The Burgess bound via a trivial delta method

# The Burgess bound via a trivial delta method

Keshav Aggarwal, Roman Holowinsky, Yongxiao Lin, and Qingfeng Sun Department of Mathematics, The Ohio State University
231 W 18th Avenue
Columbus, Ohio 43210-1174
School of Mathematics and Statistics
Shandong University, Weihai
Weihai
Shandong 264209
China
###### Abstract.

Recently, Munshi established the following Burgess bounds and , for any given , where is a fixed Hecke cusp form for , and is a primitive Dirichlet character modulo a prime . The key to his proof was his novel delta method. In this paper, we give a new proof of these Burgess bounds by using a trivial delta method.

###### Key words and phrases:
subconvexity, Dirichlet characters, Hecke cusp forms, -functions
11F66

## 1. Introduction and statement of results

Let be a primitive Dirichlet character modulo . Let be a Hecke cusp form of fixed level. In this note, we prove the Burgess bound in the level aspect for the -functions

 L(s,χ)=∞∑n=1χ(n)nsandL(s,g⊗χ)=∞∑n=1λg(n)χ(n)ns

using a “trivial delta method”. The first bound in level aspect was established by Burgess [4] for Dirichlet -functions. For a primitive Dirichlet character modulo , Burgess proved

 (1) L(12,χ)≪εM1/4−1/16+ε.

A subconvex bound of such strength is called a Burgess bound. In the setting, the Burgess bound for an -function of a Hecke cusp form twisted by a Dirichlet character of conductor is

 (2) L(12,g⊗χ)≪g,εM1/2−1/8+ε.

This was first established by Bykovskiĭ[5] in the case of a holomorphic Hecke cusp form. While for more general this was established by Blomer, Harcos and Michel [2] under the Ramanujan conjecture, and subsequently by Blomer and Harcos [3] unconditionally. In a series of papers [15, 14, 13], Munshi introduced a novel Petersson delta method to prove level-aspect subconvex bounds for -functions. In a recent preprint [12], he demonstrated that the delta method can also be applied to the classical setting of Dirichlet -functions and obtained the bounds (1) and (2) simultaneously.

While carefully studying the works of Munshi, Holowinsky and Nelson [7] discovered the following key identity hidden within Munshi’s proof [14],

 χ(n)=MRg¯χ∞∑r=1χ(r)e(n¯rM)V(rR)−1g¯χ∑r≠0Sχ(r,n;M)\widecheckV(rM/R).

Here is the Guass sum, is the generalized Kloosterman sum, is the Fourier transform of the Schwartz function which is supported on and is normalized such that , and is a parameter. This allowed them to produce a method which removed the use of the delta symbol and establish a stronger subconvex boud. Subsequently, Lin [9] was able to generalize the identity in the application to the subconvexity problem in both the Dirichlet character twist and -aspect case via the identity,

Here , is any positive constant, is a smooth compactly supported function with bounded derivatives. With this approach, Lin [9] obtained the following bound

 L(12+it,π⊗χ)≪π,ε(M(|t|+1))3/4−1/36+ε,

for a fixed Hecke-Maass cusp form for , where the corresponding convexity bound for is .

In this paper, we demonstrate that one is again able to remove the delta method and replace it in our subconvexity problem by the following trivial key identity,

 δ(n=0)=1q∑c|q∑amodc(a,c)=1e(anc), when q>|n|,

where denotes the Kronecker delta symbol. We shall establish the following bounds.

###### Theorem 1.1.

Let be a fixed Hecke cusp form for and be a Dirichlet character modulo a prime . For any ,

 L(12,g⊗χ)≪g,εM1/2−1/8+ε.
###### Theorem 1.2.

Let be a Dirichlet character modulo a prime . For any ,

 L(12,χ)≪εM1/4−1/16+ε.

## 2. Some Notations and Lemmas

For a smooth function with bounded derivatives, we define,

 \widecheckV(x)=∫RV(u)e(−xu)du.

Repeated integration by parts shows . Next, we collect some lemmas that we will use for the proof.

###### Lemma 2.1 (Trivial delta method).

One has

 (3) δ(n≡mmodq)=1q∑c|q∑∑⋆amodce(a(n−m)c).

where the star over the inner sum denotes the sum is over .

###### Lemma 2.2 (Voronoi summation formula, [8, Theorem A.4]).

Let be a Hecke cusp form of level with Fourier coefficients . Let and be such that and let be a smooth compactly supported function. For ,

 ∞∑n=1λg(n)e(anc)W(nN)=I(g;W,c,N)+Nc∑±∞∑n=1λg(n)e(∓¯¯¯anc)ˆW±g(nNc2),

where

 I(g;W,c,N)={Nc∫∞0(logxN+2γ−2logc)W(x)dx if λg is the divisor % function τ,0 otherwise.

Here is the Euler’s constant. is an integral transform of given by the following.

1. If is holomorphic of weight , then

 ˆW+g(y)=∫∞0W(x)2πikJk−1(4π√yx)dx,

and .

2. If is a Maass form with and , and is an eigenvalue under the reflection operator,

 ˆW+g(y)=∫∞0−πW(x)sinπir(J2ir(4π√yx)−J−2ir(4π√yx))dx,

and

 ˆW−g(y)=∫∞04εgcosh(πr)W(x)K2ir(4π√yx)dx.

If ,

 ˆW+g(y)=∫∞0−2πW(x)Y0(4π√yx)dx and ˆW−g(y)=∫∞04εgW(x)K0(4π√yx)dx.
3. When is the divisor function,

 ˆW+g(y)=∫∞0−2πY0(4π√xy)dx and ˆW−g(y)=∫∞04K0(4π√yx)dx.

Moreover in each case, .

Very often we will use the following bound when is a Maass form with Fourier coefficients .

###### Lemma 2.3 (Ramanujan bound on average).

Let be a smooth compactly supported function. Then

 (4) ∞∑n=1∣∣λg(n)∣∣W(nX)≪g,εX1+ε.

This follows from Cauchy–Schwarz inequality and the Rankin–Selberg estimate (see [11]).

## 3. The set-up

For any , define the following sum

 (5) S(N)=∞∑n=1λg(n)χ(n)W(nN),

where is a smooth bump function supported on with . Application of Cauchy–Schwarz inequality to the -sum in (5) followed by the estimate (4) gives the bound . Using an approximate functional equation of , one can derive the following.

###### Lemma 3.1.

For any , we have

 L(12,g⊗χ)≪MεsupN|S(N)|√N+M1/2−δ/2+ε,

where the supremum is taken over in the range .

From the above lemma, it suffices to improve the bound in the range , where is a constant to be chosen later.

Let be the set of primes in the dyadic interval , where is a parameter to be determined later. Denote . Then , as the following argument shows. For ,

 ∑ℓ∈L|λg(ℓ)|2≍1logL∑ℓ∈L(logℓ)|λg(ℓ)|2=1logL2L∑n=LΛ(n)|λg(n)|2+O(L1−ε),

where is the Von Mangoldt function. By the prime number theorem for automorphic representations (see [10, Corollary 1.2]), we have . Thus

 L⋆=∑ℓ∈L|λg(ℓ)|2≍LlogL.

Similarly, we let be a parameter and be the set of primes in the dyadic interval . Denote . We will choose and so that .

Let , and . For and , the condition is equivalent to the congruence . Since , we assume that,

 (6) P≫L1+ε.

Therefore, under the assumption

 PM≫(NL)1+ε,

by using the detection (3) with , the main sum of interest defined in (5) can be expressed as

 (7)

Here is a smooth function supported on , constantly on and satisfies . The error term arises from the Hecke relation . Our strategy would be to apply dual summation formulas to the and -sums, followed by applications of Cauchy–Schwarz inequality and Poisson summation to the -sum. A careful analysis of the resulting congruence conditions yields the final bounds.

## 4. Application of dual summation formulas

We start with an application of the Voronoi summation formula (Lemma 2.2) to the -sum in equation (7). Then,

 (8) S(N)=Ig=τ+NL⋆P⋆∑±∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈P1pM∑c|pM1c∞∑n=1λg(n)ˆW±g(nc2/Nℓ)×∞∑r=1χ(r)V(rN)S(rℓ,±n;c)+O(N1+εL),

where vanishes if is a cusp form, otherwise it is given by

 Ig=τ=NL⋆P⋆∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈P1pM∑c|pM1c∑∑⋆αmodc∞∑r=1χ(r)e(−αrℓc)V(rN)×∫∞0(logx+2γ+logNℓ−2logc)W(x)dx.

Next, we apply Poisson summation to the -sums in (8). We introduce a few notations. For , we let to be the lcm of and , to be the gcd of and , and . We note that if is squarefree, then .

Writing , the -sums in (8) are given by

 ∑r≥1χ(r)e(−αrℓc)V(rN).

Breaking the sum modulo and applying the Poisson summation formula, the -sum becomes

 (9) N[c,M]∑r∈Z⎛⎝∑βmod[c,M]χ(β)e(−αβℓc)e(rβ[c,M])⎞⎠\widecheckV(rN[c,M]).

Using the relation and reciprocity, the -sum can be rewritten as

 ∑βmodMχ(β)e((r−αℓMc)¯¯¯¯¯¯¯cMβM)×∑βmodcMe((r−αℓMc)¯¯¯¯¯¯MβcM)=¯¯¯¯χ((r−αℓMc)¯¯¯¯¯¯¯cM)gχ×cMδ(r−αℓMc≡0modcM),

where is the Gauss sum. Therefore (9) becomes

 NgχM∑|r|≪[c,M]Nε/Nr−αℓMc≡0modcM¯¯¯¯χ((r−αℓMc)¯¯¯¯¯¯¯cM)\widecheckV(rN[c,M])+O(N−2018).

Using the bound , the above sum is bounded by . Substituting the above expression into (8), we arrive at

 (10) S(N)=Mg=τ+N2gχML⋆P⋆∑±∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈P1pM∑c|pM1c∑∑⋆αmodc∞∑n=1λg(n)e(∓¯¯¯¯αnc)ˆW±g(nc2/Nℓ)∑|r|≪[c,M]Nε/Nr−αℓMc≡0modcM¯¯¯¯χ((r−αℓMc)¯¯¯¯¯¯¯cM)\widecheckV(rN[c,M])+O(N1+εL).

The term vanishes if is a cusp form. Otherwise it is given by

 (11) Mg=τ=N2gχML⋆P⋆∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈P1pM∑c|pM1c∑∑⋆αmodc∑|r|≪[c,M]Nε/Nr−αℓMc≡0modcM¯¯¯¯χ((r−αℓMc)¯¯¯¯¯¯¯cM)\widecheckV(rN[c,M])×∫∞0(logx+2γ+logNℓ−2logc)W(x)dx≪N(PM)ε√MLP∑1≤ℓ≤2L|λg(ℓ)|ℓ≪(PML)εNLPM1/2.

The last inequality is deduced by an application of Cauchy–Schwarz inequality and lemma 2.3. We similarly bound the sums in (10) corresponding to . When , we get arbitrarily small contribution because of the weight functions . When , we again get arbitrarily small contribution (because of the weight functions ) since we will choose such that

 (12) P2

When , we have

 (13) N2gχML⋆P⋆∑±∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈P1pM2∞∑n=1λg(n)ˆW±g(nM2/Nℓ)×∑|r|≪MNε/N(∑∑⋆αmodM¯¯¯¯χ(r−αℓ)e(∓¯¯¯¯αnM))\widecheckV(rNM)≪(PML)ε(MP+NL).

This bound is obtained by making use of Lemma A.1 to get . Therefore from (10), (11) and (13), we obtain,

 (14) S(N)=S⋆(N)+O((PML)ε(NLPM1/2+MP+NL))

under the condition , where

 S⋆(N)=N2gχM3L⋆P⋆∑±∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈Pχ(p)p2∑|n|≪p2M2/NLλg(n)ˆW±g(np2M2/Nℓ)×∑|r|≪pMNε/Nr−αℓ≡0modp(∑∑⋆αmodpM¯¯¯¯χ(r−αℓ)e(∓¯¯¯¯αnpM))\widecheckV(rNpM)+O(N−2018).

Since , the sum factors as

 ∑∑⋆αmodM¯¯¯¯χ(r−αℓ)e(∓¯¯¯¯¯¯αpnM)×∑∑⋆αmodpe(∓¯¯¯¯¯¯¯¯¯αMnp)=e(∓¯¯¯¯¯¯¯¯rMnℓp)∑∑⋆αmodM¯¯¯¯χ(r−αℓ)e(∓¯¯¯¯¯¯αpnM),

where we have used the condition to determine . Since , we have . Moreover, it suffices to estimate the ‘minus’ term of since the estimates of the ‘plus’ terms will be similar. By abuse of notation, we write as . Then,

 S⋆(N)=N2gχL⋆P⋆M3∑n≪NεP2M2/NLλg(n)∑ℓ∈L¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ)ℓ∑p∈Pχ(p)p2∑(r,p)=1e(−¯rnℓ¯Mp)∑∑⋆α(M)¯χ(r+α)e(¯αnℓ¯pM)\widecheckV(rpM/N)ˆW(np2M2/Nℓ).

Munshi treated a sum similar to ours in [12, P.13]. For the sake of completeness, we will carry out the details, but our arguments closely follow that of Munshi [12, Section 7].

## 5. Cauchy–Schwarz and Poisson Summation

In the following, whenever we need to bound the Fourier coefficients for a Maass form, we simply apply the Rankin–Selberg estimate , as a substitute of the Ramanujan bound for individual coefficients. We break the -sum into dyadic segments of length . Up to an arbitrarily small error,

Applying Cauchy–Schwarz inequality to the -sum and using the Ramanujan bound on average,

where

 (15)

We first calculate the trivial bound on . Applying Cauchy–Schwarz inequality to the -sum,

 S⋆(N,N0)≪(∑ℓ∈L|λg(ℓ)|2)∑ℓ∈Lℓ2∑nU(nN0)(∑p∈P1p2∑r≪(pM)1+ε/N∣∣∣∑∑⋆αmodM¯χ(r+α)e(¯αnℓ¯pM)∣∣∣)2.

Estimating trivially with the help of the bound (see Lemma A.1),

 S⋆(N,N0)≪(PM)εN0M3L4/N2.

Then,

To bound the first term, we choose for some small . This is chosen in order to effectively bound the term appearing in (21). We note that . From now on we assume is such that

 (17) P2M1+γ/N≪N0≪P2M2/NL.

Opening the square in (15) and switching the order of summations, it suffices to bound the following

 (18) S⋆(N,N0)=∑ℓ1∈L¯¯¯¯¯¯¯¯¯¯¯¯¯¯λg(ℓ1)ℓ1∑ℓ2∈Lλg(ℓ2)ℓ2∑p1∈P∑p2∈Pχ(p1)¯¯¯¯χ(p2)(p1p2)2∑(r1,p1)=1∑(r2,p2)=1\widecheckV(r1p1M/N)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\widecheckV(r2p2M/N)∑∑⋆α1(M)¯¯¯¯χ(r1+α1)∑∑⋆α2(M)χ(r2+α2)×T,

with

 T=∞∑n=1e(−¯¯¯¯¯r1nℓ1¯Mp1)e(¯¯¯¯¯r2nℓ2¯Mp2)e(¯¯¯¯¯¯α1nℓ1¯¯¯¯¯p1−¯¯¯¯¯¯α2nℓ2¯¯¯¯¯p2M)U(nN0)ˆW(np21M2/Nℓ1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ˆW(np22M2/Nℓ2).

We note that one can truncate the , -sums in (18) at , at the cost of a negligible error. For smaller values of and , we will use the trivial bounds .

Breaking the above -sum modulo and applying Poisson summation to it,

 T=N0p1p2M∑n∑bmodp1p2Me(−¯¯¯¯¯r1bℓ1¯¯¯¯¯¯Mp1)e(¯¯¯¯¯r2bℓ2¯¯¯¯¯¯Mp2)e(¯¯¯¯¯¯α1bℓ1¯¯¯¯¯p1−¯¯¯¯¯¯α2bℓ2¯¯¯¯¯p2M)e(bnp1p2M)J(np1p2M/N0),

where

 J(np1p2M/N0):=∫RU(x)e(−nN0xp1p2M)ˆW(xN0p21M2/Nℓ1)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ˆW(xN0p22M2/Nℓ2)dx.

The integral gives arbitrarily power saving in if . Hence we can truncate the dual -sum at at the cost of a negligible error. For smaller values of , we use the trivial bound . Since , we apply reciprocity to write

 T=N0p1p2M∑n∑bmodp1p2e((−¯¯¯¯¯r1ℓ1p2+¯¯¯¯¯r2ℓ2p1+n)¯¯¯¯¯¯Mbp1p2