)

n≤y τ (n) = n≤y µ(n) 2 τ (n) = Cy log y + O(y), with C = p 1 + 2 p 1 - 1 p 2 ,
which follows from the Selberg-Delange method (see for example Theorem II.5.2 of [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]), upon noting that n≥1

µ(n) 2 τ (n) n s = ζ(s) 2 G(s) where G(s) = p (1 + 2/p s )(1 - 1/p s ) 2 is analytic in the half plan Re(s) > 1/2.
Thus, it remains to find a non-trivial upper bound for the second sum on the right hand side of (2.3). Using (2.4) together with the fact that 0 ≤ g(p) ≤ 1 we obtain

n≤x τ (n) p|n g(p) ≤ p≤x 2g(p) m≤x/p τ (m) = Cx p≤x 2g(p) p log x p + O(x log log x). Now, observe that p≤x 2g(p) p log x p = p≤x 2 -α(log p/ log x) p log x p = 1≤m≤ log x log 2 2 -2 cos π m + 1 x 1/(m+1) <p≤x 1/m log(x/p) p .
By partial summation, and the prime number theorem in the form π(x) = Li(x) + O(x/(log x) 3 ), we get

x 1/(m+1) <p≤x 1/m log(x/p) p = x 1/m x 1/(m+1) log(x/t) t log t dt + O m 2 log x .
Thus, we obtain

p≤x 2g(p) p log x p = 1≤m≤ log x log 2 2 -2 cos π m + 1 x 1/m x 1/(m+1) log(x/t) t log t dt + O(1) = x 2 2 -α(log t/ log x) t log x t dt log t + O(1) = log x 1 log 2 log x 2 -α(u) u (1 -u)du + O(1)
by using the change of variables u = log t/ log x. The result follows upon noting that

1 log 2 log x 2 -α(u) u (1 -u)du ≤ 1 0 2 -α(u) u (1 -u)du = I 1 (1).
We finish this section by deducing Theorem 1.1 from Lemma 2.1 and Proposition 2.2.

Proof of Theorem 1.1. Combining Lemma 2.1 and Proposition 2.2 we deduce that if x is large enough and λ f (n) ≥ 0 for all n ≤ x, then

n≤x λ f (n) ≥ C(1 -I 1 (1))x log x + O(x log log x).
Moreover, we have

I 1 (1) = 1 0 2 -α(u) u (1 -u)du = ∞ m=1 2 -2 cos π m + 1 1/m 1/(m+1) 1 -u u du = ∞ m=1 2 -2 cos π m + 1 log m + 1 m - 1 m(m + 1)
= 0.95897... This completes the proof. In order to prove Theorem 1.5, we shall compute the moments of S f (x) as f varies in H k . When so doing, we shall use the harmonic weights that arise naturally in the Petersson trace formula (see Lemma 3.1 below). The harmonic weight of f ∈ H k is defined by

ω f = Γ(k -1) (4π) k-1 f, f = 2π 2 (k -1)L(1, Sym 2 f ) ,
where f, f is the Petersson inner product, and L(s, Sym 2 f ) is the symmetric square L-function of f . Given a sequence (α f ) f ∈H k , its harmonic average is defined as the sum

h f ∈H k α f = f ∈H k ω f α f ,
and if S ⊂ H k we will let |S| h denote the harmonic measure of S, that is

|S| h := h f ∈S 1.
Moreover, the classical estimate

(3.1) |H k | h = 1 + O k -5/6 ,
together with the bounds of Goldfeld, Hoffstein and Liemann (see the Appendix of [START_REF] Hoffstein | Coefficients of Maass forms and the Siegel zero. With an appendix by D. Goldfeld[END_REF])

(3.2) 1 k log k ω f log k k ,
show that the harmonic weight ω f is close to the natural weight

1/|H k | (since |H k | k),
and it defines asymptotically a probability measure on H k . We shall use the following consequence of the Petersson trace formula which follows from Lemma 2.1 of [START_REF] Rudnick | Lower bounds for moments of L-functions: symplectic and orthogonal examples[END_REF]. Lemma 3.1. Let k be a large even integer, and n be a positive integer such that n ≤ k 2 /10 4 . Then, we have

(3.3) 1 |H k | h h f ∈H k λ f (n) = δ(n) + O k -5/6 ,
where δ(n) = 1 if n = 1, and is 0 otherwise.

Proof. It follows from Lemma Lemma 2.1 of [15] that

h f ∈H k λ f (n) = δ(n) + O e -k .
The result follows from combining this estimate with (3.1).

The formula (3.3) can be interpreted as follows: Recall that G is the set of conjugacy classes of G = SU (2) endowed with the Sato-Tate measure µ st (the direct image of the Haar measure µ G by the canonical projection). Let n > 1 and n = p a 1 1 • • • p a be its prime factorization. Then we have the identity

(3.4) λ f (n) = λ f (p a 1 1 ) • • • λ f (p a ) = tr (Sym a 1 (g f (p 1 ))) • • • tr (Sym a (g f (p ))) ,
where

g f (p) = e iθ f (p) 0 0 e -iθ f (p) .
Fix now the primes p 1 , . . . , p . By the identity (3.4) together with the Peter-Weyl Theorem and Weyl's equidistribution criterion, the estimate (3.3) applied to integers n divisible only by the primes in {p 1 , . . . , p } yields the equidistribution of the -tuple of conjugacy classes {g f (p 1 ), . . . , g f (p )} f ∈H k (appropriately weighted by ω f ) into the product of copies of G , as k → ∞. Based on this equidistribution result, we construct a probabilistic random model for the Hecke eigenvalues λ f (n) as follows: Consider a probability space (Ω, µ) and let {g p } p prime be a sequence of independent random variables defined on (Ω, µ) and taking values in G . We assume that each random variable g p is distributed according to the measure µ st . We define X(1) := 1 and for n > 1

X(n) := tr Sym a 1 (g p 1 ) • • • tr Sym a (g p ) , if n = p a 1 1 • • • p a
is the prime factorization of n. Furthermore, one can easily check that the X(n) satisfy the Hecke relation (1.1), namely that

X(m)X(n) = d|(m,n) X mn d 2 .
We prove the following lemma. Lemma 3.2. Let n be a positive integer. Then we have

E(X(n)) = δ(n).
Proof. Let n > 1, and write the prime factorization of n as n = p a 1 1 • • • p a . First, by the independence of the random variables g p for different primes p, we have

E(X(n)) = j=1 E tr Sym a j (g p j ) .
By Weyl's integration formula, the map

θ → g (θ) = e iθ 0 0 e -iθ ,
identifies G with the interval [0, π] and µ st with the distribution dµ st (t) := Y n (t) := sin((n + 1)t) sin t form an orthonormal basis of L 2 ([0, π], dµ st ). This completes the proof.

Using Lemmas 3.1 and 3.2 we prove that in a certain range of x, the harmonic moments of S f (x) (as f varies in H k ) are very close to the moments of the sum of random variables n≤x X(n). Proposition 3.3. Let k be a large even integer. Let x ≥ 2 and be a positive integer such that x 6 ≤ k. Then, we have

1 |H k | h h f ∈H k n≤x λ f (n) 2 = E   n≤x X(n) 2   + O k -1/3 .
In order to prove this proposition, we need to understand the combinatorics of the Hecke relations (1.1). These relations can be written as

λ f (n 1 )λ f (n 2 ) = m|n 1 n 2 b m (n 1 , n 2 )λ f (m) , where b m (n 1 , n 2 ) = 1 if m = n 1 n 2 /d 2
for some d|(n 1 , n 2 ), and equals 0 otherwise. More generally, one can write

(3.5) λ f (n 1 ) • • • λ f (n r ) = m| r j=1 n j b m (n 1 , . . . , n r )λ f (m),
for some integers b m (n 1 , . . . , n r ). These coefficients have a nice interpretation in terms of the representation theory of G = SU (2). The irreducible characters of G are g → tr (Sym a (g)) ,

for a ≥ 0. Hence, for n

= p a 1 1 • • • p a , the character χ n (g p 1 , . . . , g p ) = tr (Sym a 1 (g p 1 )) • • • tr (Sym a (g p ))
is an irreducible character of the product of copies of G, and the formula

χ n 1 • • • χ nr = m| r j=1 n j b m (n 1 , . . . , n r )χ m
is the decomposition formula for the product of the r characters χ n 1 , . . . , χ nr in terms of the irreducibles χ m . In particular, the coefficients b m (n 1 , . . . , n r ) are non-negative, and we also have

(3.6) X(n 1 ) • • • X(n r ) = m| r j=1 n j b m (n 1 , . . . , n r )X(m).
Moreover, one can easily prove (either by induction on r or by exploiting the representation theory of SU (2)) that

(3.7) b m (n 1 , . . . , n r ) ≤ τ (n 1 ) • • • τ (n r ).
Lemma 3.4. Let g be a real-valued arithmetic function. For all x ≥ 2 and positive integers we have

E   n≤x X(n)g(n) 2   = n 1 ,...,n 2 ≤x b 1 (n 1 , . . . , n 2 )g(n 1 )g(n 2 ) • • • g(n 2 ).
Proof. We have

E   n≤x X(n)g(n) 2   = E n 1 ,...,n 2 ≤x X(n 1 )X(n 2 ) • • • X(n 2 )g(n 1 )g(n 2 ) • • • g(n 2 ) = n 1 ,...,n 2 ≤x g(n 1 )g(n 2 ) • • • g(n 2 )E X(n 1 )X(n 2 ) • • • X(n 2 ) .
Moreover, it follows from (3.6) that

E X(n 1 )X(n 2 ) • • • X(n 2 ) = m|n 1 n 2 •••n 2 b m (n 1 , n 2 , . . . , n 2 )E(X(m)) = b 1 (n 1 , n 2 , . . . , n 2 ),
by Lemma 3.2. This completes the proof.

We deduce the following corollary.

Corollary 3.5. Let g and h be arithmetic functions such that g(n) ≥ h(n) ≥ 0 for all n ≥ 1. Then we have

E   n≤x X(n)g(n) 2   ≥ E   n≤x X(n)h(n) 2   .
We are now ready to prove Proposition 3.3.

Proof of Proposition 3.3. By (3.5) we obtain

1 |H k | h h f ∈H k n≤x λ f (n) 2 = 1 |H k | h h f ∈H k n 1 ,...,n 2 ≤x λ f (n 1 ) • • • λ f (n 2 ) = n 1 ,...,n 2 ≤x m|n 1 •••n 2 b m (n 1 , . . . , n 2 ) 1 |H k | h h f ∈H k λ f (m).
Therefore, by Lemma 3.1 we get

1 |H k | h h f ∈H k n≤x λ f (n) 2 = n 1 ,...,n 2 ≤x b 1 (n 1 , . . . , n 2 ) + O   k -5/6 n 1 ,...,n 2 ≤x m|n 1 •••n 2 b m (n 1 , . . . , n 2 )   . Now, using that b m (n 1 , . . . , n 2 ) ≤ τ (n 1 ) • • • τ (n 2 )
we deduce that the error term above is

k -5/6 n 1 ,...,n 2 ≤x τ (n 1 ) • • • τ (n 2 )τ (n 1 • • • n 2 ) ε x 2 ε k -5/6 n≤x τ (n) 2 k -1/3 . using the bound τ (n 1 • • • n 2 ) ε (n 1 • • • n 2 ) ε ≤ x 2 ε
together with the estimate n≤x τ (n) x log x. Appealing to Lemma 3.4 completes the proof.

To complete the proof of Theorem 1.5 we need to derive lower bounds for the moments of n≤x X(n). We establish the following proposition. Proposition 3.6. Let ≥ 2 be an integer. Then, for all real numbers 2 ≤ y ≤ x we have

E   n≤x X(n) 2   ≥ Ψ(x, y; τ ) 2 exp -10 y log log x log y + O log x .
Proof. First, by Corollary 3.5 with g(n) = 1 and h(n) being the characteristic function of the y-friable numbers, we get

(3.8) E   n≤x X(n) 2   ≥ E     n≤x P (n)≤y X(n) 2     .
For a prime p, write

g p = e iθp 0 0 e -iθp ,
where θ p is a random variable taking values in [0, π] and distributed according to the Sato-Tate distribution dµ st (t) := 2 π sin 2 (t)dt . Let A(X) be the event corresponding to 2 , for all primes p ≤ y. By the independence of the θ p for different primes p, we deduce that the probability of A(X) is

|θ p | ≤ 1 (log x)
P(A(X)) = 2 π (log x) -2 0 sin 2 tdt π(y) ≥ c (log x) 6 π(y)
exp -10 y log log x log y , for some positive constant c. On the other hand, one can see that for any prime p ≤ y and all outcomes in A(X), we have 4 .

X(p a ) = tr(Sym a g p ) = sin((a + 1)θ p ) sin θ p = (a+1) 1 + O a 2 θ 2 p = τ (p a ) 1 + O a 2 (log x)
Therefore, if n ≤ x and P (n) ≤ y then for all outcomes in A(X) we have

X(n) = τ (n) 1 + O ω(n) (log x) 2 = τ (n) 1 + O 1 log x ,
where ω(n) is the number of distinct prime factors of n, which satisfies ω(n) log x. Thus, we deduce that

E     n≤x P (n)≤y X(n) 2     ≥     n≤x P (n)≤y τ (n) 1 + O 1 log x     2 P(A(X))
Ψ(x, y; τ ) 2 exp -10 y log log x log y + O log x , as desired.

We finish this section by proving Theorem 1.5, and deducing Corollary 1.4.

Proof of Theorem 1.5. Let = [log k/(6 log x)]. Then, it follows from Proposition 3.3 and Proposition 3.6 that

1 |H k | h f ∈H k ω f n≤x λ f (n) 2 ≥ Ψ(x, y; τ ) 2 exp -10 y log log x log y + O log x + O k -1/3 ≥ Ψ(x, y; τ ) 2 exp -15 log k log log log k (log log k) 2 .
Therefore, in view of (3.1) and (3.2) we obtain (3.9)

f ∈H k n≤x λ f (n) 2 ≥ Ψ(x, y; τ ) 2 • k exp -20 log k log log log k (log log k) 2 .
Let B be the set of Hecke cusp forms f ∈ H k such that

n≤x λ f (n) ≥ Ψ(x, y; τ ) 1 - 1 √ log log k . Since |H k | k we obtain f ∈H k \B n≤x λ f (n) 2 ≤ Ψ(x, y; τ ) 2 • k exp - log k 10A(log log k) 3/2 .
Combining this bound with (3.9) we get (3.10)

f ∈B n≤x λ f (n) 2 Ψ(x, y; τ ) 2 • k exp -20 log k log log log k (log log k) 2 .
On the other hand, we have

f ∈B n≤x λ f (n) 2 ≤ |B| n≤x τ (n)
Let f ∈ H k . For Re(s) > 1 we have log L(s, f ) = ∞ n=2 Λ(n)b f (n) n s log n ,
where b f (n) = (e iθ f (p) ) a + (e -iθ f (p) ) a if n = p a for some prime p, and equals 0 otherwise. For y ≥ 1 we define

L y (s, f ) = P (n)≤y λ f (n) n s = p≤y 1 - e iθ f (p) p s -1 1 - e -iθ f (p) p s -1
.

In order to approximate S f (x) by Ψ(x, y; λ f ), we shall prove that conditionally on GRH, log L(s, f ) is very well approximated by log L y (s, f ) for Re(s) ≥ 1. This will be the key ingredient in the proof of Theorem 1. Proof. Consider the circles with centre 2+it and radii r = 2-σ and R = 2-σ 0 , so that the smaller circle passes through s. By our assumption, log L(z, f ) is analytic inside the larger circle. For a point z on the larger circle, it follows from the standard convexity bound for L(s, f ) that Re log L(z, f ) log k. Finally, using the Borel-Caratheodory theorem we obtain log

L(s, f ) ≤ 2r R -r max |z-2-it|=R Re log L(z, f ) + R + r R -r | log L(2 + it, f )| log k σ -σ 0 .
Proof of Lemma 4.1. Let c 1 = 1 -σ + 1/ log y. Then it follows from Perron's formula (see [START_REF] Davenport | Multiplicative number theory[END_REF]) that 1 2πi In order to deduce Corollary 1.2, we need to prove the bound (1.9), which shows that Ψ(x, y; τ ) = o(x log x) when u = log x/ log y → ∞. Proof of Corollary 1.2. The result holds trivially for x > k by (1.6), so we may assume that x ≤ k. Then, using Theorem 1.3 with y = (log k) 3 , together with Lemma 4.3 and our assumption on x completes the proof.

c 1 +iy c 1 -iy log L(s + z, f ) y z z dz = n≤y Λ(n)b f (n) n s log n + O y c 1 ∞ n=1 1 n σ+c 1 min 1, 1 y log |y/n| = n≤y Λ(n)b f (n) n s log n + O y -σ log y ,

3 .

 3 Large sums of Hecke eigenvalues : proofs of Theorem 1.5 and Corollary 1.4

2 π sin 2 θ sin 2

 22 (t)dt. Furthermore, note that Sym a g (θ) = a prime p and a positive integer a we obtain E(X(p a )θdθ = 0, since the functions {Y n } n≥0 , defined by

2 . 4 .

 24 Moreover, by (1.7) we have(3.11) Ψ(x, y; τ) ≥ Ψ(x, x 1/(2A) ; τ ) A n≤x τ (n).Hence, we derive from (3.10) that|B| ≥ k exp -20 log k log log log k (log log k) 2 + O A log k log log k ,which completes the proof.Proof of Corollary 1.4. The result follows from Theorem 1.5 together with Eq.(3.11). Cancellations under GRH: proofs of Theorem 1.3 and Corollary 1.2

2πi c 1 +iy c 1 c 2 2 .

 122 by a standard estimation of the error term. We now move the contour to the line Re(z) = c 2 where c 2 = 1/2 -σ + 1/ log y. By our assumption, we only encounter a simple pole at z = 0 that leaves a residue of log L(s, f ). Furthermore, it follows from Lemma 4.2 with σ 0 = 1/2 that log L(s + z, f ) log k log y, uniformly for z with Re(z) ≥ c 2 and |Im(z)| ≤ y. Therefore, we deduce that1 -iy log L(s + z, f ) y z z dz = log L(s, f ) + E, +iy log L(s + z, f ) y z z dz (log y) 2 log k y σ-1/The result follows upon noting thatlog L y (s, f ) -n≤y Λ(n)b f (n) n s log n = p≤y p a >y (e iθ f (p) ) a + (e -iθ f (p)We now prove Theorem 1.3.Proof of Theorem 1.3. Without loss of generality assume that x ∈ Z + 1/2. Let c = 1 + 1/ log x. By Perron's formula together with (1.2) we haven≤x n| log(x/n)| ε ζ(1 + 1/ log x) 2 + x ε/L y (s, f ) x s s ds + O ε (x ε ). Define R y (s, f ) := L(s, f ) L y (s, f ) .Then, combining the above estimates we getn≤x λ f (n) -Ψ(x, y; λ f ) s, f ) -L y (s, f )) x s s ds + O ε (x ε ) = 1 2πi c+ix c-ix L y (s, f ) exp(log R y (s, f )) -1 x s s ds + O ε (x ε ).Moreover, using Lemma 4.1 we obtain exp(log R y (s, f )) = 1 + O (log y) 2 log k √ y , for all s with Re(s) = c and |Im(s)| ≤ x. Furthermore, note that for Re(s) = c we have L y (s, f ) = exp p≤y λ f (p) p s + O(1) (log y) 2 , since λ f (p) ≤ 2. Combining these estimates, we deduce that 1 2πi c+ix c-ix L y (s, f ) exp(log R y (s, f )) -1 x s s ds x(log x)(log y) 4 log k √ y , which completes the proof.

Lemma 4 . 3 ./ 4 τ 1 + 2p β p exp 2 p≤y 1 +

 434121 Let 10 ≤ y ≤ x be real numbers. Then we haveΨ(x, y; τ ) e -u/2 x log x.Proof. Let β = 2/(3 log y). Then, observe that Ψ(x, y; τ ) ≤n≤x 3(n) + x -3β/4x 3/4 ≤n≤x P (n)≤yn β τ (n) x 3/4 log x + e -u/2 n≤x P (n)≤y n β τ (n). Let g(n) = n β τ (n) if P (n) ≤ y, 0 otherwise.Then g is multiplicative, and for all primes p ≤ y we have g(p a ) = (a + 1)p aβ ( 19 10 ) a . Therefore, by Corollary 3.5.1 of[START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF] we obtain n≤x P (n)≤yn β τ (n)The result follows upon noting that x 3/4 xe -u/2 for y ≥ 10 and p≤y O(β log p) p (log y) 2 .

3 .

 3 Lemma 4.1. Let f ∈ H k and assume GRH for L(s, f ). Let 2 ≤ y ≤ k, and s = σ + it with 1 ≤ σ < 1 + 1/ log y and |t| ≤ 2k. Then, we have

	log L(s, f ) -log L y (s, f )	(log y) 2 log k √ y	.
	To prove this result we need the following standard bound.

Lemma 4.2. Let f ∈ H k . Let s = σ + it with 1/2 < σ ≤ 3/2 and |t| ≤ 3k. Let 1/2 ≤ σ 0 < σ, and suppose that there are no zeros of L(z, f ) inside the rectangle {z : σ 0 ≤ Re(z) ≤ 1, |Im(z) -t| ≤ 3}. Then, we have log L(s, f ) log k σ -σ 0 .

Acknowledgements

I would like to thank Emmanuel Kowalski for useful comments concerning the probabilistic random model for the Hecke eigenvalues in Section 3. I would also like to thank the anonymous referee for their comments and for suggesting a simpler proof of Theorem 1.1.

The author is partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.