Y.-J Jiang

Y.-K Lau
email: yklau@maths.hku.hk

Emmanuel Royer
email: emmanuel.royer@math.univ-bpclermont.fr

Jie Wu
email: jie.wu@math.cnrs.fr

Yujiao Jiang
email: yujiaoj@hotmail.com

Yuk-Kam Lau

Guangshi Lü

ON FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF INTEGRAL WEIGHT AT SQUAREFREE INTEGERS

Keywords: Mathematics Subject Classification. 11F30 Fourier coefficients, Half-integral weight modular forms, Dirichlet series

.

Introduction

Some modular forms are endowed with nice arithmetic properties, for which techniques in analytic number theory can be applied to unveil their extraordinary features. For instance, Matomäki and Radziwill [START_REF] Matomäki | Multiplicative functions in short intervals[END_REF] made an important progress for multiplicative functions with an application (amongst many) to give a very sharp result on the holomorphic Hecke cusp eigenforms of integral weight. The Hecke eigenforms of halfintegral weight is substantially different from the case of integral weight. A simple illustration is the multiplicativity of their Fourier coefficients. If f is a Hecke eigenform of integral weight (for SL 2 (Z)), its Fourier coefficient a f (m) will be factorized into a f (m) = p r m a f (p r ). However, for a Hecke eigenform f of half-integral weight (for Γ 0 (4)), we only have a f (tm 2 ) = a f (t) p r m a f (p 2r ) for any squarefree t, due to Shimura. (Both a f (1) = a f (1) = 1 are assumed.) This, on one hand, alludes to the mystery of {a f (t)} ♭ t 1 † and, on the other hand, provides an interesting object {a f (n)} n 1 whose multiplicativity (is limited to the square factors) has no analogue to the classical number-theoretic functions.

The classical divisor function τ (n) := d|n 1 appears to be Fourier coefficients of some Eisenstein series. In the literature there are investigations on {τ (t)} ♭ t 1 and on the associated Dirichlet series L(s) := ♭ t 1 τ (t)t -s , which is however rather obscure. Using the multiplicative properties of τ (n), L(s) is connected to the reciprocal of the Riemann zeta-function ζ(2s) 3 , and it extends analytically to a holomorphic function on (a slightly bigger region containing) the half-plane ℜe s > 1 2 except at s = 1. A further extension is equivalent to a progress towards the Riemann Hypothesis.

On the other hand, to study the sign-changes in {a f (t)} ♭ t 1 , Hulse et al. [START_REF] Hulse | The sign of Fourier coefficients of half-integral weight cusp forms[END_REF] recently considered L ♭ f (s) := ♭ t 1 λ f (t)t -s (where λ f (t) = a f (t)t -(ℓ/2-1/4) ). Interestingly they showed that L ♭ f (s) extends analytically to a holomorphic function in ℜe s > 3 4 . One naturally asks how far L ♭ f (s) can further extend to. Compared with the case of τ (n) but without adequate multiplicativity, a continuation to the region ℜe s > 1 2 is curious, non-trivial and plausibly (very close to) the best attainable with current technology.

The argument of proof in [START_REF] Hulse | The sign of Fourier coefficients of half-integral weight cusp forms[END_REF] is based on the convexity principle and includes two key ingredients:

(a) the inequality λ f (tr 2 ) ≪ ε |λ f (t)|r ε , (b) the functional equations of the twisted L-functions for f by additive characters e(un/d). The inequality (a) is a substitute for the unsettled Ramanujan Conjecture for halfintegral weight Hecke eigenforms, and this is derived from the Shimura correspondence and the Deligne bound for modular forms of integral weight. According to various d's, the functional equations of (b) involves the Fourier expansions of f at different cusps, which is detailedly computed in [START_REF] Hulse | The sign of Fourier coefficients of half-integral weight cusp forms[END_REF]. However, due to the multiplier system, the Fourier expansion at the cusp 1 2 is not of period 1, of which Hulse et al seemed not aware. We shall propose an amendment in Section 4.

Our main goal is to prove that L ♭ f (s) extends analytically to ℜe s > 1 2 . We shall not use the convexity principle but apply the approximate functional equation with the point s close to the line ℜe s = 1 2 (from right). The cancellation amongst the exponential factors and real quadratic characters arising from the twisted L-functions are explored. Without a known Ramanujan Conjecture, the inequality (a) is crucial and indeed we need more -an inequality of the same type for the Fourier coefficients at all cusps, which is done in Section 3. There we study the Fourier coefficients of a (complete) Hecke eigenform at the two cusps 0 and 1 2 , and derive some inequalities and bounds useful for analytic approaches, which are of their own interest.

Main results

Let ℓ 2 be a positive integer, and denote by S ℓ+1/2 the set of all holomorphic cusp forms of weight ℓ + 1/2 for the congruence subgroup Γ 0 (4). The Fourier expansion of f ∈ S ℓ+1/2 at ∞ is written as

(2.1) f(z) = n 1 λ f (n)n ℓ/2-1/4 e(nz) (z ∈ H ),
where e(z) = e 2πiz and H is the Poincaré upper half plane. Define

(2.2) L ♭ f (s) := ♭ t 1 λ f (t)t -s
for s = σ + iτ with σ > 1, where ♭ t 1 ranges over squarefree integers t 1. Theorem 1. Let ℓ 2 be a positive integer and f ∈ S ℓ+1/2 be a complete Hecke eigenform. The series L ♭ f (s) in (2.2) extends analytically to a holomorphic function on ℜe s > 1 2 . Moreover, for any ε > 0 we have

(2.3) L ♭ f (s) ≪ f,ε (|τ | + 1) 1-σ+2ε ( 1 2 + ε σ 1 + ε, τ ∈ R)
, where the implied constant depends on f and ε only.

Remark 1. It follows immediately the Riesz mean

♭ t x (1 -t/x)λ f (t) ≪ x 1/2+ε
, exhibiting a support towards square-root cancellation of {λ f (t)} ♭ n 1 . An application of Theorem 1 is a better lower bound (than [START_REF] Lau | Sign of Fourier coefficients of modular forms of half integral weight[END_REF]Theorem 4]) for the sign-changes of {λ f (t)} ♭ t 1 with t ∈ [1, x]. Together with a refinement on the mean square formula, we shall prove the next theorem in Section 8.

Theorem 2. Let ℓ 2 be an integer and f ∈ S ℓ+1/2 a complete Hecke eigenform such that its Fourier coefficients are real. Denote by C ♭ f (x) the number of sign changes of λ f (t) where t ranges over squarefree integers in [1, x]. ‡ Let ̺ ∈ (0, 1 4 ) be defined as in (3.13), and ϑ any number satisfying

0 < ϑ < min( 1-2̺ 3 , 1 4 ). Then for all x x 0 (f, ϑ), C ♭ f (x) ≫ f,ϑ
x ϑ where the constant x 0 (f, ϑ) and the implied constant depend on f and ϑ only.

3.

Half-integral weight cusp forms for Γ 0 (4)

We follow Shimura [START_REF] Shimura | On modular forms of half-integral weight[END_REF] to explicate the definition of f ∈ S ℓ+1/2 . The main aim is to discuss some properties of the Fourier coefficients at all cusps when f is a complete Hecke eigenform.

Let GL + 2 (R) be the set of all real 2 × 2 matrices with positive determinant. Define G to be the set of all (α, ϕ(z)) where α = a b c d

∈ GL + 2 (R) and ϕ(z) is a holomorphic function on H such that ϕ(z) 2 := ς det(α) -1/2 (cz + d), for some ς ∈ C with |ς| = 1. Then G is a group under the composition law (α, ϕ(z))(β, ψ(z)) = (αβ, ϕ(βz)ψ(z)). The projection map (α, ϕ(z)) → α is a surjective homomorphism from G to GL + 2 (R). We write (α, ϕ(z)) * = α. Let f be any complex-valued function on H . The slash operator ξ → f | [ξ] , defined as f | [ξ] := ϕ(z) -(2ℓ+1) f (αz) if ξ = (α, ϕ(z)), gives an anti-homomorphism on G, i.e. f | [ξη] = (f | [ξ] )| [η] . Define for γ = a b c d ∈ Γ 0 (4) and z ∈ H , j(γ, z) := θ(γz) θ(z) = ε -1 d c d (cz + d) 1/2 ,
where ε d = 1 or i according as d ≡ 1 or 3 (mod 4), the extended Jacobi symbol c d and the square root (cz+d) 1/2 are defined as in [START_REF] Shimura | On modular forms of half-integral weight[END_REF]. The map γ → γ * with γ * := (γ, j(γ, z)) is an one-to-one homomorphism from Γ 0 (4) to G. For γ ∈ Γ 0 (4), we will abbreviate

f | [γ * ] as f | [γ] .
A cusp form f of weight ℓ + 1/2 for Γ 0 (4) is a holomorphic function on H such that 1 • f| [γ] = f for all γ ∈ Γ 0 (4), 2 • f admits a Fourier series expansion at every cusp a ∈ {0, -1 2 , ∞},

f| [ρ] = n∈Z n+r>0
c n e((n + r)z). ‡ See [START_REF] Lau | Sign of Fourier coefficients of modular forms of half integral weight[END_REF] for explanation in detail.

Here ρ ∈ G satisfies that its projection is a scaling matrix for the cusp a, i.e. ρ * (∞) = a, and for some |t| = 1,

ρ -1 η * ρ = (η ∞ , t) with η ∞ := 1 1 1 ,
where η is a generator of the stabilizer Γ a in Γ 0 (4) for the cusp a. The value of r ∈ [0, 1) is determined by e(r) = t 2ℓ+1 . (See [15, p.444].)

3.1. Fourier expansions at the three cusps. Explicitly we take ρ = ρ a where

ρ a =                  1 1 , 1 for a = ∞, 1 -2 1 , (-2z + 1) 1/2 for a = -1 2 , -1 4 , 2 1/2 (-iz) 1/2
for a = 0.

(3.1)

Set η a = ρ a * η ∞ ρ a -1 * .
Then η a ∈ Γ 0 (4) for all the three cusps. A direct checking shows that ρ -1 a η * a ρ a = (η ∞ , t a ) where t a = 1, i, 1 for a = ∞, -1 2 , 0, respectively. (When a = -1 2 , the factor

ε -1 -1 -4 -1 inside j(η * a , z) equals i.) Hence, for f ∈ S ℓ+1/2 , f(z + 1) = f(z) (note f| [ρ∞] = f) and f| [ρ 0 ] (z + 1) = f| [ρ 0 ] (z), while for a = -1 2 , f| [ρa] (z + 1) = t 2ℓ+1 -1/2 f| [ρaη∞] (z) = i 2ℓ+1 f| [η * a ] | [ρa] (z) = i 2ℓ+1 f| [ρa] (z). (3.2) Let α = 4 1 , 2 -1/2 .
For our purpose, we set

(3.3) g(z) := f| [ρ -1/2 ] | [α] (z) = 2 ℓ+1/2 f| [ρ -1/2 ] (4z) and h(z) := f| [ρ 0 ] (z).
Their Fourier series expansions (at ∞) are of the form (3.4)

g(z) = 2 ℓ+1/2 n 0 c n e (4n + (2 + (-1) ℓ-1 ))z = 2 ℓ+1/2 n 1 λ g (n)n ℓ/2-1/4 e(nz), say,
where the sequence {λ g (n)} is supported on positive integers n ≡ (-1) ℓ (mod 4), and

(3.5) h(z) = n 1 λ h (n)n ℓ/2-1/4 e(nz). Remark 2. (i) The cusp form h(z) is f 0 (z) in [3] but g(z) = f 1 2 (4z), not f 1 2 (z), there. The Fourier expansion of f 1 2 (z) at ∞ is of the form n 1 c n e((n + 1 4 )z). (ii)
The form h is a cusp form for Γ 0 (4) but g is a cusp form for Γ 0 (16). (iii) Using the Rankin-Selberg theory, one can prove that

(3.6) n x |λ f (n)| 2 ∼ c f x (f = f, g or h)
for some constant c f > 0. See [10, Section 3], for example. (There the assumption that f is a complete Hecke eigenform is not necessary, which is clearly seen from the proof.)

Eigenform properties of a complete Hecke eigenform at various cusps.

Let N be a positive integer divisible by 4, and p ∤ N be any prime. The action of the Hecke operator T(p 2 ) on a modular form f of half-integral weight ℓ + 1/2 for Γ 0 (N) is defined as (cf. [15, p.451])

T(p 2 )f := p ℓ-3/2 0 b<p 2 f | [α ⋆ b ] + 1 h<p f | [β ⋆ h ] + f | [σ ⋆ ] ,
where

α ⋆ b := α b , p 1/2 = 1 b p 2 , p 1/2 , β ⋆ h := β h , ε -1 p -h p = p h p , ε -1 p -h p , σ ⋆ := σ, p -1/2 = p 2 1 , p -1/2 .
Suppose f is a complete Hecke eigenform, i.e. T(p 2 )f = ω p f for all prime p. One may wonder whether g and h defined as in (3.3) are eigenforms. We can prove the following.

Lemma 3.1. Let p be any odd prime. If f ∈ S ℓ+1/2 is an eigenform of T(p 2 ), then so are the forms g and h defined in (3.3) and both have the same eigenvalues as f.

Proof. Let N = 4 or 16, and ∆ 0 = Γ 0 (N) * be the image of Γ 0 (N) under the lifting map. It suffices to show that for (i)

ρ -1/2 α, N = 16 and (ii) ρ = ρ 0 , N = 4, the elements ρα ⋆ b , ρβ ⋆ h , ρσ ⋆ (0 b < p 2 , 1 h < p) form a set of representatives for ∆ 0 \ ∆ 0 σ ⋆ ρ ⊔ 1 h<p 2 ∆ 0 β ⋆ b ρ ⊔ 0 b<p 2 ∆ 0 α ⋆ b ρ .
(i) For the case ρ = ρ -1/2 α, we check by routine calculation that

∆ 0 ρσ ⋆ = ∆ 0 α ⋆ (p 2 +1)/2 ρ, ∆ 0 ρα ⋆ (p 2 -1)/8 = ∆ 0 σ ⋆ ρ, {∆ 0 ρβ ⋆ h : 1 h < p} = {∆ 0 α ⋆ d ρ : p (1 -2d)}, {∆ 0 ρα ⋆ b : p ∤ (1 + 8b)} = {∆ 0 α ⋆ d ρ : p ∤ (1 -2d)}, {∆ 0 ρα ⋆ d : p (1 + 8b)} = {∆ 0 β ⋆ h ρ : 1 h < p}. For example, from ρ = ρ -1/2 α = 4 -8 1 , 2 -1/2 (-8z + 1) 1/2 we obtain ρ * β h ρ -1 * = γα d , where γ = p + 8h (4h -d(p + 8h))p -2 -16h (p -8h + 16hd)p -2 ∈ Γ 0 (16) if we take 1 d < p 2 such that d(p + 8h) ≡ 4h ( mod p 2 ). Note that this choice implies p -8h + 16hd ≡ p(1 -2d) (mod p 2 ) and dp ≡ 4h(1 -2d) (mod p 2 ). The latter implies p | (1 -2d), so the former is ≡ 0 (mod p 2 ). Next the ϕ-part of ρβ ⋆ h ρ -1 is ε -1 p -h p (-16hz + p -8h) 1/2 p -1/2 .
To evaluate the ϕ-part of γ * α ⋆ d , we remark that j(γ, z) = j(γ -1 , γz) -1 and thus consider γ * -1 whose j-part is simply

ε -1 p+8h 16h p + 8h (16hz + p + 8h) 1/2 . Hence, the ϕ-part of γ * α ⋆ d is ε p h p (-16γα d z + p + 8h) -1/2 p 1/2 . From γα d = ρ * β h ρ -1 * and ε p -1 p = ε -1
p , we easily verify this case. The other cases are checked in the same way.

(ii) For the case ρ = ρ 0 , we find similarly that

∆ 0 ρσ ⋆ = ∆ 0 α ⋆ 0 ρ, ∆ 0 ρα ⋆ 0 = ∆ 0 σ ⋆ ρ, {∆ 0 ρβ ⋆ h : 1 h < p} = {∆ 0 α ⋆ pd ρ : 1 d < p}, {∆ 0 ρα ⋆ d : p b} = {∆ 0 β ⋆ h ρ : 1 h < p}, {∆ 0 ρα ⋆ b : p ∤ b} = {∆ 0 α ⋆ d ρ : p ∤ d}.
3.3. Shimura's correspondence and bounding coefficients. Let f ∈ S ℓ+1/2 , not necessarily a complete Hecke eigenform. By Shimura's theory [15, Section 3], for any squarefree t 1, there is a cusp form Sh t f of weight 2ℓ for Γ 0 (2) such that

(3.7) t ℓ/2-1/4 L(s + 1 2 , χ t ) n 1 λ f (tn 2 )n -s = L(s, Sh t f),
where L(•, χ t ) is the Dirichlet L-function associated to the character

χ t (n) = χ 0 (n) -1 n ℓ t
n (χ 0 is the principal character mod 4) and L(s, F ) := n 1 λ F (n)n -s is the L-function for the cusp form of integral weight 2ℓ with nebentypus χ 2 0 ,

F (z) = n 1 λ F (n)n (2ℓ-1)/2 e(nz).
The Shimura lift f → Sh t f commutes with Hecke operators: Sh t (T(p 2 )f) = T (p)(Sh t f) for all primes p. § It follows that the coefficients λ f (mp 2r ) satisfy a recurrence relation in r when f is a T(p 2 )-Hecke eigenform. Moreover, if f is a Hecke eigenform of T(p 2 ) for all p / ∈ S (where S is any set of primes), the right-hand side of (3.7) will admit a factorization (see Corollary 1.8 and Main Theorem in [START_REF] Shimura | On modular forms of half-integral weight[END_REF])

(3.8) t ℓ/2-1/4 L(s + 1 2 , χ t ) n 1 λ f (tn 2 ) n s = n 1 p|n⇒p∈S λ Sh t f (n) n s p / ∈S 1 - ω p p s + χ 0 (p) p 2s -1
, where T(p 2 )f = ω p p (2ℓ-1)/2 f. Remark that the product p / ∈S remains the same for lifts of different squarefree t's.

The commutativity between Sh t and T(p 2 ) implies that ω p is also an eigenvalue of the Hecke operator T (p) for Sh t f. Decompose (3.9)

Sh t f(z) = i c i f i (ℓ i z)
where each f i is a newform (of perhaps lower level) and f i (ℓ i z)'s are linearly independent. Let S ′ be the set of all prime p dividing the level of Sh t f, so S ′ = {2} in our case. [6, (2.14)] and [START_REF] Iwaniec | Analytic number theory[END_REF]Section 14.7].) Applying T (p) on both sides of (3.9), we thus see that ω p is the T (p)-eigenvalue of some newform (for p ∤ level of Sh t f) and hence

If p / ∈ S ′ , then T (p)(f i (ℓ i z)) = (T (p)f i )(ℓ i z), ∀ i. (See
(3.10) |ω p | 2 ∀ p / ∈ S ∪
|λ f (m)| |λ f (q)|τ (r) 2 .
Remark 3. Every integer m 1 decomposes uniquely into the desired form: Decompose m uniquely into m = tn 2 where t is squarefree, write n = ur such that p | u implies p ∈ Q and p | r implies p / ∈ Q, and then set q = tu 2 .

Proof. Let m = qr 2 = tu 2 r 2 be decomposed as in Remark 3. By (3.8), we see that

t ℓ/2-1/4 λ f (tu 2 r 2 ) = ab=u λ Sh t f (a)µ(b) χ t (b) √ b cd=r ω c µ(d) χ t (d) √ d ,
where ω c is the coefficient of c -s in p / ∈Q (1 -ω p p -s + χ 0 (p)p -2s ) -1 and µ(d) is the Möbius function. The case r = 1 tells that the first bracket is t ℓ/2-1/4 λ f (tu 2 ), i.e. t ℓ-1/4 λ f (q). Next, since |ω c | τ (c) (by (3.10) and its definition), the absolute value of the second bracket is τ (r) 2 .

3.4. Bounds for coefficients of a complete Hecke eigenform at all cusps. ¶ In case f is a complete Hecke eigenform, we may express (3.8) as

(3.11) t ℓ/2-1/4 L(s + 1 2 , χ t ) n 1 λ f (tn 2 )n -s = t ℓ/2-1/4 λ f (t)L(s, F ),
where the Shimiura lift F is a cusp form independent of t. As the Ramanujan's conjecture holds for holomorphic newforms of integral weight, the rth Fourier coefficients of F are ≪ f τ (r)r ℓ-1/2 , where the implied constant is independent of t. Consequently the question of the size of λ f (m) is reduced to the size at the squarefree part of m:

λ f (m) ≪ f |λ f (t)|τ (r) 2 (3.12)
if m = tr 2 and squarefree t. Due to Iwaniec [START_REF] Iwaniec | Fourier coefficients of modular forms of half-integral weight[END_REF] or Conrey & Iwaniec [START_REF] Conrey | The cubic moment of central values of automorphic L-functions[END_REF], etc, there are good estimates for

(3.13) λ f (t) ≪ f,̺ t ̺ ∀ squarefree t,
for some 0 < ̺ < 1 4 . The value of ̺ is 1 6 + ε by [START_REF] Conrey | The cubic moment of central values of automorphic L-functions[END_REF]. ¶ The content of this subsection is not used in the remaining part of the paper but we would include here for its own interest and for applications in other occasions.

We know from Lemma 3.1 that for a complete Hecke eigenform f, the forms g and h are eigenforms of T(p 2 ) with the same corresponding eigenvalue for all odd prime p. But for p = 2, we do not get the same conclusion. This may result in an unpleasant situation of without (3.12). Note that Lemma 3.2 gives at most a bound of the form |λ f (t2 2j )|τ (r) 2 (where f = g, h). Now we attempt to clarify as much as possible.

In view of [15, Proposition 1.5], the Hecke operator T( 22 ) is the same as the operator U 4 whose action is

(f |U 4 )(z) = 1 4 ν (mod 4) f z + ν 4 = n 1 a(4n) e(nz) (3.14) if f (z) = n 1 a(n) e(nz
f| [ρ -1/2 ] (z + ν) = 2 ℓ-3/2 f| [ρ -1/2 ] (z) 0 ν 3 i ν(2ℓ+1)
where the sum is obviously zero. Thus g is also a complete Hecke eigenform although it takes the different eigenvalue 0 for T(2 2 ), implying the validity (3.12) for g as well.

However for the case of h, we cannot get the conclusion of T( 22 )-eigenform and we shall get the analogous bound via some bypass. To its end, let us recall Niwa's result in [START_REF] Niwa | On Shimura's trace formula[END_REF], cf. Kohnen [8, p. 250], saying that U 4 W 4 is Hermitian operator on S ℓ+1/2 and (Note that the operator acts on f from right.) In particular we observe that c = 0, because otherwise, -2µ 2 f = 0 implying f = 0. As h = f|W 4 and W 4 is an involution (i.e. W 2 4 is the identity), we deduce that (h|U 4 ) = µf + 2µ 2 c -1 h. (3.17)

U 4 W 4 U 4 W 4 -µU 4 W 4 -2µ 2 = 0, (3.16) where µ = 2 2ℓ+1 2 ℓ-1 = (-1) ℓ(ℓ+1)/2 2 ℓ-1 and (f |W 4 )(z) = (-2iz) -(ℓ+1/2) f (-1 4z ) = f | [ρ 0 ] (z). Suppose f|T(2 2 ) = cf
We separate into two cases:

• Case 1: c 2 = 2µ 2 .
We set α := cµ/(2µ 2 -c 2 ) and consider the form H := h + αf ∈ S ℓ+1/2 . Then cα + µ = 2αµ 2 /c and thus by (3.17),

H|T(2 2 ) = H|U 4 = 2µ 2 c -1 H.
i.e. The cusp form H is an eigenform of T(2 2 ), and by Lemma 3.1, H is also an eigenform of T(p 2 ) for all odd primes p. (Note that f and h have the same T(p 2 )eigenvalue.) Consequently, both coefficients λ H (m) and λ f (m) satisfy (3.12). As λ h (m) = λ H (m) -αλ f (m), we establish (3.18) for h.

Here we write f|T(p 2 ) for T(p 2 )f.

• Case 2: c 2 = 2µ 2 .

We infer from (3.17) and (3.14) that for all integers n 1,

4 ℓ/2-1/4 λ h (4n) = µλ f (n) + cλ h (n).
Let d = c/4 ℓ/2-1/4 . This recurrence relation gives

λ h (4 J n) = d J λ h (n) + µ 4 ℓ/2-1/4 1 j<J d j λ f (4 J-j n).
Note µ 2 = 2 2ℓ-2 , so |d| = 1, and (3.12) holds for λ f (4 J-j n). Hence, for any integer m = tr 2 4 J where t is squarefree and r is odd, 2 and consequently

λ h (m) ≪ |λ h (tr 2 )| + J 2 |λ f (t)|τ (r) 2 . By Lemma 3.2 with Q = {2}, we get |λ h (tr 2 )| ≪ |λ h (t)|τ (r)
λ h (m) ≪ (|λ h (t)| + |λ f (t)|)J 2 τ (r) 2 ≪ (|λ h (t)| + |λ f (t)|)τ (r2 J ) 2 .
In summary, we have proved the following. Lemma 3.3. Let f be a complete Hecke eigenform, g and h be defined as in (3.3). For any integer m = tr 2 where t 1 is squarefree, we have

λ f (m) ≪ f |λ f (t)|τ (r) 2 + |λ f (t)|τ (r) 2 ≪ f,̺ t ̺ τ (r) 2 (3.18)
for f = f, g, h respectively, where ̺ satisfies (3.13). The first implied ≪-constant depends only f and the second implied ≪-constant depends at most on f and ̺.

Remark 4. When f lies in the Kohnen plus space, the Hecke operator T + (2 2 ) := 3 2 U 4 pr is taken in place of T( 22 ), where pr is the orthogonal projection onto the plus space, cf. [9, p. 42-43]. If f is an eigenform of T + (2 2 ) and T(p 2 ) for all odd primes p, then Lemma 3.3 will still be valid. Firstly Lemma 3.1 and (3.11) hold for f and hence (3.12). Next we claim (3.12) holds for g and h. For g, (3.15) holds if f ∈ S ℓ+1/2 , thus g is a complete Hecke eigenform so (3.12) holds. Note 2µh = f| U 4 once f is in the plus space, see [8, Proposition 2]; thus 2µλ h (m) = λ f (4m), the claim follows from (3.12) for f.

A preparation

We start with the method of proof in [START_REF] Hulse | The sign of Fourier coefficients of half-integral weight cusp forms[END_REF] for the set-up. Meanwhile we amend, for the case 2 d, the functional equation to relate f with g (not f 1 2 in [START_REF] Hulse | The sign of Fourier coefficients of half-integral weight cusp forms[END_REF]), cf. [3, (4.5)] and our Remark 2 (i). Lastly we indicate the vital components for improvement with a first attempt (see Proposition 1 and Remark 6).

Define ½ r 2 (n) = 1 if r 2 | n and 0 otherwise. Replace the divisibility condition with additive characters, we can write

½ r 2 (n) = 1 r 2 u(mod r 2 ) e nu d 2 = 1 r 2 d|r 2 * u(mod d) e nu d ,
where * u(mod d) runs over u(mod d) coprime to d. Recall µ(n) 2 = r 2 |n µ(r). When σ > 1, one thus has Consequently they proved that

L ♭ f (s) = ∞ r=1 µ(r) r 2
(4.4) D r (s) ≪ r 2-4σ+ε (1 + |τ |) 1-σ+2ε (-ε σ 1 + ε).
To obtain the functional equation of L f (s, u/d), one considers for rational q,

Λ(f, q, s) := ∞ 0 f(iy + q)y s+ ℓ 2 -1 4 dy y = Γ(s + ℓ 2 -1 4 ) (2π) s+ ℓ 2 -1 4 m 1 λ f (m) e(mq) m s •
The integral is absolutely convergent for every s ∈ C. We define Λ(g, q, s) and Λ(h, q, s) in the same way.

λ(n; d) ̟ d (n, v) 4 | d λ f (n) ε 2ℓ+1 v d v e -nv d 2 d λ g (n) ε 2ℓ+1 v d v e -nv 4d 2 ∤ d λ h (n) i ℓ+1/2 ε -(2ℓ+1) d v d e -4nv d with 44 ≡ 1 (mod d). Write (4.7) L ∞ (s) := (2π) -s Γ s + ℓ 2 -1 4
) and

(4.9) q s d L ∞ (s)L f (s, u/d) = i -(ℓ+1/2) q 1-s d L ∞ (1 -s) L f (1 -s, v/d) where uv ≡ 1 (mod d).
Remark 5. For the case 2 d, the right-side of the equation (4.9) is of period d or probably its divisor in the parameter v, which is not obvious in view of the factor e(-nv 4d ). Indeed, one checks that

̟ d (n, v + d) = ̟ d (n, v) by using (i) if d = 2h where h is odd, then d v = -2 h h v ; (ii) n ≡ (-1) ℓ (mod 4
) in light of the support of {λ g (n)}. Assume σ < 0. Applying the functional equation (4.9) to (4.3), we obtain

(4.10) D r (s) = i -(ℓ+1/2) r -2 L ∞ (1 -s) L ∞ (s) d|r 2 q 1-2s d n 1 λ(n; d) n 1-s * u (mod d) ̟ d (n, v).
With

̟ d (n, v) ≪ d 1/2 τ (d)(d, n) 1/2 . (4.11)
But in fact it carries more arithmetic properties, as shown below.

̟ 2 e a 2 b (m, v) = G e,b (m)a 2 f |a 2 µ(f ) f ½ a 2 /f (m),
where Let ε > 0 be small and σ = -ε. Applying (4.11) and Stirling's formula to (4.10), it follows that (recalling

q d = d or 2d) D r (-ε + iτ ) ≪ r -2 (1 + |t|) 1+ε d|r 2 d 3/2+3ε n 1 |λ(n; d)|(d, n) 1/2 n -(1+ε) ≪ (r(1 + |t|)) 1+ε because |λ(n; d)|(n, d) 1/2 |λ(n; d)| 2 + (n, d), implying that the last summation is ≪ n 1 |λ(n; d)| 2 n -(1+ε) + ℓ|d n 1 n -(1+ε) ≪ d ε .
By [3, Lemma 4.2], we have D r (1+ε+iτ ) ≪ r -2 . An application of Phragmén-Lindelöf principle gives D r (σ + iτ ) ≪ r 1-3σ+ε (1 + |τ |) ε . To assure the convergence in (4.1), we require 1 -3σ < -1 and hence conclude the following.

Proposition 1. L ♭ f (σ + iτ ) ≪ f,ε (|τ | + 1)
1-σ+2ε for 2 3 + ε σ 1 + ε and τ ∈ R. Remark 6. We have applied only the mean square estimate (3.6) for g and h, and only the Hecke eigenform property of f is used. In the next section, we will invoke the arithmetic property revealed in (4.12), the eigenform properties of all f, g, h and the approximate functional equation to prove the main result.

Proof of Theorem 1

We begin with the approximate functional equation for L f (s, u/d) below, whose proof is given in Section 7.

L f (s, u/d) = m 1 λ f (m) e(mu/d) m s V m q d T + i -(ℓ+1/2) (q d T ) 1-2s m 1 λ(m; d)̟ d (m, v) m 1-s V s,T m q d T
where uv ≡ 1 (mod d), V (y) and V s,T (y) are smooth functions on (0, ∞) and satisfy the following: for any

0 < η < 1 4 , V (y) = 1 + O η (y η ), V s,T (y) = L ∞ (1 -s) T 1-2s L ∞ (s) + O η (y η ) ≪ η 1 + y η ,
and for any η > 0, both V (y) and V s,T (y) ≪ η y -η . Now we deal with L ♭ f (s). In (4.1), we replace the even squarefree r by 2r and thus

L ♭ f (s) =
♭ f (s) into three pieces, L ♭ f (s) = M ∞ (f, s) + M 1/2 (f, s) + M 0 (f, s), where M ∞ (f, s) := - 1 4 ∞ r=1 odd µ(r) r 2 d|r 2 * u(mod 4d) L f (s, u/4d), M 1/2 (f, s) := - 1 4 ∞ r=1 odd µ(r) r 2 d|r 2 * u(mod 2d) L f (s, u/2d), M 0 (f, s) := 3 4 ∞ r=1 odd µ(r) r 2 d|r 2 * u(mod d) L f (s, u/d).
We shall verify the uniform convergence for the three series of holomorphic functions in ℜe s > 1 2 , and concurrently obtain the desired upper estimate (2.3). Let σ 0 = 1 2 + ε 0 where ε 0 > 0 is arbitrarily small but fixed, and T 1 be any integer. Consider s = σ + iτ where σ 0 σ σ 0 +

L f s, u 2 e a 2 b
where e = 0, 1, 2. Now we apply Lemma 5.1 and observe, as before, the set of v given by uv ≡ 1 (mod d) runs through a reduced residue class as u varies. We are led to

Σ 1 = a,b,c µ(2abc) (abc) 2 m 1 λ f (m) m s V m a 2 bT e * u (mod 2 e a 2 b) e mu 2 e a 2 b , (5.1)
Σ 2 = T 1-2s e a,b,c µ(2abc) a 4s b 1+2s c 2 m 1 λ f,e (m) m 1-s V s,T m a 2 bT e * v(mod 2 e a 2 b) ̟ 2 e a 2 b (m, v), (5.2)
where λ f,e (m) := λ(m; 2 e ), see (4.6), and T e = 2T or 4T according as e = 0 or not (so that q 2 e a 2 b T = a 2 bT e , see (4.5)).

Inserting (4.12) into Σ 2 in (5.2), we further decompose a = f g and m = f g 2 h in light of the squarefreeness of f and the conditions f | a 2 and (a 2 /f ) | m.

(5.3) Σ 2 = T 1-2s e f,g,b,c µ(f )µ(2f gbc) f 3s g 2s b 1+2s c 2 h 1 λ f,e (f hg 2 ) h 1-s V s,T h f bT e G e,b (f hg 2 ).
To justify the uniform convergence, it suffices to consider the sum over dyadic ranges: (f, g, b, c) ∼ (F, G, B, C), meaning F f < 2F , etc. Denote by Σ F,G,B,C 2 the expression on the right-side of (5.3) under this range restriction. We estimate each summand trivially with the bound G e,b (m) ≪ √ b in Lemma 4.2. A little simplification leads to

(5.4) Σ F,G,B,C 2 ≪ T 1-2σ e (f,g,b,c) ∼(F,G,B,C) |µ(2f gbc)| f 3σ g 2σ b 1/2+2σ c 2 h 1 |λ f,e (f hg 2 )| h 1-σ V s,T h f bT e .
Next we treat the sum over h in order for the following estimate * * :

(5.5)

h 1 |λ f,e (f hg 2 )| h 1-σ V s,T h f bT e ≪ G ε (T F B) σ-1/2+ε h (F BT ) 1+ε |λ f,e (f h)| √ h •
To establish (5.5), we invoke Lemmas 3.1 and 3.2, to remove g inside λ f,e (f hg 2 ), and the estimate for V s,T . Set Q to be the set of all primes not dividing g, and write h = qr 2 where p 2 |q implies p ∈ Q and p|r implies p / ∈ Q (see Remark 3). As (2f, g) = 1, Q contains 2 and all the prime factors of f . Thus p 2 |f q implies p ∈ Q. Thus, |λ f,e (f hg 2 )| = |λ f,e (f q(gr) 2 )| ≪ ε |λ f,e (f q)|(gr) ε . From Lemma 5.1, we deduce the estimate

V s,T h f bT e ≪ ε (F BT ) ε for h (F BT ) 1+ε , h -2 otherwise.
The sum over h (F BT ) 1+ε is negligible, in fact ≪ (T F GB) ε (for which we may use the crude bound |λ f,e (f q)| ≪ (f q) 1/2 by (3.6)). Consequently, the left side of (5.5) is

≪ (F BT ) ε qr 2 (F BT ) 1+ε g ε r 2(σ-1)+ε |λ f,e (f q)|q -(1-σ) ≪ G ε (F BT ) σ-1/2+ε q (F BT ) 1+ε
|λ f,e (f q)|q -1/2

(recalling σ σ 0 > 1/2) which is (5.5) after renaming q into h. Inserting (5.5) into (5.4), we deduce that

Σ F,G,B,C 2 ≪ (T F GB) ε T -σ+1/2 F -2σ G -2σ+1 B -σ C -1 f ∼F h (F BT ) 1+ε |λ f,e (f h)|(f h) -1/2 .
Write m = f h and note the divisor function τ (m) ≪ ε m ε . The double sum is

≪ ε (F 2 BT ) ε m≪(F 2 BT ) 1+ε |λ f,e (m)|m -1/2 ≪ ε (F 2 BT ) 1/2+ε
, by (3.6). In summary, we get

Σ F,G,B,C 2 ≪ ε (T F GB) ε T 1-σ F 1-2σ G 1-2σ B -σ+1/2 C -1 .
Recall σ 0 = 1 2 + ε 0 and take ε ε 0 /2. Consequently, uniformly for σ 0

σ σ 0 + 1 2 and T -1 |τ | T , we have Σ F,G,B,C 2 → 0 as max(F, G, B, C) → ∞, concluding the uniform convergence. Moreover, as T 1-σ ≪ (1 + |s|) 1-σ , it follows that Σ 2 ≪ F,G,B,C Σ F,G,B,C 2 ≪ ε (1 + |s|) 1-σ+ε 0 ,
recalling the multiple summations range over powers of two.

We turn to Σ 1 in (5.1) which is plainly treated in the same fashion and indeed easier. The inner exponential sum in (5.1) equals

2 e a 2 b δ|2 e a 2 b µ(δ) δ ½ 2 e a 2 b/δ (m).
Noting that a, b are squarefree and (a, b) = 1, we write δ = 2 j f k, a = f g and b = kl where j = 0 or 1. Then the summation over m will be confined to run over the sequence of m = 2 e-j f g 2 lh for positive integers h. Explicitly we have

(5.6) Σ 1 = j=0,1 f,g,k,l,c 2 (e-j)(1-s) µ(2 j f k)µ(2f gklc) f 1+s g 2s k 2 l 1+s c 2 h 1 λ f (2 e-j f lhg 2 ) h s V h 2 j f kT .
Analogously we divide the summation ranges into dyadic intervals and consider the subsum of f,g,k,l,c with (f, g, k, l, c) ∼ (F, G, K, L, C). Repeating the above argument † † , † † For the calculation as in (5.5), there is a little variant since the exponent σ of

|h s | is > 1 2 .
correspondingly we obtain

Σ F,G,K,L,C 1 ≪ ε (T F GK) ε F -σ-1 G 1-2σ K -1 L -σ-1 C -1 max j=0,1,2 (f,l)∼(F,L) h≪(F KT ) 1+ε |λ f (2 j f lh)|h -σ ≪ ε (T F GKL) ε F -1 G 1-2σ K -1 L -1 C -1 m≪(F 2 KLT ) 1+ε |λ f (m)|m -σ ≪ ε (T F GKL) ε T 1-σ F 1-2σ G 1-2σ K -σ L -σ C -1 ,
which assures the uniform convergence and the upper estimate. Our proof is complete by changing ε 0 into 2ε.

Proof of Lemma 4.2

First consider e = 1 or 2, and take the complex conjugate of the left side to simplify a bit the exponential factor. Then by ( 4 We write v = α8b + βa 2 . Note that v runs over a reduced residue class mod 2 e a 2 b when α (mod a 2 ) and β (mod 2 e b) run over the respective reduced residue classes, since a is odd and (a, 2b) = 1. Our substitution choice implies v ≡ β (mod 4) and thus ε v = ε β . Moreover, the extended Jacobi symbol may be written as, cf. [15, p.442 We separate the sum in (6.2) into two subsums, whose summands take the same value of ε β , as follows: * β (mod 2 e b) β≡1(mod 4)

+ i -(2ℓ+1) * β (mod 2 e b) β≡-1(mod 4)
.

With the primitive character χ 4 mod 4 (given by χ 4 (n) = (-1) (n-1)/2 for odd n), we relax the extra conditions with the factors 1 2 (1+χ 4 (β)) and 1 2 (1-χ 4 (β)). Consequently, letting θ ± = 1 2 (i ℓ+1/2 ± i -(ℓ+1/2) ), we rearrange the terms to have

G ′ e,b (m) = θ + * β (mod 2 e b) ψ 2,b (β)χ b ′ (β) e mβ 2
* v(mod a 2 b) v b e -4mv a 2 b = a 2 f |a 2 µ(f ) f ½ a 2 /f (m) * β(mod b) χ b ′ (β) e - 4mβ b .
Take G 0,b (m) to be the product of i ℓ+1/2 ε -(2ℓ+1) d

and the character sum (over β). This gives, with (6.3), the desired result in (4.12), completing the proof.

Proof of Lemma 5.1

Let H(z) be an entire function such that H(z) ≪ η,A (1 + |z|) -A for ℜe z = η and any A > 0, H(0) = 1 and H(z) = H(-z). (See [START_REF] Harcos | Uniform approximate functional equation for principal L-functions[END_REF] for its construction.) We infer with the residue theorem that

L f (s, u/d) = 1 2πi (2)
-

(-2) L f (s + z, u/d)(q d T ) z H(z) dz z =: I 1 + I 2 , (say).
Changing z to -z and invoking the functional equation, we transform

I 2 into i -(ℓ+1/2) (q d T ) 1-2s 1 2πi (2) L f (1 -s + z, v/d) L ∞ (1 -s + z) T 1-2s+2z L ∞ (s -z) (q d T ) z H(z) dz z • Set V (y) := 1 2πi (2) y -z H(z) dz z , V s,T (y) := 1 2πi (2) L ∞ (1 -s + z) T 1-2s+2z L ∞ (s -z) y -z H(z) dz z •
The formula follows readily after inserting the Dirichlet series of L f (s, u/d) and L f (s, v/d) in (4.2) and (4.8). It remains to check the properties of V (y) and V s,T (y). The case of V (y) is quite obvious, and for V s,T (y), we recall the estimate in [2, Lemma 3.2]: For

α > -σ, Γ(z + σ) Γ(z) ≪ α,σ |z + σ| σ (ℜe z α).
Recalling (4.7), this yields

(7.1) L ∞ (1 -s + z) T 1-2s+2z L ∞ (s -z) ≪ η 1 -s + z + ℓ 2 -1 4 T 1-2σ+2η ≪ 1 + |z| T 1-2σ+2η
.

We shift the line of integration to the right, yielding V s,T (y) ≪ η y -η for any η > 0 and shift to the left to derive

V s,T (y) = L ∞ (1 -s) T 1-2s L ∞ (s) + O η (y η )
for any 0 < η < 1 4 . The main term is O(1) by (7.1). The proof of Lemma 5.1 ends.

Proof of Theorem 2

We start with a lemma which supersedes [10, (14)] with an improvement on the exponent of the O-term from 3 4 + ̺ + ε to 3 4 + ε. Lemma 8.1. Let ℓ 2 be a positive integer and f ∈ S ℓ+1/2 be a complete Hecke eigenform. Then for any ε > 0 and all x 2, we have • trivially w ± (s) ≪ X σ and (8.3) w ± (1) = X + O(Y ).

Obviously we have (8.4)

n |λ f (n)| 2 w -(n) X<n 2X |λ f (n)| 2 n |λ f (n)| 2 w + (n).
Let the Dirichlet series associated with |λ f (n)| 2 be defined as (see e.g. [10, (11)])

D(f ⊗ f, s) = ∞ n=1 |λ f (n)| 2 n -s .
By the Mellin inversion formula ( 1 2 < σ 3), we derive

n |λ f (n)| 2 w ± (n) = D f X + O f,ε Y + X 1+κ Y -1 .
Taking κ = 1 2 + ε and Y = X 3/4 , and combining the obtained estimation with (8.4), we find that

X<n 2X |λ f (n)| 2 = D f X + O f,ε X 3/4+ε ,
which implies (8.1) after a dyadic summation. Now we are ready to prove Theorem 2 along the same line of argument in [START_REF] Lau | Sign of Fourier coefficients of modular forms of half integral weight[END_REF]. Take h = x η where η > 3 4 is specified later. Lemma 8.1 gives (i) Ch and subsequently improve the upper bound O(h 3/4 x ε ) in [10, (21)] to O(h 1/2 x ε ). Ultimately the lower bound in [10, (27)], which relies on [10, (21) and (26)], is sharpened to

x -1-̺ h 2 + O(h 1/2 x ε ). (8.6) The optimal choice of η for the positivity of (8.6) is 2 3 (1 + ̺) + ε. Together with the constraint η >

  for some scalar c. By (3.16) and U 4 = T(2 2 ), we get c(f|W 4 U 4 W 4 ) -cµ(f|W 4 ) -2µ 2 f = 0.

d|r 2 *L

 2 u(mod d) L f (s, u/d) f (s, u/d). Now each summand L f (s, u/d) extends to an entire function (explained below), so the task is to establish the (uniform) convergence of the series in r. Hence this leads to the estimation of L f (s, u/d) in terms of r. The method of Hulse et al. is to derive the functional equation of L f (s, u/d) and then apply the convexity principle to D r (s). They gave an estimate for D r (s) on the line σ = -ε by bounding L f (s, u/d) individually.

1 2 2 ,

 12 Let q = u/d where (u, d) = 1 and d 1. By [3, Lemma 4.3], Λ(f, u/d, s) satisfies a functional equation in connection with Λ(f, -u/d, 1 -s) and Λ(h, -4u/d, 1 -s) respectively according as 4 | d or 2 ∤ d, where xx ≡ 1 (mod d). For the case 2 d, we revise f to be g, which causes a minor change of Λ(f 1 -u/d, 1 -s) into Λ(g, -u/(4d), 1 -s). We would unite the three functional equations into one. Let us introduce q d = d or 2d according to 4 | d or not, (4.5) and the symbols λ(n; d) and ̟ d (n, v) defined as: (4.6)

(4. 8 ) 1 λ

 81 L f (s, v/d) := n (n; d)̟ d (n, v)n -s . Now we rephrase [3, Lemma 4.3] of Hulse et al. with the above modification for 2 d.

Lemma 4 . 1 .

 41 Let f ∈ S ℓ+1/2 where ℓ 1 be an integer, d ∈ N and (u, d) = 1. Then L f (s, u/d) extends analytically to an entire function and satisfies the functional equation:

Lemma 4 . 2 .

 42 For e ∈ {0, 1, 2}, b an odd squarefree integer and (a, 2b) = 1, we have

( 4 .

 4 12) * v(mod 2 e a 2 b)

  G e,b (m) ≪ √ b with an absolute ≪-constant, and ½ d (n) = 1 if d | n and 0 otherwise. (Recall that we are confined to m ≡ (-1) ℓ (mod 4) in the case of e = 1.) Lemma 4.2's proof is postponed to Section 6. Now we apply (4.11) to give a technically lightweight improvement on the result (4.4) of Hulse et al.

Lemma 5 . 1 . 2 σ 3 2

 5123 Let T 1 be any number and s = σ + iτ . Suppose 1 and |τ | T . We have

  )D 2r (s). Next we separate the sum over d according as 4 | d, 2 d or 2 ∤ d, and hence obtain a decomposition of L

1 2 and T - 1

 1 |τ | T . In view of the condition d | r 2 for squarefree r, we decompose into d = a 2 b and r = abc where a, b, c are pairwise coprime and squarefree. It is equivalent to consider the series a,b,c µ(2abc) (abc) 2 * u(mod 2 e a 2 b)

  .6), * v(mod 2 e a 2 b) ̟ 2 e a 2 b (m, v) = * v(mod 2 e a 2 b) ε -(2ℓ+1)v 2 e a 2 b v e mv 2 4-e a 2 b . (6.1)

(- 1 )

 1 (b-1)/2 b β = ψ 2,b (β)χ b ′ (β), (say), where b ′ = (-1) (b-1)/2 b (is a quadratic discriminant) and χ b ′ (•) is the primitive quadratic character of conductor b. (Note b is odd squarefree.) Thus, we express the right side of (6.1) asG ′ e,b (m) * α (mod a 2 ) e 2 e-1 mα a 2 = G ′ e,b (m)a 2 f |a 2 µ(f ) f ½ a 2 /f (m) (cf. [7, p.44 (3.2)] and recalling a is odd) where(6.2) G ′ e,b (m) = * β (mod 2 e b) ε -(2ℓ+1) β ψ 2,b (β)χ b ′ (β) e mβ24-e b . This gives (4.12) with G e,b (m) = G ′ e,b (m), and thus it remains to show G ′ e,b (m) ≪ √ b so as to finish the proof.

  4-e b + θ - * β (mod 2 e b) ψ ′ 2,b (β)χ b ′ (β) e mβ 2 4-e b where ψ ′ 2,b (β) = χ 4 ψ 2,b . Both ψ 2,b and ψ ′ 2,b are characters (not necessarily primitive) modulo 8. Repeating the argument of writing β = 8β 1 + bβ 2 , we infer that (6.3) G ′ e,b (m) ≪ * β (mod b) χ b ′ (β) e 2 e-1 mβ b ≪ b 1/2 by the primitivity of χ b ′ , see [7, p.47 (3.12) and p.48 (3.14)]. Next we come to the case e = 0. In this case, we set v = bα + a 2 β with α (mod a 2 ), (α, a) = 1 and β (mod b), (β, b) = 1, then

  f (n)| 2 = D f x + O f,ε x 3/4+ε ,where D f is a positive constant depending on f. Proof. We choose two smooth compactly supported functions w ± such that• w -(x) = 1 for x ∈ [X + Y, 2X -Y ], w -(x) = 0 for x 2X and x X; • w + (x) = 1 for x ∈ [X, 2X], w + (x) = 0 for x 2X + Y and x X -Y ; • w(j)± (x) ≪ j Y -j for all j 0; • the Mellin transform of w(x)

  w ± (s)x -s ds, we writen |λ f (n)| 2 w ± (n) = 1 2πi (2) w ± (s)D(f ⊗ f, s) ds.With the help of Cauchy's residue theorem, we obtain that (8.5)n λ f (n) 2 w ± (n) = D f w ± (1) + 1 2πi (κ) w ± (s)D(f ⊗ f, s) ds,where 1 2 < κ < 1 and D f := Res s=1 D(f ⊗ f, s). By (8.3), (8.2) with j = 2 and the convexity bound [10, Proposition 7] D(f ⊗ f, s) ≪ f,ε (1 + |τ |) 2 max(1-σ,0)+ε

√ x+h τ (m) 4 ♭x/m 2 t 4 ♭x/m 2 t

 4242 λ f (n) 2 and (ii) x/m 2 t (x+h)/m 2 λ f (n) 2 ≪ hm -3/2for any m √ x + h, where the positive constant C and the implied ≪-constant depend on f and η only. Combining (i) with the bound λ f (tm 2 ) ≪ λ f (t)τ (m) 2 (cf.[START_REF] Lau | Sign of Fourier coefficients of modular forms of half integral weight[END_REF] Lemma 6]) leads toCh x n x+h λ f (n) 2 C ′ m (x+h)/m 2 λ f (t) 2where C ′ > 0 is a constant depending at most on f. By (ii) and the factm A τ (m) 4 m -3/2 ≫ A -1/2+ε , we conclude that for a large enough constant A, (x+h)/m 2 λ f (t) 2 {C/C ′ + O(A -1/2+ε )}h ≫ hwhich is[10, (23)]. Thus, repeating the same argument (in [10, (24)-(26)]), we obtain[10, (26)] with a smaller admissible h = x η (here only η >3 4 is required instead of η > 3 4 + ̺, which is due to the improved O-term in Lemma 8.1). Next we apply the new bound (2.3) of Theorem 1 to M(f, s) (= L ♭ f (s) here, cf. [10, (12)]) in Equation (20) of [10, Section 4.2]. Then we get ♭ x t x+h λ f (t) min log x

3 4 (

 4 from the new smaller admissible range of h), we set η = max 2 3 (1 + ̺), 3 4 + ε. The proof is complete with the same argument in remaining part of [10, Section 4.2].

  S ′ , by Deligne's bound. Consequently we have the following estimate for f ∈ S ℓ+1/2 . Lemma 3.2. Let Q be a (not necessarily finite) set of primes with 2 ∈ Q. Suppose f is an Hecke eigenform of T(p 2 ) for all p / ∈ Q. Let m 1 be any integer decomposed into m = qr 2 such that p | r implies p / ∈ Q, and p 2 | q implies p ∈ Q. Then we have

  ). Then it follows easily that g|T(2 2 ) = 0, because by(3.14)

	and (3.2),
	(3.15)	(g|U 4 )(z) = 2 ℓ-3/2
		ν (mod 4)

§ This commutativity is pointed out in[START_REF] Ono | The web of modularity: arithmetic of the coefficients of modular forms and q-series[END_REF] Corollary 3.16] under the extra condition p ∤ 4tN , which is relaxed to all primes p in[START_REF] Purkait | On Shimura's decomposition[END_REF].

* * Throughout the proof, ε denotes an arbitrarily small positive number whose value may differ, up to our disposal, at each occurrence.

Acknowledgments. Jiang is supported by the China Postdoctoral Science Foundation (No. 2017M620285), the Natural Science Foundation of Shandong Province (No. ZR2018QA004) and NSFC (No. 11801318), and Lü is supported by NSFC (Nos. 11771252, 11531008), IRT16R43, and Taishan Scholars Project. Lü, Royer and Wu are partially supported by the program PRC 1457 -AuForDiP (CNRS-NSFC). Lau is supported by GRF (Project no. 17313616 and 17305617) of the Research Grants Council of Hong Kong. The preliminary form of this paper was finished during the visit of E. Royer and J. Wu at The University of Hong Kong in 2015. They would like to thank the department of mathematics for hospitality and excellent working conditions.


domain: phys chim chim.mate spi

text:
Eli Sutter
email: psutter@unl.edu

Jacob S French

Akshay Balgarkashi

Nicolas Tappy

Anna Fontcuberta

Morral

Juan Carlos Idrobo

Peter Sutter

Single-Crystalline γ-Ga 2 S 3 Nanotubes via Epitaxial Conversion of GaAs Nanowires

Keywords: Nanowires, Kirkendall effect, cathodoluminescence, electron energy loss spectroscopy, optoelectronics

The chemical transformation of nanowire templates into nanotubes is a promising avenue toward hollow one-dimensional (1D) nanostructures. To date, high-quality single crystalline tubes of non-layered inorganic crystals have been obtained by solid-state reactions in diffusion couples of nanowires with deposited thin film shells, but this approach presents issues in achieving singlephase tubes with a desired stoichiometry. Chemical transformations with reactants supplied from the gas-or vapor-phase can avoid these complications, allowing single-phase nanotubes to be obtained through self-termination of the reaction once the sacrificial template has been consumed. Here, we demonstrate the realization of this scenario with the transformation of zincblende GaAs nanowires into single-crystalline cubic γ-Ga 2 S 3 nanotubes by reaction with sulfur vapor. The conversion proceeds via the formation of epitaxial GaAs-Ga 2 S 3 core-shell structures, vacancy injection and aggregation into Kirkendall voids, elastic relaxation of the detached Ga 2 S 3 shell, and finally complete incorporation of Ga in a crystalline chalcogenide tube.

Absorption and luminescence spectroscopy on individual nanotubes show optoelectronic properties, notably a ~3.1 eV bandgap and intense band-edge and near band-edge emission consistent with high-quality single crystals, along with transitions between gap-states due to the inherent cation-vacancy defect structure of Ga 2 S 3 . Our work establishes the transformation of nanowires via vapor-phase reactions as a viable approach for forming single-crystalline hollow 1D nanostructures with promising properties.

High-quality nanostructures, such as nanocrystals, -wires, -tubes, -plates, etc., have been synthesized by a host of different approaches, including for instance colloidal synthesis (nanocrystals), [START_REF] Turkevich | A study of the nucleation and growth processes in the synthesis of colloidal gold[END_REF][START_REF] Murray | Synthesis and characterization of nearly monodisperse CdE (E = sulfur, selenium, tellurium) semiconductor nanocrystallites[END_REF] vapor-liquid-solid growth (semiconductor nanowires), [START_REF] Wagner | Vapor-Liquid-Solid Mechanism of Single Crystal Growth[END_REF][START_REF] Morales | A Laser Ablation Method for the Synthesis of Crystalline Semiconductor Nanowires[END_REF][START_REF] Güniat | Vapor Phase Growth of Semiconductor Nanowires: Key Developments and Open Questions[END_REF] arc evaporation (fullerenes, carbon nanotubes), [START_REF] Kroto | Buckminsterfullerene[END_REF] or chemical vapor deposition (inorganic fullerenes and nanotubes). [START_REF] Tenne | Polyhedral and cylindrical structures of tungsten disulphide[END_REF][START_REF] Feldman | High-Rate, Gas-Phase Growth of MoS 2 Nested Inorganic Fullerenes and Nanotubes[END_REF] An alternative to such direct synthesis approaches, for example for materials that pose challenges in controlled synthesis or to obtain specific morphologies, such as hollow nanoparticles or -tubes, [START_REF] Wang | Properties, and Applications of Hollow Micro-/Nanostructures[END_REF] is the use of a suitable nanostructured template as a substrate for epitaxial growth [START_REF] Goldberger | Single-crystal gallium nitride nanotubes[END_REF] or as a sacrificial scaffold that can be replaced or transformed via chemical reactions. [START_REF] Jin Fan | Monocrystalline spinel nanotube fabrication based on the Kirkendall effect[END_REF] For metal (or metal oxide) [START_REF] Oh | Galvanic Replacement Reactions in Metal Oxide Nanocrystals[END_REF] nanostructures, galvanic replacement [START_REF] Xia | 25th Anniversary Article: Galvanic Replacement: A Simple and Versatile Route to Hollow Nanostructures with Tunable and Well-Controlled Properties[END_REF] offers a way of obtaining hollow shells of metals, alloys, or metal oxides. Solid-state, gas-, or vapor-phase reactions have been used to convert nanoparticle or nanowire templates into core-shell or hollow structures, notably for oxides and chalcogenides. A classic example is the transformation of metal (e.g., Co, Cd) nanocrystals into hollow metal-oxide or -chalcogenide nanocrystals via the nanoscale Kirkendall effect, [START_REF] Smigelskas; Kirkendall | Zinc Diffusion in Alpha Brass[END_REF][START_REF] Yin | Formation of Hollow Nanocrystals Through the Nanoscale Kirkendall Effect[END_REF][START_REF] Cabot | Sulfidation of Cadmium at the Nanoscale[END_REF] which involves the fast outdiffusion of metal ions through a growing oxide or chalcogenide shell accompanied by the injection, accumulation and coalescence of vacancies in the metallic core, so that the core is completely consumed and a hollow shell results. A similar approach has been used to produce a wide variety of hollow nanostructures, [START_REF] El Mel | The Kirkendall effect and nanoscience: hollow nanospheres and nanotubes[END_REF][START_REF] Wang | Hollow Nanocrystals through the Nanoscale Kirkendall Effect[END_REF][START_REF] Yu | The Design and Synthesis of Hollow Micro-/Nanostructures: Present and Future Trends[END_REF] including spinel nanotubes formed by reaction of ZnO nanowires with Al 2 O 3 shells, [START_REF] Jin Fan | Monocrystalline spinel nanotube fabrication based on the Kirkendall effect[END_REF] cation exchange of ZnO nanowires to TiO 2 nanotubes, [START_REF] Yu | Evolution of Hollow TiO2 Nanostructures via the Kirkendall Effect Driven by Cation Exchange with Enhanced Photoelectrochemical Performance[END_REF] transformation of Cu nanowires into CuS 2 and CuS nanotubes, [START_REF] Lee | Selective transformation of Cu nanowires to Cu2S or CuS nanostructures and the roles of the Kirkendall effect and anion exchange reaction[END_REF] conversion of on-surface Cd nanowires into hemicylindrical CdS tubes, [START_REF] Li | Photoconductive Cadmium Sulfide Hemicylindrical Shell Nanowire Ensembles[END_REF] transformation of sacrificial hydroxide (Co(CO 3 ) 0.35 Cl 0.20 (OH) 1.10 ) nanorods into Co 9 S 8 nanotubes, reaction of β-FeOOH nanorods with ZrO 2 shells to Zr-Fe 2 O 3 nanotubes, [START_REF] Li | Surviving High-Temperature Calcination: ZrO2-Induced Hematite Nanotubes for Photoelectrochemical Water Oxidation[END_REF] and the formation of hollow one-dimensional metal oxide shells by oxidation of Ag [START_REF] Yu | Quantitatively in Situ Imaging Silver Nanowire Hollowing Kinetics[END_REF] or Cu [START_REF] Xu | Comparison Study on the Stability of Copper Nanowires and Their Oxidation Kinetics in Gas and Liquid[END_REF] nanowires.

With few exceptions, the hollow nanostructures resulting from chemical transformations are polycrystalline even when starting from single-crystalline templates. Single crystalline nanotubes obtained through chemical transformations were typically obtained by solid-state reactions in diffusion couples, e.g., spinel ZnAl 2 O 4 by reaction of ZnO nanowires with an atomic-layer deposition (ALD) Al 2 O 3 shell or cubic spinel LiMn 2 O 4 by annealing β-MnO 2 nanotubes coated with LiOH. [START_REF] Ding | Single-Crystalline LiMn2O4 Nanotubes Synthesized Via Template-Engaged Reaction as Cathodes for High-Power Lithium Ion Batteries[END_REF] While these approaches realized attractive properties, such as a well-defined hollow geometry and perfect monocrystallinity, obtaining a single-phase product is complicated by the fact that the solid reactants -typically a sacrificial core and a deposited shell -need to have a precise molar ratio to completely react to the desired product crystal. This can be especially challenging if the template, e.g., a semiconductor nanowire, is tapered with a varying diameter. [START_REF] Colombo | Ga-assisted catalyst-free growth mechanism of GaAs nanowires by molecular beam epitaxy[END_REF][START_REF] Küpers | Diameter evolution of selective area grown Ga-assisted GaAs nanowires[END_REF] Furthermore, in solid-state reactions it is often unclear which of the solid reactants acts as the template that determines the orientation and crystal phase of the product. [START_REF] Jin Fan | Monocrystalline spinel nanotube fabrication based on the Kirkendall effect[END_REF] Chemical transformations with reactants supplied from the gas-or vapor-phase can avoid these complications. Single-phase nanotubes can be obtained easily since the reaction self-terminates once the sacrificial template has been exhausted. And the crystal phase and lattice orientation of the single-crystalline hollow nanostructures are determined by strict rules of epitaxy over the template crystal at the initial stages of the transformation, i.e., before larger Kirkendall voids appear and the shrinking core detaches from the growing shell/tube.

Here, we demonstrate the realization of this scenario with the transformation of GaAs nanowires into hollow, single-crystalline nanotubes of gallium sulfide (Ga 2 S 3 ). Ga 2 S 3 can crystallize in three different phases, monoclinic α-Ga 2 S 3 , 29,30 hexagonal β-Ga 2 S 3 , [START_REF] Medvedeva | Chalcogenides of Elements of Subgroup IIIB of the Periodic System[END_REF][START_REF] Jones | Observation of a Phase Transformation of Ga2S3 in a Quartz Effusion Cell above 1230 K by Means of Neutron Scattering[END_REF] and cubic (zincblende) γ-Ga 2 S 3 . [START_REF] Barbouth | The First Steps of the Sulfurization of III-V Compounds[END_REF][START_REF] Pardo | Diagramme de phases gallium-soufre et études structurales des phases solides[END_REF] This polymorphism is caused by an inherent defect structure, in which one third of Ga sites are vacant and the stable crystal phases differ in the ordering of the cation sublattice. [START_REF] Paorici | An open-tube technique to grow Ga2S3 crystals[END_REF][START_REF] Kokh | Study of Ga2S3 crystals grown from melt and PbCl2 flux[END_REF] Wide bandgaps close to 3 eV make these materials interesting for optoelectronics, such as light emission and photodetection in the short-wavelength visible range, [START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF][START_REF] Ho | Synthesis of In2S3 and Ga2S3 crystals for oxygen sensing and UV photodetection[END_REF] as well as second-harmonic generation [START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF][START_REF] Zhang | Two phases of Ga2S3: promising infrared second-order nonlinear optical materials with very high laser induced damage thresholds[END_REF] benefitting from high laser-induced damage thresholds. [START_REF] Zhang | Two phases of Ga2S3: promising infrared second-order nonlinear optical materials with very high laser induced damage thresholds[END_REF] Light emission is often governed by defect levels, e.g., due to the abundant cation vacancies, and thus occurs across a wider range of the visible spectrum. [START_REF] Kokh | Study of Ga2S3 crystals grown from melt and PbCl2 flux[END_REF][START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF][START_REF] Ho | Synthesis of In2S3 and Ga2S3 crystals for oxygen sensing and UV photodetection[END_REF][START_REF] Ho | Optically decomposed near-band-edge structure and excitonic transitions in Ga2S3[END_REF][START_REF] Liu | Synthesis and Phase Evolutions in Layered Structure of Ga2S3 Semiconductor Thin Films on Epiready GaAs (111) Substrates[END_REF][START_REF] Liu | Epitaxial Synthesis, Band Offset, and Photoelectrochemical Properties of Cubic Ga2S3 Thin Films on GaAs (111) Substrates[END_REF] We show that zincblende GaAs nanowires exposed to sulfur vapor transform into GaAs-Ga 2 S 3 core-shell nanowires, followed by the retraction of the cores by injection and coalescence of vacancies into Kirkendall voids, and ultimately the complete reaction of Ga from the GaAs cores, congruent desorption of As, and formation of hollow Ga 2 S 3 nanotubes. The templating by high-quality GaAs nanowires enables the formation of single-crystalline zincblende γ-Ga 2 S 3 tubes, whose lattice structure -down to the arrangement of crystal twins -is entirely inherited from the template crystals. Optical absorption measurements on individual Ga 2 S 3 nanotubes by electron energy loss spectroscopy in monochromated scanning transmission electron microscopy (STEM-EELS) identify a bandgap of 3.1 eV. Nanoscale luminescence spectroscopy shows band-edge emission, intense near bandedge luminescence at ~460 nm (i.e., ~2.67 eV photon energy), and additional light emission between ~560 -750 nm from transitions involving defect states in the gap. These characteristics document the promising optoelectronic properties of single-crystalline Ga 2 S 3 nanotubes.

The approach for preparing high-quality zincblende γ-Ga 2 S 3 nanotubes is summarized in Figure 1. Single-crystalline GaAs nanowires grown by self-catalyzed molecular beam epitaxy using Ga drops as catalysts (see Methods for details) are used as templates for the epitaxial conversion to hollow nanotubes. Fig. 1 (a) shows a transmission electron microscopy (TEM) image of one of the GaAs nanowires. The striped contrast stems from a sequence of twins in the zincblende GaAs nanowires. [START_REF] Spirkoska | Structural and optical properties of high quality zinc-blende/wurtzite GaAs nanowire heterostructures[END_REF] The darker liquid Ga catalysts appear as homogeneous spherical caps without any internal contrast. [START_REF] Sutter | Supercooling of nanoscale Ga drops with controlled impurity levels[END_REF] Electron diffraction (Fig. 1 (a), inset) demonstrates that the nanowires are single crystalline with symmetry axis oriented along the [111] direction. While prior work showed alternative mechanisms for the hollowing-out of nanoscale structures, such as shell-induced Ostwald ripening, [START_REF] Yu | Shell-Induced Ostwald Ripening: Simultaneous Structure, Composition, and Morphology Transformations during the Creation of Hollow Iron Oxide Nanocapsules[END_REF] several observed characteristics support the assignment of the process to a nanoscale Kirkendall effect. Although we were not able to obtain a direct comparison of the same nanowire/tube before and after the sulfurization reaction, ensemble statistics show the increase in diameter from the initial GaAs nanowire template to the final Ga 2 S 3 nanotube expected due to void formation in the core and Ga mass conservation. For example, thin GaAs nanowires with diameters of 32-35 nm are converted to Ga 2 S 3 nanotubes with diameters of 38-40 nm. Wires with varying degree of conversion at different positions (quantified via the projected void fraction in TEM images) show a constant inner diameter of the Ga 2 S 3 tube, and an outward sidewall growth with a wall thickness correlated with the amount of reacted core material (Fig. S4), which are clear indicators for a preferential outdiffusion of Ga.

Finally, the observed protrusions attached to the outer tube wall are also consistent with a Kirkendall mechanism.

While it is plausible that the initially ultrathin shell is strained in the interface plane to the lattice parameter of the GaAs template (Fig. 1 (d), stage (ii)), we were unable to identify conditions under which this state is realized experimentally and the strained ultrathin Ga 2 S 3 shell would be detectable in electron diffraction (Fig. S8). The analysis of Fig. S4 (c)-(e) suggests the formation of a thin Ga 2 S 3 shell during the initial reaction with sulfur without substantial Ga mass transport, with vacancy injection and hollowing out of the core beginning later. Figure 2 summarizes the diffraction analysis of the strain relaxation of the Ga 2 S 3 shell after it detaches from the GaAs core via insertion of Kirkendall voids between core and shell (Fig. 1 the entire core-shell structure (Fig. S2). The only exception is the nanowire tip, where the Ga catalyst drop is typically converted into polycrystalline cubic Ga 2 S 3 .

Further reaction with sulfur vapor converts these core-shell nanowires into hollow, single-crystalline γ-Ga 2 S 3 nanotubes (Fig. 1 (d), stage (iv)). The final structure and morphology resulting from the sulfurization reaction are documented in Figure 3. TEM images of a representative Ga 2 S 3 nanotube (Figs. 3 (a),(b)) and HAADF-STEM (Fig. 3 (c)) clearly show the hollow core and uniform ~25 nm sidewall, the latter with darker and brighter contrast in TEM and HAADF-STEM, respectively, due to its larger overall projected thickness of solid Ga 2 S 3 .

The outer surface of the tube now appears more corrugated than that of the core-shell wire shown in Fig. 2, primarily due to local differences in the thickening of the Ga 2 S 3 shell producing characteristic arrays of fin-like protrusions. Nevertheless, the hollow tubes obtained by reaction with sulfur above 600°C are invariably single crystalline and aligned along the [111] direction, as dictated by the templating effect of the [111]-oriented GaAs nanowires. This is shown by selected-area electron diffraction (Fig. 3 (d)) and corroborated by series of nanobeam diffraction patterns obtained along the tubes (Fig. S3), which now contain only a single set of spots indexed to monocrystalline γ-Ga 2 S 3 . The diffraction spots from GaAs observed in the core-shell structures at earlier reaction stages (Fig. 2) are now absent, confirming the complete transformation of the GaAs core into a Ga 2 S 3 nanotube. No sulfur is detectable, as expected after cooling of the reacted nanotubes under vacuum, i.e., at negligible S vapor pressure. Highresolution (HR) TEM illustrates the high crystalline quality of the tube (Fig. 3 (e)) and shows a sequence of twins as the predominant defect in the tubes (Fig. S5). The similarity of the twinning patterns in the initial zincblende GaAs nanowires and the Ga 2 S 3 nanotubes suggests that the sulfurization reaction transcribes not only the lattice structure and orientation, but even the sequence of planar defects from the GaAs nanowire templates to the final Ga 2 S 3 nanotubes (Figs.

S5-S7).

Whereas ordered monocrystalline γ-Ga 2 S 3 nanotubes are obtained by sulfurization at temperatures between 600-620°C, lower reaction temperatures still produce hollow nanotubes, albeit consisting of polycrystalline γ-Ga 2 S 3 . Figures S9 andS10 illustrate the morphology of representative GaAs nanowires after reaction with sulfur vapor at 450°C. TEM imaging and electron diffraction (Fig. S9 (b)) demonstrate that the GaAs nanowires remain intact after the brief sulfur exposure at these conditions. The liquid Ga drop at the tip undergoes a drastic change, and after sulfurization is replaced by a thick shell surrounding a hollow core. Electron diffraction (Fig. S9 (b')) identifies its structure and that of the thin shell on the nanowires as cubic Ga 2 S 3 . Similar morphologies were observed after brief sulfurization at temperatures between 350°C and 550°C. Longer exposures allowed us to access the entire sequence of regimes shown in Fig. 1 (d), however without achieving a long-range ordered monocrystalline structure. Note that the exposure times are correlated to the GaAs nanowire diameter, i.e., thinner GaAs nanowires progress toward the final hollow tube morphology faster than thicker wires (Fig. S11). Reactions at higher temperatures (550°C) show etching of the GaAs core preferentially along [110] directions, and the formation of smooth, polycrystalline γ-Ga 2 S 3 shells and nanotubes (Figs. S11-S13).

Single-crystalline Ga 2 S 3 nanotubes formed by reaction with sulfur may be of interest for applications in optoelectronics (e.g., photodetection [START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF] and short wavelength light emission), second harmonic generation, [START_REF] Zhang | Two phases of Ga2S3: promising infrared second-order nonlinear optical materials with very high laser induced damage thresholds[END_REF] and sensing. [START_REF] Ho | Synthesis of In2S3 and Ga2S3 crystals for oxygen sensing and UV photodetection[END_REF] To explore this potential, we probed the optoelectronic properties of individual high-quality γ-Ga 2 S 3 tubes using nanometer-scale STEM-EELS combined with cathodoluminescence spectroscopy in STEM (STEM-CL) and scanning electron microscopy (SEM-CL). In contrast to other phases of Ga 2 S 3 , particularly monoclinic α-Ga 2 S 3 (Refs. 38,40,46), the optoelectronic properties of cubic γ-Ga 2 S 3 remain poorly understood.

Reported absorption measurements on single crystals showed a bandgap of ~2.74 eV, smaller than that reported for the other Ga 2 S 3 phases (3.0-3.4 eV). [START_REF] Kokh | Study of Ga2S3 crystals grown from melt and PbCl2 flux[END_REF][START_REF] Ho | Synthesis of In2S3 and Ga2S3 crystals for oxygen sensing and UV photodetection[END_REF][START_REF] Ho | Optically decomposed near-band-edge structure and excitonic transitions in Ga2S3[END_REF] Recent CL measurements on thin γ-Ga 2 S 3 flakes showed pronounced defect luminescence centered between 550-800 nm -far from the band-edge emission expected below ~450 nm -with a temperature-dependent intensity that was strongly attenuated at room temperature. [START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF] Figure 4 summarizes measurements of the optoelectronic properties on individual singlecrystalline γ-Ga 2 S 3 nanotubes formed by sulfurization of GaAs nanowire templates at 620°C (as shown in Fig. 3). Local absorption measurements by monochromated STEM-EELS (Fig. 4 (a)-(b)) show a narrow zero-loss peak (full-width at half maximum: FWHM ~ 68 meV) and a clear onset of increased losses due to interband transitions, allowing us to extract a bandgap E g = 3.1 eV (close to the reported value for monoclinic α-Ga 2 S 3 ). [START_REF] Ho | Synthesis of In2S3 and Ga2S3 crystals for oxygen sensing and UV photodetection[END_REF] energies below E g . These different features were extracted by least-squares fits to a minimal set of Gaussian components, which were used for the analysis of all luminescence spectra. Panchromatic luminescence maps in STEM-CL show intense emission from the sulfurized Ga drop at the nanotube tip as well as from the tube sidewalls, where the projected volume of Ga 2 S 3 excited by the electron beam is largest (Fig. 4 (c),(d)). Resolved features such as the fin-like protrusions along the tube surface attest to the high spatial resolution achieved in these measurements (Fig. 4 (d)). Besides the band-edge luminescence, STEM-CL spectra obtained both at the tip and sidewalls show a dominant peak due to near band-edge luminescence (centered at λ = 464 nm, 2.67 eV) and prominent green (558 nm) and red (670 nm) defect luminescence. The defect emissions are consistent with photoluminescence peaks of γ-Ga 2 S 3 prepared by sulfurizing bulk GaAs, [START_REF] Liu | Synthesis and Phase Evolutions in Layered Structure of Ga2S3 Semiconductor Thin Films on Epiready GaAs (111) Substrates[END_REF] and the red emission reported for cubic Ga 2 S 3 flakes grown on mica. [START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF] The green luminescence has been attributed to transitions from the conduction band minimum or shallow donor levels to an acceptor level due to excited V Ga +2 vacancies positioned ~0.7 eV above the valence band edge; [START_REF] Liu | Synthesis and Phase Evolutions in Layered Structure of Ga2S3 Semiconductor Thin Films on Epiready GaAs (111) Substrates[END_REF][START_REF] Aono | Green photoemission of α-Ga2S3 crystals[END_REF] the red emission was assigned to transitions from a deep donor level to the V Ga +2 acceptor. [START_REF] Liu | Synthesis and Phase Evolutions in Layered Structure of Ga2S3 Semiconductor Thin Films on Epiready GaAs (111) Substrates[END_REF] While showing similar defect emission components, our luminescence spectra contrast in two key aspects with previous reports: [START_REF] Zhou | Nonlayered Two-Dimensional Defective Semiconductor γ-Ga2S3 toward Broadband Photodetection[END_REF][START_REF] Liu | Synthesis and Phase Evolutions in Layered Structure of Ga2S3 Semiconductor Thin Films on Epiready GaAs (111) Substrates[END_REF] In conclusion, we have demonstrated a process -based on reactions with vapor-or gasphase species -for the controlled conversion of sacrificial nanowire templates into high-quality single-crystalline nanotubes, exemplified here by the transformation of GaAs nanowires into γ-Ga 2 S 3 nanotubes by reaction with chalcogen (sulfur) vapor. Similar to previous approaches that used diffusion couples of nanowires and deposited solid shells to form hollow tubes via solidstate reactions, the method reported here combines structural templating by nanowires with hollowing by the nanoscale Kirkendall effect to create single-crystalline nanotubes. But in contrast to solid diffusion couples, where a precisely tuned local molar ratio of the reactants is crucial and which therefore face limitations, e.g., in case of inhomogeneous diameters of the starting nanowires, our reactions with species supplied from the vapor-or gas-phase are selfteminating and robust in providing single-phase tubes. For the particular system considered here, the transformation of GaAs nanowires into Ga

Figure 1 .

 1 Figure 1. Single-crystalline Ga 2 S 3 nanotubes by solid-state transformation of a sacrificial GaAs nanowire template. (a) Zincblende GaAs nanowire grown by MBE using Ga catalyst drops. The nanowire axis is aligned along the GaAs [111] direction. Inset: Selected-area electron diffraction pattern. (b) GaAs-Ga 2 S 3 core-shell nanowire following sulfurization of a GaAs nanowire for 15 min. at 620°C (sulfur vapor pressure, p~10 -4 Torr). (c) Single-crystalline hollow γ-Ga 2 S 3 nanotube obtained by complete transformation of the sacrificial GaAs nanowire template after annealing in sulfur for 60 min. at 620°C. Inset: Selected-area electron diffraction pattern indexed to cubic (zincblende) γ-Ga 2 S 3 . The tube axis is aligned with the Ga 2 S 3 [111] direction. Note: Panels (a) -(d) do not show the same nanowire/tube. (d) Schematic reaction sequence: (i) Sacrificial GaAs nanowire template; (ii) Initial stage of solid-state transformation during sulfur exposure: Formation of a strained epitaxial Ga 2 S 3 shell on the GaAs nanowire core; (iii) Progressive outdiffusion and reaction of Ga from the GaAs core, causing the detachment and elastic relaxation of the thickening Ga 2 S 3 shell; (iv) Final stage: Single-crystalline (zincblende) γ-Ga 2 S 3 nanotube. GaAs nanowires were progressively transformed by reaction with sulfur vapor at different temperatures (see Methods), following the sequence of reaction steps illustrated schematically in Fig. 1 (d). The initial sulfur exposure converted the near-surface region of the nanowires into a thin epitaxial Ga 2 S 3 shell that is strained within the cylindrical interface to the GaAs lattice (Fig. 1 (d), (ii)). Further exposure to sulfur drives the diffusion of Ga ions and vacancies through this shell, causing a thickening of the Ga 2 S 3 and its detachment from the

  (d), stage (iii)). TEM (Fig. 2 (a)) and STEM images (Fig. 2 (b)) show a clear separation between the core and shell (see Fig. 2 (b), inset). Selected-area electron diffraction patterns of this monocrystalline core-shell nanowire along two different zone axes ( , Fig. 2 (c); , Fig. 2 (d)) consist of a superposition of two zincblende single-crystal patterns originating from GaAs (lattice constant 5.565 Å) and γ-Ga 2 S 3 (5.181 Å), respectively. Comparison with simulated diffraction patterns of Ga 2 S 3 and GaAs (Figs. 2 (c'), (d')) shows an excellent match when assuming unstrained but aligned GaAs and Ga 2 S 3 crystal lattices. The epitaxial alignment is evident from the well-defined offset directions of the two sets of diffraction spots, with larger reciprocal spacings corresponding to γ-Ga 2 S 3 (Fig. 2 (c, c'), Fig. 2 (d, d'); HRTEM: Fig. 2 (e)-(f)). Linear arrays of nanobeam electron diffraction patterns obtained along the nanowire show that these characteristics -relaxed, lattice-aligned monocrystalline core and shell -are obtained across

Figure 2 .

 2 Figure 2. Core-shell nanowires consisting of detached, unstrained GaAs templates and Ga 2 S 3 shells. (a) TEM and (b) HAADF-STEM of a partially reacted core-shell nanowire (stage (iii), Fig. 1 (d)) following exposure to sulfur (vapor pressure, p ~10 -4 Torr) at 620°C. The inset in (b) shows a magnified view of the dashed area. (c), (d) Selected-area electron diffraction along two different zone axes, (c) and (d), with superimposed patterns of GaAs and relaxed Ga 2 S 3 showing the epitaxial relationship between the GaAs core and the single crystalline Ga 2 S 3 shell. (c'), (d') Simulated diffraction patterns of epitaxially aligned GaAs and Ga 2 S 3 , both with their equilibrium (unstrained) lattice constants. (e), (f) HRTEM images of the GaAs-Ga 2 S 3 core-shell nanostructure.

Figure 3 .

 3 Figure 3. Single crystalline Ga 2 S 3 nanotubes by sulfurization of sacrificial GaAs nanowire templates. (a) TEM image of a characteristic Ga 2 S 3 nanotube formed during sulfurization at 620ºC for 1 hour (sulfur vapor pressure, p ~10 -4 Torr). (b) Higher magnification TEM and (c) HAADF-STEM images of a section of the nanotube in (a). The inset in (b) shows a magnified view of the marked rectangular area. (d) Electron diffraction pattern along the zone axis, showing single crystalline cubic γ-Ga 2 S 3 . (e) HRTEM image of the sidewall of the Ga 2 S 3 nanotube, marked in the inset of (b).

  Some of the EELS spectra show an additional absorption onset in the gap, consistent with sub-bandgap (defect) emissions in luminescence. Local luminescence was measured by STEM-CL between 300-1000 nm (Figs. 4 (c)-(f)), and by SEM-CL with extended ultraviolet range down to 200 nm (Figs. 4 (g)-(i)). The spectra generally show contributions from band-edge luminescence centered near 400 nm (3.1 eV), along with intense near band-edge emission and additional radiative transitions with

Figure 4 .

 4 Figure 4. Optoelectronics of individual single-crystalline Ga 2 S 3 nanotubes. (a)-(b) Single nanotube absorption measurements by monochromated STEM-EELS at 300 K. (a) HAADF-STEM image of a typical Ga 2 S 3 nanotube. (b) Monochromated STEM-EELS spectra obtained at two locations of the nanotube, marked in (a). FWHM of the zero-loss peak (ZLP): 68 meV. Measured bandgap: E g = 3.1 eV. Inset: Zoomed spectra in the bandgap region, showing additional absorption from gap states (red arrow). (c)-(f) STEM cathodoluminescence, measured at 300 K. (c) HAADF-STEM image of a typical Ga 2 S 3 nanotube. (d) Panchromatic STEM-CL map, showing predominant photon emission from the Ga drop at the tip (transformed by sulfurization to Ga 2 S 3 ) as well as the Ga 2 S 3 nanotube sidewalls. (e) STEM-CL spectrum obtained with the exciting electron beam positioned at the Ga 2 S 3 tip (blue circle in (c)). (f) STEM-CL spectrum from the Ga 2 S 3 nanotube sidewall (red diamond in (c)). Colored regions in (e) and (f) represent components of a Gaussian peak shape analysis of the CL spectra, with center wavelengths of the Gaussians as indicated (in nm). (g)-(i) SEM cathodoluminescence, measured at T = 10 K. (g) SEM image of a Ga 2 S 3 nanotube. (h) Panchromatic SEM CL map showing enhanced emission from the tip and wire end.(i) SEM-CL spectra obtained at positions along the Ga 2 S 3 nanotube marked in (g). Colored regions represent components of a Gaussian peak shape analysis of the CL spectra, with center wavelengths fixed at the same values used in the deconvolution of the STEM-CL spectra shown in (e) and (f), except for the highest energy (band-edge) peaks, which were shifted to lower wavelength to account for bandgap widening between 300 K and 10 K. (j) Comparison of STEM-CL and SEM CL intensity profiles, obtained along the dashed lines in the panchromatic maps shown in (d) and (h), respectively. Both intensity profiles show enhanced emission from the sidewall and a dip near the center of the hollow Ga 2 S 3 nanotubes.

  (i) they show bright band-edge and near band-edge luminescence; and (ii) STEM-CL measurements at variable temperature show the emission intensity remaining nearly unchanged across a wide temperature range between 110 K and room temperature.Panchromatic SEM-CL maps (Fig.4(g), (h)) are in excellent agreement with measurements using high-energy electron excitation in STEM and show the same intensity distribution with maxima at the tube walls and a dip over the hollow core (Fig.4 (j)). SEM-CL spectra obtained at T = 10 K at different positions along single-crystalline γ-Ga 2 S 3 nanotubes (Fig.4(i), Fig.S14) were fitted with the same overall line shape observed in STEM-CL, i.e., deconvoluted into the same Gaussian components assigned to band-edge and defect luminescence. To account for the bandgap widening at low temperature, the Gaussians representing the band-edge peaks were blue-shifted by ~80 meV relative to the same fit components in the room-temperature STEM-CL spectra. With its extended spectral range, SEM-CL clearly confirms the absence of short-wavelength emissions beyond those found in STEM-CL.

  2 S 3 nanotubes, our results establish several interesting structural features, notably the selection among several known Ga 2 S 3 phases of a specific (γ, zincblende) polymorph matching the structure of the zincblende GaAs template, and the transcription of other characteristics such as the arrangement of twins from the nanowire template. Nanoscale luminescence spectroscopy on individual γ-Ga 2 S 3 nanotubes shows, inThe morphology of the GaAs nanowires after sulfurization was investigated at UNL by TEM, HAADF-STEM, and nanobeam electron diffraction in an FEI Talos F200X field-emission microscope. For these investigations the nanowires/tubes were dispersed on amorphous C films supported by Cu grids. The optoelectronic properties of individual Ga 2 S 3 nanotubes were characterized by both local absorption and luminescence spectroscopy. Absorption was measured by mapping the onset of losses in the low-loss region by monochromated STEM-EELS in a Nion Hermes microscope (at Oak Ridge) at 60 keV electron energy and 0.12 nm probe size. Cathodoluminescence (CL) measurements were performed in STEM mode (STEM-CL at UNL) and SEM mode (SEM-CL at EPFL). STEM-CL was performed using a Gatan Vulcan CL holder at room temperature and 200 keV electron energy, at an incident beam current between 300-400 pA. Local STEM-CL spectra were acquired by positioning the focused electron beam excitation at chosen locations and integrating the spectrometer output detected by a Si CCD array detector for 10-20 seconds. SEM-CL measurements were carried out at T = 10 K in an Attolight system at 5 keV electron energy and beam currents of 5-20 nA.

ACKNOWLEDGEMENTS

The authors thank E. Uccelli for contributions to the synthesis of high-quality GaAs nanowire templates. This work was supported by the National Science Foundation, Division of Materials Research, Solid State and Materials Chemistry Program under Grant No. DMR-1607795. A.F. i M. acknowledges support from the Swiss National Science Foundation (Projects No. 40B2-0_176680 and 200020_137648) and from the European Union through Horizon-2020 (project LIMQUET).

addition to the ubiquitous transitions between gap states due to the inherent defect structure with abundant cation vacancies, strong band-edge and near band-edge emissions, providing further support for the high quality of these hollow nanostructures and paving the way for their application in areas such as optoelectronics or sensing.

MATERIALS AND METHODS

GaAs nanowires were synthesized at EPFL by molecular beam epitaxy as described in detail elsewhere. [START_REF] Dufouleur | P-Doping Mechanisms in Catalyst-Free Gallium Arsenide Nanowires[END_REF][START_REF] Spirkoska | The use of molecular beam epitaxy for the synthesis of high purity III-V nanowires[END_REF] Briefly, Ga-catalyzed growth was carried out in an ultrahigh vacuum reactor at 630°C on GaAs substrates covered with SiO 2 . As-grown nanowires on the original substrates were transferred through air to UNL, where they were exposed to sulfur vapor in a quartz-tube reactor with two temperature zones. For the sulfurization reactions, sulfur powder (99.9995%, Alfa Aesar) was loaded into a quartz boat and placed in the center of the first (upstream) zone and samples were positioned in the second zone. The reactor was evacuated by a mechanical pump and Ar:H 2 (ratio 98:2) carrier gas was introduced at 50 sccm flow rate and 20 mTorr pressure. GaAs nanowire sulfurization was carried out at different sample temperatures between 350°C and 650°C. Heating the sulfur reservoir to a temperature of 80°C established a vapor pressure of ~4×10 -4 Torr. [START_REF] Meyer | Elemental sulfur[END_REF] To compare the effects of sulfur vapor pressure, three different sulfur reservoir temperatures (80°C, 120°C, and 170°C) corresponding to sulfur vapor pressures between 4×10 -4 Torr and 0.72 Torr were used. Constant temperatures during the reaction were ensured by ramping the two furnace zones so as to reach the chosen settings at the same time.

Sulfur exposure time ranged between 15 minutes and 4 hours (as indicated in the paper). Upon completion the system naturally cooled to room temperature.