Sign of Fourier coefficients of modular forms of half integral weight

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Sign of Fourier coefficients of modular forms of half

integral weight

Yuk-Kam Lau, Emmanuel Royer, Jie Wu

To cite this version:

Yuk-Kam Lau, Emmanuel Royer, Jie Wu. Sign of Fourier coefficients of modular forms of half integral weight. Mathematika, University College London, 2016, 62 (3), pp.866-883. �hal-01167163v2�

SIGN OF FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF INTEGRAL WEIGHT

YUK-KAM LAU, EMMANUEL ROYER, AND JIE WU

ABSTRACT. We establish lower bounds for (i) the numbers of positive and negat-ive terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

1. INTRODUCTION

1.1. Results. Let𝓁 _{≥ 4 be a positive integer. Denote by 𝔖𝓁+1∕2}the vector space of all cusp forms of weight𝓁 + 1∕2 for the congruence subgroup Γ₀(4) . The Fourier expansion of 𝔣 ∈ 𝔖_𝓁+1∕2at ∞ can be written as

𝔣(𝑧) = ∞ ∑

𝑛=1

𝜆_𝔣(𝑛)𝑛𝓁∕2−1∕4e(𝑛𝑧) (𝑧 ∈ℋ ), (1)

where e(𝑧) = e2𝜋i𝑧andℋ is the Poincaré upper half plane. For any squarefree integer 𝑡 Waldspurger [17] proved the following elegant formula

𝜆_𝔣(𝑡)2= 𝐶_𝔣𝐿(1₂,Sh𝑡𝔣, 𝜒𝑡), (2)

where Sh𝑡𝔣 is the Shimura lift of 𝔣 associated to 𝑡 (this is a cusp form of weight

2𝓁 and of level 2), 𝜒𝑡(𝑛) is a real character modulo 𝑡 (defined in Section 2) and 𝐶𝔣 is a constant depending on 𝔣 only. In the following, the letter 𝑡 will always be a squarefree integer and ∑♭a sum over squarefree integers.

In view of (2), Kohnen [10] posed the following question: in the case where 𝜆_𝔣(𝑡) is a real number, what is its sign? Very recently, Hulse, Kairal, Kuan & Lim made a significant progress toward this question by proving that 𝜆_𝔣(𝑡) changes sign infinitely often if 𝔣 ∈ 𝔖_𝓁+1∕2 is an eigenform of all the Hecke operators (see [4, Theorem 1.1]).

In order to describe the order of magnitude of 𝜆_𝔣(𝑡), we choose 𝛼 a non negative real number such that the inequality

𝜆_𝔣(𝑡) ≪_𝔣,𝛼 𝑡𝛼 (3)

Date: 27/01/2016– 15:01.

2010 Mathematics Subject Classification. Primary 11F30 ; Secondary 11F37,11M41,11N25.

Key words and phrases. Half integral weight modular forms, sign of Fourier coefficients, Dirichlet series.

This work was supported by a grant from France/Hong Kong Joint Research Scheme, Procore, sponsored by the Research Grants Council of Hong Kong (F-HK026/12T) and the Consulate General of France in Hong Kong & Macau (PHC PROCORE 2013, N◦_{28212PE). Lau is also supported by}

GRF 17302514 of the Research Grants Council of Hong Kong.

holds for all squarefree integers 𝑡. The implied constant depends on 𝔣 and 𝛼 only. It is conjectured that one can take

𝛼= 𝜀

for any 𝜀 > 0. This could be regarded as an analogue of the Ramanujan conjecture on cusp forms of integral weight. Conrey & Iwaniec [3, Corollary 1.3] proved that one can take

𝛼= 1

6+ 𝜀 for any 𝜀 > 0.

The main aim of this paper is to establish a quantitative version of the result of Hulse, Kairal, Kuan & Lim. Define

_𝔣+(𝑥) = #{𝑡_{≤ 𝑥, 𝑡 squarefree ∶ 𝜆𝔣}(𝑡) > 0} and

_𝔣−(𝑥) = #{𝑡_{≤ 𝑥, 𝑡 squarefree ∶ 𝜆𝔣}(𝑡) < 0}.

We establish the following results.

Theorem 1 – Let 𝓁 ≥ 4 be a positive integer and 𝔣 ∈ 𝔖𝓁+1∕2an eigenform of all the Hecke operators such that the 𝜆_𝔣(𝑛) are real for all 𝑛_{≥ 1. Then for any 𝜀 > 0,}

we have

_𝔣+(𝑥)_{≥ 𝑥}1−2𝛼−𝜀, _𝔣−(𝑥)_{≥ 𝑥}1−2𝛼−𝜀

for all 𝑥_{≥ 𝑥0}(𝔣, 𝜀), where 𝛼 is given by (3) and 𝑥₀(𝔣, 𝜀) is a positive real number

depending only on 𝔣 and 𝜀.

Remark 2 – In particular, the Conrey & Iwaniec bound leads to _𝔣+(𝑥)≥ 𝑥2∕3−𝜀, _𝔣−(𝑥)≥ 𝑥2∕3−𝜀 for all 𝑥≥ 𝑥0(𝔣, 𝜀).

Remark 3 – The study about the sign equidistribution of the sequence(𝜆_𝔣(𝑡𝑛2₎)

𝑛∈ℕ was investigated in [2], [10], [9], [5] and [6]. In particular, Inam & Wiese proved in [5] that, if 𝑡 is a fixed squarefree integer, then

lim 𝑥→+∞ #{𝑝 prime ∶ 𝑝≤ 𝑥, 𝜆𝔣(𝑡𝑝2) > 0} #{𝑝 prime ∶ 𝑝_{≤ 𝑥}} = 1 2 and lim 𝑥→+∞ #{𝑝 prime ∶ 𝑝≤ 𝑥, 𝜆𝔣(𝑡𝑝2) < 0} #{𝑝 prime ∶ 𝑝≤ 𝑥} = 1 2.

Let us precise what we call number of squarefree sign changes of the sequence

𝜆_𝔣 = (𝜆_𝔣(𝑡))_𝑡

≥0 (where 𝜆𝔣(0) = 0) restricted to squarefree indexes 𝑡. From this sequence of Fourier coefficients, we build a sequence of pairs of squarefree integers (𝑡+_𝑛, 𝑡−_𝑛), that may be finite or even void, in the following way: for any integer 𝑛, we have

max(𝑡+_𝑛, 𝑡−_𝑛) < min(𝑡+_𝑛+1, 𝑡−_𝑛+1),

and 𝜆_𝔣(𝑡) = 0 for all squarfree integer 𝑡 between 𝑡+_𝑛 and 𝑡−_𝑛. The number of squarefree sign changes of 𝜆_𝔣is the function defined by

𝔣(𝑥) = #{𝑛_{≥ 1 ∶ max(𝑡}+_𝑛, 𝑡−_𝑛)≤ 𝑥}.

Theorem 4 – Let 𝓁 ≥ 4 be a positive integer and 𝔣 ∈ 𝔖𝓁+1∕2be an eigenform of all the Hecke operators such that the 𝜆_𝔣(𝑛) are real for all 𝑛≥ 1. For any 𝜀 > 0, the

number of squarefree sign changes of 𝜆_𝔣satisfies

𝔣(𝑥) ≫_𝔣,𝜀𝑥1−4𝛼5 −𝜀

for all 𝑥_{≥ 𝑥0}(𝔣, 𝜂), where the constant 𝑥₀(𝔣, 𝜂) and the implied constant depends

on 𝔣 and 𝜀.

Remark 5 – In particular, the Conrey & Iwaniec bound leads to 𝔣(𝑥) ≫_𝔣,𝜀𝑥

1 15−𝜀

for all 𝑥≥ 𝑥0(𝔣, 𝜀).

1.2. Methods. To prove Theorem 1, we detect signs with |𝜆𝔣(𝑡)| + 𝜆𝔣(𝑡)

2 =

{

𝜆_𝔣(𝑡) if 𝜆_𝔣(𝑡) > 0

0 otherwise.

Bounding the Fourier coefficients with (3), we get plainly ∑♭ 𝑡_≤𝑥 (| ||𝜆𝔣(𝑡)||| + 𝜆𝔣(𝑡) ) log(𝑥 𝑡 ) ≪_{𝔣,𝛼 𝔣}+(𝑥)𝑥𝛼log 𝑥

(recall that the letter 𝑡 is for squarefree integers hence the sum is restricted to squarefree integers). Then we use the analytic properties of the Dirichlet series

𝑀(𝔣, 𝑠) = ∑♭

𝑡_≤𝑥

𝜆_𝔣(𝑡)𝑡−𝑠 and 𝐷(𝔣 ⊗ 𝔣, 𝑠) =∑

𝑛_≥1

𝜆_𝔣(𝑛)2𝑛−𝑠

in Lemma 8 and Proposition 7 of §2.2 to make an auxiliary tool – Lemma 9. (Note that Lemma 8 is due to [4].) More precisely, we utilize that the Dirichlet series defining 𝑀 (𝔣, 𝑠) and 𝐷(𝔣 ⊗ 𝔣, 𝑠) are absolutely convergent for Re 𝑠 > 1. The function

𝑀(𝔣, 𝑠) has an analytic continuation to Re 𝑠 > 3∕4 whereas the function 𝐷(𝔣 ⊗ 𝔣, 𝑠) has a meromorphic continuation to Re 𝑠 > 1∕2 with a unique pole; this pole is at 1 and it is simple. Thus we can easily derive Lemma 9 and then the lower bound

∑♭ 𝑡≤𝑥 (| ||𝜆𝔣(𝑡)||| + 𝜆𝔣(𝑡) ) log (_𝑥 𝑡 ) ≫ 𝑥1−𝛼.

Theorem 4 rests on the following delicate device of Soundararajan [15]: let 𝑐 > 0 and 𝛿 > 0, then 1 2𝜋i∫ 𝑐+i∞ 𝑐−i∞ (e𝛿𝑠−1)2 𝑠2 𝜉 𝑠_d𝑠 = {

min(log(e2𝛿𝜉),log (1∕𝜉)) if e−2𝛿_{≤ 𝜉 ≤ 1}

0 otherwise. (4)

(Thanks to the referee for suggesting this device.) Using it with the analytic properties of 𝑀 (𝔣, 𝑠) and 𝐷(𝔣⊗𝔣, 𝑠), some weighted first and second moments on short intervals are evaluated. We use these moments to detect the sign changes via the positivity of

∑ 𝑚_≤𝐴 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 (| ||𝜆𝔣(𝑡)||| + 𝜀𝑚𝜆𝔣(𝑡) ) min ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) for all (𝜀₁,… , 𝜀𝐴) ∈ {−1, 1}𝐴.

The paper is organized as follows. Section 2 is devoted to the background on half-integral weight modular forms (§2.1) and the establishment of the analytic properties for the Dirichlet series we need (§2.2). Theorem 1 is proven in Section 3. Theorem 4 is proven in Section 4.

Acknowledgement. We express our hearty gratitude to the anonymous referee for his/her insightful advice that led to the current much better version of Theorem 4 as well as the helpful comments on presentation. The preliminary form of this paper was finished during the visit of E. Royer and J. Wu at Hong Kong University in 2014. They would like to thank the department of mathematics for hospitality and excellent working conditions.

2. BACKGROUND

2.1. Modular forms of half-integral weight. In this section, we want to recall the basic facts we need on modular forms of half-integral weight on the congruence subgroup Γ₀(4). All the content of this section is classical and is to be found in the main references [14] and [13]. It contains however the very few that the non-specialist reader will need.

The theta function is defined on the upper half planeℋ by

𝜃(𝑧) = 1 + 2 +∞ ∑

𝑛=1 e(𝑛2𝑧)

for any 𝑧 ∈ℋ . Since the 𝜃 function does not vanish on ℋ , we can define the theta multiplier: for any 𝛾 ∈ Γ0(4) and 𝑧 ∈ℋ , let

𝑗(𝛾, 𝑧) = 𝜃(𝛾𝑧)

𝜃(𝑧) . If 𝛾 =(𝑎 𝑏

𝑐 𝑑

)

, it can be shown that 𝑗(𝛾, 𝑧)2= 𝑐𝑧 + 𝑑. For any complex number 𝜉, let

𝑑)1∕2 is called the theta multiplier. It does not depend on 𝑧 and can be explicitly described in terms of 𝑐 and 𝑑 (see, for example, [7, §2.8]).

Let𝓁 be a non negative integer. A modular form of weight 𝓁 + 1∕2 is a holo-morphic function 𝔣 onℋ satisfying

𝔣(𝛾𝑧) = 𝑗(𝛾, 𝑧)2𝓁+1𝔣(𝑧)

for all 𝛾 ∈ Γ₀(4) and 𝑧 ∈ ℋ , and that is holomorphic at the cusps of Γ₀(4). If moreover 𝔣 vanishes at the cusps of Γ₀(4), then 𝔣 is called a cusp form of weight𝓁 + 1∕2. The congruence subgroup has three cusps: 0, −1∕2 and ∞. The corresponding scaling matrices are respectively

𝜎₀= ( 0 −1∕2 2 0 ) , 𝜎_−1∕2 = ( 1 0 −2 1 ) and 𝜎_∞= ( 1 0 0 1 ) .

Then, if 𝔣 is a cusp form of weight𝓁 + 1∕2, the following functions have a Fourier expansion vanishing at ∞: 𝔣|_𝜎 0(𝑧) = (2𝑧) −𝓁−1∕2_𝔣(₋ 1 4𝑧 ) and 𝔣|_𝜎 −1∕2(𝑧) = (−2𝑧 + 1) −𝓁−1∕2_𝔣(₋ 1 2𝑧 − 1 ) . We shall write 𝔣(𝑧) = +∞ ∑ 𝑛=1 ̂_{𝔣(𝑛) e(𝑛𝑧) (5)}

for the Fourier expansion of 𝔣. The set 𝔖_𝓁+1∕2of modular forms of weight𝓁 + 1∕2 is a finite dimensional vector space over ℂ. If𝓁 _{≤ 3, then 𝔖𝓁+1∕2} = {0}. In the following, we shall assume𝓁 _{≥ 4.}

Shimura established a correspondence between half-integral cusp forms and integral weight cusp forms on a congruence subgroup. Niwa [12] gave a more direct proof of this correspondence and lowered the level of the congruence group involved. Fix a squarefree integer 𝑡. We write 𝜒₀for the principal character of modulus 2 and define a character 𝜒𝑡by 𝜒_𝑡(𝑛) = 𝜒₀(𝑛) (₋₁ 𝑛 )𝓁(_𝑡 𝑛 ) .

Let 𝔣 ∈ 𝔖_𝓁+1∕2. Then, the Dirichlet series defined by the product

𝐿(𝜒_𝑡, 𝑠−𝓁 + 1) +∞ ∑ 𝑛=1 ̂_{𝔣(𝑡𝑛}2₎ 𝑛𝑠

is the Dirichlet series of a cusp form of integral weight 2𝓁 over the congruence subgroup Γ₀(2). We denote by Sh𝑡𝔣 this cusp form and 𝑆2𝓁the vector space of cusp forms of weight 2𝓁 over Γ₀(2). At this point, the dependence in 𝑡 of Sh_𝑡𝔣 is not really clear. It will become clearer after we introduce the Hecke operators.

The Hecke operator of half-integral weight𝓁 + 1∕2 and order 𝑝2is the linear endomorphism 𝔗𝑝2 on 𝔖_𝓁+1∕2that sends any cusp form with Fourier coefficients

(̂𝔣(𝑛))𝑛≥1to the cusp form with Fourier coefficients defined by

̂ 𝔗_𝑝2(𝔣)(𝑛) = ̂𝔣(𝑝2𝑛) + 𝜒₀(𝑝) ( (−1)𝓁𝑛 𝑝 ) 𝑝𝓁−1𝑓̂(𝑛) + 𝜒₀(𝑝)𝑝2𝓁−1̂𝔣 ( 𝑛 𝑝2 ) .

If 𝑛∕𝑝2is not an integer, then ̂𝔣(𝑛∕𝑝2) is considered to be 0. Hecke operators and the Shimura correspondence commute, meaning that if 𝑇𝑝is the Hecke operator of

order 𝑝 over 𝑆2𝓁, then

Sh_𝑡(𝔗_𝑝2𝔣) = 𝑇_𝑝(Sh_𝑡𝔣)

for any 𝔣 ∈ 𝔖𝓁+1∕2. In particular, if 𝔣 is an eigenform of 𝔗𝑝2, then Sh_𝑡𝔣 is an

eigenform of 𝑇_𝑝 with same eigenvalue. Let 𝔣 be an eigenform of all the Hecke operators 𝔗𝑝2: denote by 𝑤_𝑝the corresponding eigenvalue. One has

𝐿(𝜒𝑡, 𝑠−𝓁 + 1) +∞ ∑ 𝑛=1 ̂_{𝔣(𝑡𝑛}2₎ 𝑛𝑠 = ̂𝔣(𝑡) ∏ 𝑝 ( 1 −𝜔𝑝 𝑝𝑠 + 𝜒₀(𝑝) 𝑝2𝑠−2𝓁+1 )−1 (6)

the product being over all prime numbers. This product is the 𝐿-function of a cusp form in 𝑆_2𝓁. We denote by Sh 𝔣 this cusp form. Remark that it does not depend on 𝑡 and that Sh𝑡𝔣 = ̂𝔣(𝑡) Sh 𝔣.

Let 𝜓 be the arithmetic function defined by

𝜓(𝑛) =∏

𝑝∣𝑛 (

1 + 𝑝−1∕2)

the product being on prime numbers. We write 𝜏 for the divisor function and clearly

𝜓(𝑛)_{≤ 𝜏(𝑛) for every 𝑛 ∈ ℕ}∗. The next Lemma improves slightly Lemma 4.1 in [4]. Lemma 6 – Let 𝔣 ∈ 𝔖_𝓁+1∕2be an eigenform of all the Hecke operators 𝔗_𝑝2. There

exists a constant 𝐶 >0 such that, for any squarefree integer 𝑡 and any integer 𝑛 we

have

||

|̂𝔣(𝑡𝑛2)||| ≤ 𝐶|||̂𝔣(𝑡)|||𝑛𝓁−1∕2𝜏(𝑛)𝜓(𝑛).

Proof. From (6) we get

̂_{𝔣(𝑡𝑛}2_{) = ̂𝔣(𝑡)}∑ 𝑑∣𝑛 𝜒_𝑡 (_𝑛 𝑑 ) 𝜇 (_𝑛 𝑑 ) (_𝑛 𝑑 )𝓁−1 ̂ Sh 𝔣(𝑑). (7)

By the Deligne estimate, there exists 𝐶 > 0 such that ||

|Sh 𝔣(𝑑)|̂ || ≤ 𝐶𝑑(2𝓁−1)∕2𝜏(𝑑) (8)

for any 𝑑. It follows from (7) and (8) that || |̂𝔣(𝑡𝑛2)||| ≤ 𝐶|||̂𝔣(𝑡)|||𝑛𝓁−1 ∑ 𝑑∣𝑛 || ||𝜇 (_𝑛 𝑑 )| || |𝑑 1∕2_𝜏(𝑑) ≤ 𝐶|||̂𝔣(𝑡)|||𝑛𝓁−1∕2𝜏(𝑛)𝜓(𝑛).  The size of the Fourier coefficients of a half integral weight modular form is therefore controlled by the size of its Fourier coefficients at squarefree integers. Deligne’s bound for integral weight modular forms does not apply, although it conjecturally does. Let 𝛼 be a positive real number such that, if 𝔣 ∈ 𝔖𝓁+1∕2, then

|̂𝔣(𝑡)| ≤ 𝐶𝑡(𝓁+1∕2−1)∕2+𝛼

for any squarefree integer 𝑡 (and 𝐶 is a real number depending only on 𝔣 and 𝛼). Ramanujan-Petersson conjecture asserts that 𝛼 can be taken arbitrarily small. The

best proven result is due to Conrey & Iwaniec [3] (see also the Appendix by Mao in [1] for an uniform value of 𝐶). Their result implies that we can take 𝛼 = 1∕6 + 𝜀 with any real positive 𝜀. If 𝔣 ∈ 𝔖_𝓁+1∕2is an eigenform of all the Hecke operators, we have by comparison of (1) and (5)

𝜆_𝔣(𝑛) =

̂_𝔣(𝑛) 𝑛(𝓁+1∕2−1)∕2. For any squarefree integer 𝑡 and integer 𝑛, we have then

||

|𝜆𝔣(𝑡𝑛2)||| ≤ 𝐶1||_|𝜆𝔣(𝑡)|||𝜏(𝑛)𝜓(𝑛) ≤ 𝐶2𝑡𝛼𝜏(𝑛)𝜓(𝑛) (9) with the admissible choice 𝛼 = 1∕6 + 𝜀, where 𝐶₁and 𝐶₂are positive real numbers not depending on 𝑡 or 𝑛.

2.2. Some associated Dirichlet series. Let 𝔣 ∈ 𝔖_𝓁+1∕2, and assume it is an eigen-form of all the Hecke operators. We define

𝐷(𝔣 ⊗ 𝔣, 𝑠) = +∞ ∑

𝑛=1

𝜆_𝔣(𝑛)2𝑛−𝑠. (10)

Write 𝜎 = Re 𝑠 and 𝜏 = Im 𝑠.1According to (9), we know it is absolutely convergent as soon as 𝜎 > 1 + 2𝛼. We state analytical informations on this function. The proof is quite standard, but since we have not found a handy proof in the literature for this case, we provide the details for completeness.

Proposition 7 – Let 𝔣 ∈ 𝔖_𝓁+1∕2, and assume it is an eigenform of all the Hecke operators. The Dirichlet series(10) converges absolutely as soon as Re 𝑠 > 1. It

can be continued analytically to a meromorphic function in the half planeRe 𝑠 > 1₂

with the only pole at 𝑠= 1 . This pole is simple. Further for any 𝜀 > 0 we have

𝐷(𝔣 ⊗ 𝔣, 𝑠) ≪_𝔣,𝜀|𝜏|2 max(1−𝜎,0)+𝜀 ( 1 2 + 𝜀≤ 𝜎 ≤ 3,|𝜏| ≥ 1 ) . The implied constant depends on 𝔣 and 𝜀 only.

Proof. Let 𝔞 be a cusp of Γ = Γ₀(4). We denote by Γ_𝔞its stability group, and by 𝜎_𝔞 its scaling matrix (see [7, §2.3]). The Eisenstein series associated to 𝔞 is

𝐸_𝔞(𝑧, 𝑠) = ∑ 𝛾∈Γ_𝔞∖Γ Im(𝜎_𝔞−1𝛾𝑧)𝑠 = ∑ 𝛾∈Γ_∞∖Γ Im(𝛾𝜎_𝔞−1𝑧)𝑠= 𝐸_∞(𝜎−1_𝔞 𝑧, 𝑠).

We take {0, −1∕2, ∞} as a representative set of cusps and obtain

𝐸₀(𝑧, 𝑠) = 𝐸_∞ ( − 1 4𝑧, 𝑠 ) and 𝐸_−1∕2(𝑧, 𝑠) = 𝐸_∞ ( − 𝑧 2𝑧 − 1, 𝑠 ) .

These series converge absolutely for Re 𝑠 > 1 (see, for example [11, Theorem 2.1.1]). Moreover, 𝔣_|𝜎

𝔞admits a Fourier expansion

𝔣_|𝜎 𝔞(𝑧) = +∞ ∑ 𝑛=1 𝑛(𝓁+1∕2−1)∕2𝜆_𝔣,𝔞(𝑛) e(𝑛𝑧). Let 𝐷(𝔣_𝔞⊗𝔣_𝔞, 𝑠) = +∞ ∑ 𝑛=1 || |𝜆𝔣,𝔞(𝑛)||| 2 𝑛−𝑠. (11)

Classically (see, for example, [7, §13.2]), we have (4𝜋)𝑠+𝓁−1∕2Γ(𝑠+𝓁 − 1 2 ) 𝐷(𝔣_𝔞⊗𝔣_𝔞, 𝑠) = ∫_Γ∖ℋ𝑦 𝓁+1∕2_|𝔣(𝑧)|2 𝐸_𝔞(𝑧, 𝑠)d𝑥 d𝑦 𝑦2 for Re 𝑠 large enough. The right hand side provides an analytic continuation in the region Re 𝑠 > 1. By Landau Lemma, this implies that the Dirichlet series (11) is absolutely convergent for Re 𝑠 > 1. The general theory implies that 𝑠 ↦ 𝐸_𝔞(𝑧, 𝑠) has a meromorphic continuation to the whole complex plane and satisfies the functional equation

⃗

𝐸(𝑧, 𝑠) = Φ(𝑠) ⃗𝐸(𝑧, 1 − 𝑠)

where ⃗𝐸is the transpose of (𝐸_∞, 𝐸₀, 𝐸_−1∕2) and Φ =(𝜑_𝔞,𝔟)_{(𝔞,𝔟)∈{∞,0,−1∕2}}2 is the

scattering matrix. Indeed,

𝜑_𝔞,𝔟(𝑠) = 𝜋1∕2 Γ(𝑠 −1 2) Γ(𝑠) ∑ 𝑐>0  (𝑐)𝑐−2𝑠

where (𝑐) is the number of 𝑑, incongruent modulo 𝑐 such that, there exist 𝑎 and 𝑏 satisfying 𝜎_𝔞 ( 𝑎 𝑏 𝑐 𝑑 ) 𝜎_𝔟−1∈ Γ₀(4). This leads to Φ(𝑠) = Λ(2𝑠 − 1) Λ(2𝑠) 21−2𝑠 22𝑠_{− 1} ⎛ ⎜ ⎜ ⎝ 1 22𝑠−1− 1 22𝑠−1− 1 22𝑠−1− 1 1 22𝑠−1− 1 22𝑠−1− 1 22𝑠−1− 1 1 ⎞ ⎟ ⎟ ⎠ = Λ(2𝑠 − 1) Λ(2𝑠) Ψ(𝑠), say,

where Λ(𝑠) = 𝜋−𝑠∕2Γ(𝑠∕2)𝜁 (𝑠). On the half plane Re 𝑠≥ 1∕2, 𝐸𝔞and 𝜑_𝔞,𝔞have the same poles of the same orders [11, Theorems 4.4.2, 4.3.4, 4.3.5]. The only pole on Re 𝑠≥ 1∕2 is then 𝑠 = 1 and it is simple. Note that this follows also from the general theory since we are working on a congruence subgroup ([8, Theorem 11.3]).

Let ⃗𝐿(𝔣 ⊗ 𝔣, 𝑠) be the transpose of ( 𝐷(𝔣 ⊗ 𝔣, 𝑠), 𝐷(𝔣₀⊗𝔣₀, 𝑠), 𝐷(𝔣_−1∕2⊗𝔣_−1∕2, 𝑠) ) and ⃗ Λ(𝔣, 𝑠) = (2𝜋)−2𝑠Γ(𝑠)Γ(𝑠 +𝓁 − 1∕2)𝜁 (2𝑠) ⃗𝐿(𝔣 ⊗ 𝔣, 𝑠).

We proved that

∙ ⃗Λ(𝔣, 𝑠) = Ψ(𝑠) ⃗Λ(𝔣, 1 − 𝑠)

∙ in the half plane Re 𝑠_{≥ 1∕2, the function 𝐷(𝔣𝔞}⊗𝔣_𝔞, 𝑠) has only a simple pole at 𝑠 = 1.

Now, let‖⋅‖ denote the Euclidean norm in ℝ3. Using‖𝐷(𝔣𝔞⊗𝔣_𝔞,1 + 𝜀 + i𝜏)‖ ≪𝔣,𝜀1 for any 𝜏 ∈ ℝ and any fixed 𝜀 > 0, we deduce

|𝜁(−2𝜀 + 2i𝜏)| ⋅ ‖‖‖⃗𝐿(𝔣 ⊗ 𝔣,−𝜀 + i𝜏)‖‖‖≪𝔣,𝜀(1 +|𝜏|)2+𝜀 from the functional equation, and the estimate

|𝜁(2𝑠)| ⋅ ‖‖‖⃗𝐿(𝔣 ⊗ 𝔣,𝑠)‖‖‖≪𝔣,𝜀(1 +|𝜏|)2(1−𝜎)+𝜀 (𝑠 = 𝜎 + i𝜏, 𝜎 ∈ [0, 1], |𝜏| ≥ 1) by the standard argument with the convexity principle.2 This leads to the desired

result. 

Another useful Dirichlet series is

𝑀(𝔣, 𝑠) = ∑♭

𝑡_≥1

𝜆_𝔣(𝑡)𝑡−𝑠. (12)

The series 𝑀 (𝔣, 𝑠) is absolutely convergent for Re 𝑠 > 1 by the Cauchy-Schwarz inequality and Proposition 7. The next lemma is due to Hulse, Kiral, Kuan & Lim [4, Proposition 4.4].

Lemma 8 – Let 𝓁 ≥ 4 be a positive integer and 𝔣 ∈ 𝔖𝓁+1∕2be an eigenform of all the Hecke operators. The series 𝑀(𝔣, 𝑠), given by (12), converges for Re 𝑠 > 3

4.

Further for any 𝜀 >0 we have

𝑀(𝔣, 𝜎 + i𝜏) ≪_𝔣,𝜀(|𝜏| + 1)max(1−𝜎,0)+2𝜀 (3

4+ 𝜀≤ 𝜎 ≤ 3,|𝜏| ≥ 1)

where the implied constant depends on 𝔣 and 𝜀 only.

Proof. We only sketch the proof since it is nearly the same as in [4, Proposition 4.4]. By the relation 𝜇(𝑚)2=∑ 𝑟2_∣𝑚 𝜇(𝑟) we have 𝑀(𝔣, 𝑠) = +∞ ∑ 𝑟=1 𝜇(𝑟)𝐷𝑟(𝑠) (13) where 𝐷_𝑟(𝑠) = +∞ ∑ 𝑚=1 𝑚≡0 (mod 𝑟2₎ 𝜆_𝔣(𝑚)𝑚−𝑠.

2_{One needs the estimate|𝜁(2𝑠)| ⋅ ‖ ⃗𝐿(𝔣 ⊗ 𝔣, 𝑠)‖ ≪ e}e𝜂|𝜏|

for some 𝜂 > 0 in the strip so as to apply the convexity principle. This can be easily verified by the Fourier expansion of 𝐸𝔞(𝑧, 𝑠) and [11,

This series is absolutely convergent for Re 𝑠 > 1 by Cauchy-Schwarz inequality and Proposition 7. Then, introducing additive characters to remove the congruence condition and applying the Mellin transform, we get

𝐷_𝑟(𝑠) = (2𝜋) 𝑠+(𝓁+1∕2−1)∕2 Γ(𝑠 + (𝓁 + 1∕2 − 1)∕2)⋅ 1 𝑟2 ∑ 𝑑∣𝑟2 ∑ 𝑢(mod 𝑑) (𝑢,𝑑)=1 Λ ( 𝔣, 𝑢 𝑑, 𝑠 ) with Λ(𝔣, 𝑞, 𝑠) = ∫ +∞ 0 𝔣(i𝑦 + 𝑞)𝑦𝑠+(𝓁−1∕2)∕2d𝑦 𝑦

for any rational number 𝑞. Using the functional equation for Λ(𝔣, 𝑞, 𝑠) (see [4, Lemma 4.3]), we obtain

𝐷_𝑟(−𝜀 + i𝜏) ≪𝜀,𝔣 (1 +|𝜏|)1+2𝜀𝑟2+5𝜀. From (9), we have also

𝐷_𝑟(1 + 𝜀 + i𝜏) ≪_𝜀,𝔣 1

𝑟2. Finally, by the Phrägmen-Lindelöf principle, we deduce

𝐷_𝑟(𝜎 + i𝜏) ≪_𝜀,𝔣 (1 +|𝜏|)1−𝜎+𝜀𝑟2−4𝜎+𝜀.

Reinserting this bound into (13) leads to the result. 

3. PROOF OFTHEOREM 1

We begin by establishing mean value results for the Fourier coefficients at square-free integers.

Lemma 9 – Let 𝔣 ∈ 𝔖_𝓁+1∕2, and assume it is an eigenform of all the Hecke operators. Let 𝜀 >0. There exist positive real numbers 𝐶₁, 𝐶₂and 𝐶₃such that, for any 𝑥_{≥ 1, we have} ∑♭ 𝑡_≤𝑥 𝜆_𝔣(𝑡) log(𝑥 𝑡 ) ≤ 𝐶1𝑥3∕4+𝜀 and 𝐶₂𝑥_≤ ∑♭ 𝑡_≤𝑥 𝜆_𝔣(𝑡)2_{≤ 𝐶3}𝑥 for any 𝑥_{≥ 𝑥0}(𝔣).

Proof. Using the Perron formula [16, Theorem II.2.3], we write ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡) log (_𝑥 𝑡 ) = 1 2𝜋i∫ 2+i∞ 2−i∞ 𝑀(𝔣, 𝑠)𝑥𝑠d𝑠 𝑠2.

We move the line of integration to Re 𝑠 = 3∕4 + 𝜀 and use Lemma 8 to have ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡) log (_𝑥 𝑡 ) ≤ 𝐶1𝑥3∕4+𝜀.

For the second formula, we use an effective version of the Perron formula [16, Corollary II.2.2.1]: ∑ 𝑛≤𝑥 𝜆_𝔣(𝑛)2= 1 2𝜋i∫ 𝜅+i𝑇 𝜅−i𝑇 𝐷(𝔣 ⊗ 𝔣, 𝑠)𝑥𝑠d𝑠 𝑠 + 𝑂 ( 𝑥1+2𝛼+𝜀 𝑇 )

for any 𝑇 ≤ 𝑥 and 𝜅 = 1 + 1∕ log 𝑥. Proposition 7 allows to shift the line of integration to Re 𝑠 = 1∕2 + 𝜀. We get 1 2𝜋i∫ 𝜅+i𝑇 𝜅−i𝑇 𝐷(𝔣 ⊗ 𝔣, 𝑠)𝑥𝑠d𝑠 𝑠 = 𝑟𝔣𝑥+ 1 2𝜋i∫_𝐷(𝔣 ⊗ 𝔣, 𝑠)𝑥 𝑠d𝑠 𝑠

where 𝑟𝔣 is the residue at 𝑠 = 1 of 𝐷(𝔣 ⊗ 𝔣, 𝑠) and  is the contour made from segments joining in order the points 𝜅 − i𝑇 , 1∕2 + 𝜀 − i𝑇 , 1∕2 + 𝜀 + i𝑇 and 𝜅 + i𝑇 . With the convexity bound in Proposition 7 we have

∫ 𝜅±i𝑇 1∕2+𝜀±i𝑇 𝐷(𝔣 ⊗ 𝔣, 𝑠)𝑥𝑠d𝑠 𝑠 ≪ 𝑥1+𝜀 𝑇 if 𝑇 _{≤ 𝑥}1∕2and ∫ 1∕2+𝜀+i𝑇 1∕2+𝜀−i𝑇 𝐷(𝔣 ⊗ 𝔣, 𝑠)𝑥𝑠d𝑠 𝑠 ≪ 𝑥 1∕2+𝜀_{𝑇 .}

We choose 𝑇 = 𝑥1∕4+𝛼and obtain ∑

𝑛≤𝑥

𝜆_𝔣(𝑛)2 = 𝑟_𝔣𝑥+ 𝑂(𝑥3∕4+𝛼+𝜀). (14) Each positive integer 𝑛 may be decomposed uniquely as 𝑛 = 𝑡𝑚2with squarefree 𝑡. Using (9) we have ∑ 𝑛_≤𝑥 𝜆_𝔣(𝑛)2≪_𝔣 ∑♭ 𝑡_≤𝑥 𝜆_𝔣(𝑡)2 ∑ 𝑚_{≤(𝑥∕𝑡)}1∕2 𝜏(𝑚)𝜓(𝑚) ≪_𝔣𝑥1∕2 ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡)2 𝑡1∕2 log (_𝑥 𝑡 ) .

Combining this with (14) we find ∑♭ 𝑡_≤𝑥 𝜆_𝔣(𝑡)2 𝑡1∕2 log (_𝑥 𝑡 ) ≥ 𝑐1𝑥1∕2 (𝑥_{≥ 𝑥0}(𝔣)) (15)

where the constant 𝑐₁depends only on 𝔣. On the other hand, (14) leads to ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡)2 𝑡1∕2 log (_𝑥 𝑡 ) ≤∑ 𝑛≤𝑥 𝜆_𝔣(𝑛)2 𝑛1∕2 log (_𝑥 𝑛 ) ≤ 𝑐2𝑥1∕2 (16)

where 𝑐₂depends only on 𝔣. Let 𝑐₃∈]0, 1[. From (15) and (16), it follows that log(1∕𝑐₃) (𝑐₃𝑥)1∕2 ∑♭ 𝑐3𝑥<𝑡≤𝑥 𝜆_𝔣(𝑡)2 ≥ ∑♭ 𝑐3𝑥<𝑡≤𝑥 𝜆_𝔣(𝑡)2 𝑡1∕2 log (_𝑥 𝑡 ) = ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡)2 𝑡1∕2 log (_𝑥 𝑡 ) − ∑♭ 𝑡≤𝑐3𝑥 𝜆_𝔣(𝑡)2 𝑡1∕2 log (_𝑥 𝑡 ) ≥(𝑐₁− 𝑐₂𝑐₃1∕2 ) 𝑥1∕2. We deduce ∑♭ 𝑐₃𝑥<𝑡_≤𝑥 𝜆_𝔣(𝑡)2 _≥ 𝑐 1∕2 3 log(1∕𝑐₃) ( 𝑐₁− 𝑐₂𝑐₃1∕2 ) 𝑥.

Choosing 𝑐₃<min(1, 𝑐₁2∕𝑐₂2) we have ∑♭ 𝑡_≤𝑥 𝜆_𝔣(𝑡)2 ≫ ∑♭ 𝑐₃𝑥<𝑡_≤𝑥 𝜆_𝔣(𝑡)2 ≫ 𝑥. Finally, (14) gives ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡)2_≤∑ 𝑛≤𝑥 𝜆_𝔣(𝑛)2≪ 𝑥 hence ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡)2≍ 𝑥.  With this Lemma, we can complete the proof of Theorem 1. From (9) we derive

∑♭ 𝑡≤𝑥 || |𝜆𝔣(𝑡)|||log (_𝑥 𝑡 ) ≫ 𝑥−𝛼 ∑♭ 𝑡≤𝑥 || |𝜆𝔣(𝑡)||| 2 log (_𝑥 𝑡 ) ≫ 𝑥−𝛼 ∑♭ 𝑡≤𝑥∕2 || |𝜆𝔣(𝑡)||| 2 .

Hence, Lemma 9 implies ∑♭ 𝑡≤𝑥 || |𝜆𝔣(𝑡)|||log (_𝑥 𝑡 ) ≫_𝔣,𝛼 𝑥1−𝛼. (17)

We detect signs of Fourier coefficients with the help of |𝜆𝔣(𝑡)| + 𝜆𝔣(𝑡) 2 = { 𝜆_𝔣(𝑡) if 𝜆_𝔣(𝑡) > 0 0 otherwise. Using (9), we have ∑♭ 𝑡≤𝑥 ( |𝜆𝔣(𝑡)| + 𝜆𝔣(𝑡))log (_𝑥 𝑡 ) ≪_+ 𝔣 (𝑥)𝑥 𝛼_{log 𝑥. (18)}

Moreover, (17) and Lemma 9 imply ∑♭ 𝑡≤𝑥 ( |𝜆𝔣(𝑡)| + 𝜆𝔣(𝑡))log (_𝑥 𝑡 ) = ∑♭ 𝑡≤𝑥 |𝜆𝔣(𝑡)| log (_𝑥 𝑡 ) + ∑♭ 𝑡≤𝑥 𝜆_𝔣(𝑡) log (_𝑥 𝑡 ) ≫ 𝑥1−𝛼+ 𝑂(𝑥3∕4+𝜀) ≫ 𝑥1−𝛼. (19)

Finally, equations (18) and (19) give

_𝔣+(𝑥) ≫ 𝑥 1−2𝛼 log 𝑥⋅ Similarly, using |𝜆𝔣(𝑡)| − 𝜆𝔣(𝑡) 2 = { −𝜆_𝔣(𝑡) if 𝜆_𝔣(𝑡) < 0 0 otherwise we obtain _𝔣−(𝑥) ≫ 𝑥 1−2𝛼 log 𝑥⋅ This finishes the proof of Theorem 1.

4. PROOF OFTHEOREM 4

The basic idea of proof is the same as for Theorem 1, although here we localize on short intervals. The device (4) with the analytic properties of 𝑀 (𝔣, 𝑠) gives a nice mean value estimate for 𝜆𝔣(𝑡) over the squarefree integers in a short interval, see (20). However our series 𝐷(𝔣 ⊗ 𝔣, 𝑠) runs over all positive (not just squarefree) integers. We cannot obtain a counterpart for|𝜆𝔣(𝑡)|2. To get around, we consider a bundle of short intervals and lead to two moment estimates (21) and (26) in §4.1. Then we can enumerate the sign changes in §4.2.

4.1. Computation of moments of order 1 and 2. Let

0≤ 𝛼 < 1∕4 and 1 > 𝜂 > 3∕4 + 𝛼.

Suppose that 𝑥 is sufficiently large. We set ℎ = 𝑥𝜂 and define 𝛿 by e2𝛿= 1 + ℎ∕𝑥. We have 𝛿 ≍ ℎ∕𝑥.

For all 𝑠 ∈ ℂ such that|Re 𝑠| ≤ 2, we have (e𝛿𝑠−1)2_∕𝑠2_≪min(_𝛿2_,1∕|𝑠|2)_{. It} follows then by Lemma 8 and (4) that

∑♭ 𝑥≤𝑡≤𝑥+ℎ 𝜆_𝔣(𝑡) min ( log (_{𝑥+ ℎ} 𝑡 ) ,log (_𝑡 𝑥 )) = 1 2𝜋i∫ 3∕4+𝜀+i∞ 3∕4+𝜀−i∞ 𝑀(𝔣, 𝑠)(e 𝛿𝑠₋₁₎2 𝑠2 𝑥 𝑠_d𝑠 ≪ 𝑥3∕4+𝜀 ∫ +∞ −∞ (|𝜏| + 1)1∕4+𝜀min ( 𝛿2, 1 1 +|𝜏|2 ) d𝜏 ≪ ℎ3∕4𝑥𝜀. (20)

For any integer constant 𝐴 > 0, let (𝜀₁,… , 𝜀𝐴) ∈ {−1, 1}𝐴. The bound for the

moment of order 1 follows from (20), that is ∑ 𝑚≤𝐴 𝜀_𝑚 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 𝜆_𝔣(𝑡) min ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) ≪ ℎ3∕4𝑥𝜀. (21)

We turn to the evaluation of the moment of order 2. Since 𝜂 > 3∕4 + 𝛼, by (14) and Lemma 6, we obtain for some positive constant 𝐶,

𝐶ℎ_{≤ 𝐶}′ ∑ 𝑥<𝑛_≤𝑥+ℎ 𝜆_𝔣(𝑛)2 ≤ ∑ 𝑚≤√𝑥+ℎ 𝜏(𝑚)4 ∑♭ 𝑥 𝑚2≤𝑡≤ 𝑥+ℎ 𝑚2 𝜆_𝔣(𝑡)2.

Next we prove that√𝑥+ ℎ can be replaced by some constant 𝐴 in the outer sum up to the cost of a replacement of a smaller 𝐶. Indeed we will prove, for any fixed

𝐴 >0, ∑ 𝐴<𝑚_≤√𝑥+𝑥𝜂 𝜏(𝑚)4 ∑♭ 𝑥 𝑚2≤𝑡≤ 𝑥+𝑥𝜂 𝑚2 𝜆_𝔣(𝑡)2≪ 𝑥𝜂𝐴−1+𝜀. Note that ∑ √ 𝑥≤𝑚≤√𝑥+𝑥𝜂 𝜏(𝑚)4 ∑♭ 𝑥 𝑚2≤𝑡≤ 𝑥+𝑥𝜂 𝑚2 𝜆_𝔣(𝑡)2= ∑ √ 𝑥≤𝑚≤√𝑥+𝑥𝜂 𝜏(𝑚)4 ∑♭ 𝑡≤𝑥+𝑥𝜂 𝑚2 𝜆_𝔣(𝑡)2 ≪ 𝑥1∕2+𝜀 (22)

by (14). In light of (22), (14) and (3), it suffices to evaluate ∑ 𝐴<𝑚≤√𝑥 𝜏(𝑚)4min { max [ 𝑥𝜂 𝑚2, ( 𝑥 𝑚2 )3∕4+𝛼+𝜀] , ( 1 + 𝑥 𝜂 𝑚2 ) 𝑥2𝛼 𝑚4𝛼 } .

Write 𝑦 = 𝑥𝜂∕𝑚2and 𝑌 = 𝑥∕𝑚2, then 0 < 𝑦 < 𝑌 and 𝑌 ≫ 1. Note 2𝛼 < 3∕4 + 𝛼. The term min{⋯} in the preceding formula is then handled by observing

min{max(𝑦, 𝑌3∕4+𝛼+𝜀), (1 + 𝑦)𝑌2𝛼}≪ ⎧ ⎪ ⎨ ⎪ ⎩ 𝑌2𝛼 if 𝑦≤ 1, 𝑦𝑌2𝛼 if 1 < 𝑦≤ 𝑌3∕4−𝛼_, 𝑌3∕4+𝛼+𝜀 if 𝑌3∕4−𝛼 < 𝑦_{≤ 𝑌}3∕4+𝛼+𝜀, 𝑦 if 𝑌3∕4+𝛼+𝜀 < 𝑦 < 𝑌 .

We split the sum over 𝑚 into 4 subsums with the ranges of summation dividing at the points for which 𝑦 = 1, 𝑦 = 𝑌3∕4−𝛼and 𝑦 = 𝑌3∕4+𝛼+𝜀respectively. Write

𝜂₃ = 𝜂 2 , 𝜂2= 𝜂−3∕4+𝛼 1∕2+2𝛼 , 𝜂1= 𝜂−(3∕4+𝛼+𝜀) 1∕2−2(𝛼+𝜀)

(note 𝜂₃> 𝜂₂> 𝜂₁ >0). The 4 subsums are evaluated via the following summations: ∑ 𝑥𝜂3<𝑚≤√𝑥 𝜏(𝑚)4𝑥 2𝛼 𝑚4𝛼 ≪ 𝑥 2𝛼+(1−4𝛼)∕2+𝜀_{= 𝑥}1∕2+𝜀_{= 𝑜(𝑥}𝜂 ), ∑ 𝑥𝜂2<𝑚≤𝑥𝜂3 𝜏(𝑚)4𝑥 𝜂+2𝛼 𝑚2+4𝛼 ≪ 𝑥 𝜂+2𝛼−𝜂2(4𝛼+1)+𝜀_{= 𝑥}𝜂− (𝜂−3∕4)(1+4𝛼) 1∕2+2𝛼 +𝜀_, ∑ 𝑥𝜂1<𝑚≤𝑥𝜂2 𝜏(𝑚)4 ( 𝑥 𝑚2 )3∕4+𝛼+𝜀 ≪ 𝑥3∕4+𝛼−𝜂1(2𝛼+1∕2)+𝜀_{= 𝑥}𝜂− 𝜂−(3∕4+𝛼+𝜀) 1∕2−2(𝛼+𝜀)+𝜀_, ∑ 𝐴<𝑚≤𝑥𝜂1 𝜏(𝑚)4𝑥 𝜂 𝑚2 ≪ 𝑥 𝜂_𝐴−1+𝜀_.

By taking a large enough constant 𝐴, we infer that ∑ 𝑚≤𝐴 𝜏(𝑚)4 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 𝜆_𝔣(𝑡)2_≥(𝐶− 𝑂(𝐴−1+𝜀))ℎ ≫ ℎ.

This equation remains true if we replace (𝑥, ℎ) by (𝑥 + ℎ∕4, ℎ∕2), so ∑ 𝑚≤𝐴 𝜏(𝑚)4 ∑♭ 𝑥+ℎ∕2 𝑚2 <𝑡≤ 𝑥+3ℎ∕4 𝑚2 𝜆_𝔣(𝑡)2 ≫ ℎ. (23) Moreover ∑ 𝑚≤𝐴 𝜏(𝑚)4 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 𝜆_𝔣(𝑡)2min ( log(𝑥+ ℎ 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) ≥ ∑ 𝑚≤𝐴 𝜏(𝑚)4 ∑♭ 𝑥+ℎ∕4 𝑚2 <𝑡≤ 𝑥+3ℎ∕4 𝑚2 𝜆_𝔣(𝑡)2min ( log(𝑥+ ℎ 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) (24) and, if 𝑡 ∈ [ 𝑥+ℎ∕4 𝑚2 , 𝑥+3ℎ∕4 𝑚2 ] then 𝑥 ℎmin ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) ≫1. (25)

We deduce from (24), (25) and (23) that ∑ 𝑚≤𝐴 𝜏(𝑚)4 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 𝜆_𝔣(𝑡)2min ( log(𝑥+ ℎ 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) ≫ ℎ 2 𝑥 . (26)

4.2. Implication on the number of sign changes. We use (21) and (9) to write ∑ 𝑚_≤𝐴 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 (| ||𝜆𝔣(𝑡)||| + 𝜀𝑚𝜆𝔣(𝑡) ) min ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) ≫ ∑ 𝑚_≤𝐴 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 𝑡−𝛼𝜆_𝔣(𝑡)2min ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) + 𝑂(ℎ3∕4+𝜀) ≫ 𝑥−1−𝛼ℎ2+ 𝑂(ℎ3∕4+𝜀) (27) by (26). If 𝜂 > 4₅(1 + 𝛼), we deduce ∑ 𝑚≤𝐴 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 (| ||𝜆𝔣(𝑡)||| + 𝜀𝑚𝜆𝔣(𝑡) ) min ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) ≫ 𝑥2𝜂−1−𝛼.

Assume that, for all 𝑚 ∈ {1, … , 𝐴}, there exists 𝜀𝑚 ∈ {−1, 1} such that the sign

of 𝜆𝔣(𝑡) is −𝜀𝑚 for every squarefree 𝑡 ∈

] 𝑥 𝑚2, 𝑥+ℎ 𝑚2 [ . Then, ∑ 𝑚_≤𝐴 ∑♭ 𝑥 𝑚2<𝑡< 𝑥+ℎ 𝑚2 (| ||𝜆𝔣(𝑡)||| + 𝜀𝑚𝜆𝔣(𝑡) ) min ( log (_{𝑥+ ℎ} 𝑡𝑚2 ) ,log ( 𝑡𝑚2 𝑥 )) = 0

in contradiction with (27). Consequently, there exists 𝑚 ∈ {1, … , 𝐴} such that the interval ] 𝑥 𝑚2, 𝑥+ℎ 𝑚2 [

contains squarefree integers 𝑡 and 𝑡′satisfying ||

|𝜆𝔣(𝑡)||| = 𝜆𝔣(𝑡)≠ 0 and ||_|𝜆𝔣(𝑡′)||| = −𝜆𝔣(𝑡′)≠ 0

i.e. 𝜆_𝔣(𝑡)𝜆_𝔣(𝑡′_{) < 0.}

Let 𝑋 be any sufficiently large number. Write 𝐵 = (1 + 1∕𝐴)2, 𝐻 = (𝐵𝑋)𝜂 and

𝐽 =⌊(𝐵 − 1)𝑋∕𝐻⌋. For any 𝑗 ∈ {0, … , 𝐽 − 1} and any 𝑚 ∈ {1, … , 𝐴}, let

𝐼_𝑗(𝑚) = ] 𝑋+ 𝑗𝐻 𝑚2 , 𝑋+ (𝑗 + 1)𝐻 𝑚2 [ .

The interval 𝐼𝐽(𝑚 + 1) is on the left side of 𝐼0(𝑚). Moreover, if 𝑗 ≠ 𝑘, then

𝐼_𝑗(𝑚) ∩ 𝐼_𝑘(𝑚) = ∅. It follows that the 𝐴𝐽 intervals 𝐼_𝑗(𝑚) are disjoint. Since, for any

𝑗, there exists 𝑚 such that 𝐼𝑗(𝑚) contains a sign change, we obtain at least 𝐽 ≫ 𝑋1−𝜂

sign changes over the interval [1, 𝑋]. The proof is complete after replacing 𝜂 by

𝜂+ 𝜀.

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YUK-KAMLAU. DEPARTMENT OFMATHEMATICS, THEUNIVERSITY OFHONGKONG, POKFU

-LAMROAD, HONGKONG, HONGKONG

E-mail address: yklau@maths.hku.hk

EMMANUELROYER. (1) UNIVERSITÉCLERMONTAUVERGNE, UNIVERSITÉBLAISEPASCAL, LABORATOIRE DEMATHÉMATIQUES, BP 10448, F-63000 CLERMONT-FERRAND, FRANCE. (2) CNRS, UMR 6620, LM, F-63178 AUBIÈRE, FRANCE

E-mail address: emmanuel.royer@math.univ-bpclermont.fr

JIEWU. (1) SCHOOL OFMATHEMATICS, SHANDONGUNIVERSITY, JINAN, SHANDONG250100, CHINA. (2) CNRS, INSTITUTÉLIECARTAN DELORRAINE, UMR 7502, UNIVERSITÉ DELORRAINE, F-54506 VANDŒUVRE-LÈS-NANCY, FRANCE. (3) UNIVERSITÉ DELORRAINE, INSTITUTÉLIE

CARTAN DELORRAINE, UMR 7502, F-54506 VANDŒUVRE-LÈS-NANCY, FRANCE