The Burgess bound via a trivial delta method
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, functions2010 Mathematics Subject Classification:
11F661. 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
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) 
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) 
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 levelaspect 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],
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
for a fixed HeckeMaass 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,
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 ,
Theorem 1.2.
Let be a Dirichlet character modulo a prime . For any ,
2. Some Notations and Lemmas
For a smooth function with bounded derivatives, we define,
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) 
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 ,
where
Here is the Euler’s constant. is an integral transform of given by the following.

If is holomorphic of weight , then
and .

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

When is the divisor function,
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) 
This follows from Cauchy–Schwarz inequality and the Rankin–Selberg estimate (see [11]).
3. The setup
For any , define the following sum
(5) 
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
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 ,
where is the Von Mangoldt function. By the prime number theorem for automorphic representations (see [10, Corollary 1.2]), we have . Thus
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) 
Therefore, under the assumption
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) 
where vanishes if is a cusp form, otherwise it is given by
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
Breaking the sum modulo and applying the Poisson summation formula, the sum becomes
(9) 
Using the relation and reciprocity, the sum can be rewritten as
where is the Gauss sum. Therefore (9) becomes
Using the bound , the above sum is bounded by . Substituting the above expression into (8), we arrive at
(10) 
The term vanishes if is a cusp form. Otherwise it is given by
(11) 
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) 
When , we have
(13) 
This bound is obtained by making use of Lemma A.1 to get . Therefore from (10), (11) and (13), we obtain,
(14) 
under the condition , where
Since , the sum factors as
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,
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,
Estimating trivially with the help of the bound (see Lemma A.1),
Then,
(16) 
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) 
Opening the square in (15) and switching the order of summations, it suffices to bound the following
(18) 
with
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,
where
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