Subconvex bounds on GL(3) via degeneration to frequency zero
Roman Holowinsky Paul D. Nelson
math.NT 11F66, 11M41



  For a fixed cusp form π on _3(ℤ)
  and a varying Dirichlet character χ of prime conductor
  q,
  we prove that the subconvex bound
  
    L(π⊗χ, 12) ≪ q^3/4 - δ

  holds for any δ < 1/36.  This improves upon the earlier
  bounds δ < 1/1612 and δ < 1/308 obtained by
  Munshi using his _2 variant of the δ-method.  The
  method developed here is more direct.
  We first express χ
  as the degenerate zero-frequency contribution of a carefully
  chosen summation formula à la Poisson.  After an
  elementary “amplification” step exploiting the
  multiplicativity of χ, we then apply
  a sequence of standard  manipulations
  (reciprocity, Voronoi, Cauchy–Schwarz and the
  Weil bound) to bound the contributions of the nonzero
  frequencies and of the dual side of that formula.

MmWave Channel Estimation via Atomic Norm Minimization for Multi-User Hybrid Precoding
    Junquan Deng1, Olav Tirkkonen1 and Christoph Studer2
1Department of Communications and Networking, Aalto University, Finland 
2School of Electrical and Computer Engineering, Cornell University, NY, USA 

    December 30, 2023
===============================================================================================================================================================================================================









§ INTRODUCTION


We consider
the
problem of bounding
L(π⊗χ,12),
where


  * π is a fixed cusp form on _3(ℤ),
  not necessarily self-dual, and

  * χ
  traverses a sequence of Dirichlet characters χ of (say) prime
  conductor q tending off to ∞.

Munshi <cit.> recently 
established the first subconvex bound
in this setting
by showing
that if
π satisfies
the Ramanujan–Selberg
conjecture,
then
for any fixed δ < 1/1612,
the estimate

    |L(π⊗χ,12)|
      ≤ C q^3/4 - δ

holds
for some positive quantity C that may
depend upon δ and π, but
not upon χ.
In the preprint <cit.>,
he
improves the exponent range to δ < 1/308
and
removes the Ramanujan–Selberg assumption.

A striking feature
of his work is the introduction of a novel “_2
δ-symbol method,” whereby one detects an equality of
integers n_1 = n_2 by averaging
several instances of the Petersson trace formula.
We summarize this approach in Appendix <ref>,
referring to <cit.> and <cit.> for
details,
to <cit.> and <cit.> for other recent applications of the _2
δ-symbol method, and to <cit.> for general discussion of the spectral decomposition of the δ-symbol.


It is natural to ask about the true strength of the _2
δ-symbol method.
How does it compare to the classical δ-symbol
method of Duke–Friedlander–Iwaniec <cit.> and
Heath-Brown <cit.>?  For which problems does one fail
and the other succeed? For which problems are the two methods
“identical” or “equivalent”? Can the _2 δ-symbol
method be simplified or removed in certain applications?

In pondering such questions, we were able to better understand
the arithmetical structure and mechanisms underlying Munshi's argument and construct a more direct proof of the following quantitative strengthening of Munshi's bound.

  The subconvex bound (<ref>)
  holds for any δ < δ_0 := 1/36.


The proof is surprisingly short compared to earlier proofs of
related estimates.
Indeed, we regard the primary novelty of
this work as not in the numerical improvement of the exponent
δ but rather in the drastic simplification obtained for
the proof of any subconvex bound (<ref>).

Our point of departure is a formula (see
<ref>), derived via Poisson summation, that
expresses χ in terms of additive characters and twisted
Kloosterman sums.  We insert this into an approximate functional
equation for L(π⊗χ, 1/2).  After an elementary
“amplification” step exploiting the multiplicativity of
χ, we then conclude via standard manipulations.  We discuss
in Appendix <ref> how we arrived at this approach
through a careful study of Munshi's arguments.

We hope that the technique described here may be applied to many
other problems.  For instance, it seems natural to ask whether
it allows a simplification or generalization of the
arguments of <cit.>
for bounding symmetric square L-functions.

The works
<cit.>
bound twisted L-functions on _3 in other aspects.  In the
preprint <cit.>, Yongxiao Lin has
generalized our method to incorporate the t-aspect.  The
preprint <cit.> applies a simpler
technique to the corresponding problem for _2.




§ PRELIMINARIES




 §.§ Asymptotic notation


We work throughout this article
with a 
cusp form π  on _3(ℤ)
and a
sequence of primitive Dirichlet characters χ_𝔧
to prime moduli q_𝔧, indexed
by 𝔧∈ℤ_≥ 1, with q_𝔧→∞.
To simplify notation, we drop the subscripts and write simply
χ := χ_𝔧 and q := q_𝔧.
Our convention is that any
object (number, set, function, ...)  considered below
may depend implicitly upon 𝔧
unless we designate it as fixed;
it must then be independent of 𝔧.
Thus π is understood as fixed, while χ is not.
All assertions are to be understood
as holding
after possibly passing to
some subsequence q_𝐣_k
of the original sequence q_𝐣,
and in particular,
for 𝔧 sufficiently large.

We define standard asymptotic notation accordingly: A = Ø(B)
or A ≪ B or B ≫ A means that |A| ≤ c |B| for some
fixed c ≥ 0, while A = o(B) means |A| ≤ c |B| for
every fixed c > 0 (for 𝔧 large enough, by convention).
We write A ≍ B for A ≪ B ≪ A.
We write A = Ø(q^-∞)
to denote that A = Ø(q^-c) for each fixed c ≥ 0.
Less standardly,
we write A ≺ B or
B ≻ A as shorthand for A ≪ q^o(1) B,
or equivalently,
|A| ≤ q^o(1) |B|.
Our goal is then to show that

    L(π⊗χ, 12)
      ≺ q^3/4-δ_0.


We say that 
V ∈ C_c^∞(ℝ^×_+)
is inert
if it satisfies the support condition

    V(x) ≠ 0  x ≍ 1

and the value and derivative bounds

    (x ∂_x)^j V(x) ≺ 1  for each fixed  j ≥ 0.





 §.§ General notation


We write e(x) := e^2 π i x,
and denote by ∑_n a sum over integers n.
Let c ∈ℤ_≥ 1.
We write ∑_a(c) and ∑_a(c)^*
to denote sums over a ∈ℤ/c
and a ∈ (ℤ/c)^*, respectively.
We denote the inverse of  x ∈ (ℤ/c)^*
by x^-1 or 1/x.
We denote by e_c : ℤ/c →ℂ^×
the additive character
given by e_c(a) := e^2 π i a/c,
by
S(a,b;c) := ∑_x(c)^*
e_c(a x + b x^-1)
the Kloosterman sum,
by
K_c(a) := c^-1/2
S(a,1;c) the normalized Kloosterman sum,
by S_χ(a,b;q) := ∑_x(q)^*χ(x) e_q(a x + b x^-1)
the twisted Kloosterman sum,
and
by
(χ) :=
q^-1/2∑_a(q)^*χ(a) e_q(a)
the normalized Gauss sum
(of magnitude one).

We define the Fourier coefficients λ(m,n) of π
as in <cit.>,
so that
L(π⊗χ,s) = ∑_n ∈ℤ_≥ 1λ(1,n) χ(n) n^-s for complex numbers s with large
enough real part,
and λ(n,m) = λ(m,n).

For a condition C,
we define 1_C to be 1 if C holds
and 0 otherwise.
For instance, 1_a = b is 1 if a=b
and 0 if a ≠ b.

We denote by
V̂(ξ) := ∫_x ∈ℝ V(x) e(-ξ x)   d x
the Fourier transform
of a Schwartz function
V on ℝ.

For a pair of integers a,b,
we denote by (a,b) and [a,b] their the greatest common divisor
and least common multiple, respectively.



 §.§ Voronoi summation formula


By <cit.> (cf. <cit.>
for the formulation used here),
we have for V ∈ C_c^∞(ℝ^×_+),
m, c ∈ℤ_≥ 1, a ∈ (ℤ/c)^*, and X > 0
that

    ∑_n
      V(n/X)
      λ(m,n)
      e_c(a n)
      =
      c
      ∑ _±, n 
     d | c mℐ_±V ( n d^2/c^3 m/X)
      λ(n,d)/n d
      S (m/a,
      ± n;
      m c/d),

for integral transforms
V ↦ℐ_± V ∈ C^∞(ℝ^×_+)
of the shape

    ℐ_±V(x)
    = ∫_(s) = 1
    x^-s𝒢^±(s+1) (∫_y ∈ℝ_+^× V(y) y^-s d y/y )
     d s/2 π i ,

where 𝒢^± is meromorphic on ℂ
and 
holomorphic in the domain (s) > 5/14,
where it
satisfies
𝒢^±(s) ≪ (1 + |s|)^Ø(1)
for fixed (s).
(The indices n and d in
(<ref>) are implicitly
restricted to be positive integers.)
Set θ = 5/14 + for some sufficiently small
fixed
> 0.
By shifting the contour to (s) = θ-1
and to (s) = A,
we see that
if V is inert,
then

    (x ∂_x)^j ℐ_± V(x) ≪min(x^1-θ, x^-A)

for all fixed j,A ≥ 0.

In the special case m = 1,
we have
S (m/a, ± n; m c/d)
= (c/d)^1/2 K_c/d(± n/a),
and so

    ∑_n
    V(n/X)
    λ(1,n)
    e_c(a n)
    =
    c^3/2∑ _±, n 
     d | cℐ_±V ( n d^2/c^3/X)
    λ(n,d)/n d^3/2
    K_c/d(± n/a).




 §.§ Rankin–Selberg bounds

By <cit.>,
we have for each fixed > 0
and all X ≥ 1 that

    ∑_n ≤ X
      |λ(n,1)|^2
      =
      ∑_n ≤ X
      |λ(1,n)|^2
      ≤∑_m,n: m^2 n ≤ X
      |λ(m,n)|^2
      ≪
      X^1+.

Using the Hecke relations
as in the proof of <cit.>,
we deduce that
for all M, N ≥ 1,

    ∑_m ≤ M, n ≤ N
      |λ(m,n)|^2
      
      
      
      ≪
      (M N)^1+.

(Indeed, we may
reduce to considering
the dyadic sums
over M/2 < m ≤ M, N/2 < n ≤ N,
and then to establishing
that ∑_X/8 < m^2 n ≤ X m |λ(m,n)|^2 ≪ X^1+,
which is shown in loc. cit.)




§ DIVISION OF THE PROOF





 §.§ Approximate functional equation

Recall our main goal (<ref>).
By <cit.>,
we may write

    L(π⊗χ, 12) = ∑_n λ(1,n) χ(n)/√(n)
    V_1 (n/q^3/2) + η∑_n λ(1,n)
        χ(n)/√(n) V_2 (n/q^3/2),

for some η∈ℂ with |η| = 1 and some smooth
functions
V_1, V_2 : ℝ^×_+ →ℂ
satisfying (x ∂_x)^j V_i(x) ≪min(1, x^-A) for all
fixed j, A ∈ℤ_≥ 0.
By a smooth dyadic partition of unity
and the Rankin–Selberg estimate (<ref>),
it will suffice to show for each 0 < N ≺ q^3/2
and each inert V ∈ C_c^∞(ℝ^×_+)
that
the normalized sum

    Σ := ∑_nV (n/N)/Nλ(1,n) χ(n)

satisfies the estimate

    Σ≺ N^-1/2 q^3/4 - δ_0.

By further application of (<ref>),
we may and shall assume further that

    q^3/2 - 2 δ_0≤  N ≺ q^3/2.


The proof of (<ref>) will involve positive parameters
R,S,T
satisfying

    q^≪ R,S,T ≪ q^1- for some fixed  > 0.

Thus every integer in 
[R, 2 R] ∪ [S, 2 S] ∪ [T, 2 T] is coprime to q.




 §.§ A formula for χ

Fix a smooth function W on ℝ
supported in the interval [1,2] with ∫ W(x)   d x = 1.
Then Ŵ(0) = 1.
Observe that 1/r ∈ℤ/q is defined
for all integers r for which W(r/R) ≠ 0.
Set 
    H := q/R.

By Poisson summation,
we have

    √(q)/R∑_rW (r/R)
      χ(r)
      e_q (u/r)
      = 
      ∑_h
      Ŵ (h/H)
      1/√(q)∑_r(q)^*χ(r)
        e_q (u/r)
        e_q(h r)
      _
        = S_χ(h,u;q)
      .

For h ≡ 0 q,
we have
S_χ(h,u;q) = √(q)(χ) χ(u).
Setting
α_r :=    (χ)^-1 R^-1 W(r/R) χ(r),
we deduce by rearranging (<ref>)
that

    χ(u)
      =
      q^1/2∑_r
      α_r
      e_q (u/r)
      -
      (χ)^-1∑_h ≠ 0Ŵ (h/H)
      S_χ(h,u;q)/√(q).

The properties of the sequence
α to be used in what follows
are that it is supported on [R,2 R]
and satisfies
the estimates α_r ≺ R^-1 and ∑_r |α_r| ≍ 1.



 §.§ “Amplification”

We choose sequences of complex numbers β_s
and γ_t
supported on (say) primes in the intervals [S, 2 S] and [T, 2 T],
respectively,
so that

    β_s ≺ S^-1,   γ_t ≺ T^-1,
        ∑_s β_s χ(s) = ∑_t γ_t χ(t) =
      1.

Then

    Σ =
      ∑_n,s,tV (n/N)/Nλ(1,n)
      β_s γ_t χ (t n/s).

The properties of β_s and γ_t
just enunciated,
rather than an explicit choice,
are all that will be used;
one could take, for instance
β_s := 
χ(s)
|𝒫∩ [S, 2 S]|^-1 1_s ∈𝒫∩ [S,2 S],
where 𝒫 denotes the set of primes,
and similarly for γ_t.




 §.§ A formula for Σ

Substituting (<ref>)
with u = t n / s
into (<ref>)
gives
Σ = ℱ - (χ)^-1𝒪,
where

    ℱ =
    q^1/2∑ _r ,s,tα_r β_s γ_t
      ∑_n
      V (n/N)/Nλ(1,n)
      e_q (t n/r s),


    𝒪 = ∑_n
    V (n/N)/Nλ(1,n)
    ∑ _s,t β_s γ_t
    ∑ _h ≠ 0Ŵ (h/H)
    S_χ(h,t n/s;q)/√(q).




 §.§ Main estimates

We prove these in the next two sections.

  Assume that
  
    q R S ≻ T N.

  Then
  
    |ℱ|^2 ≺q/N( q R S/T N)^3
        + 
        q (R S)^3 /N^2( 1/S T
          + 1 + N/R^2 S/R^1/2 S).



  As explained in the remark of
  <ref>, the first term on the RHS
  of (<ref>) is unnecessary.
  Including it simplifies slightly our proofs without affecting our final estimates.


  Assume that
  
    S T ≤ q^- R

  for some fixed > 0.
  Then
  
    |𝒪|^2 ≺
        H^2 1/S T H.





 §.§ Optimization

Our goal reduces to establishing
that
ℱ, 𝒪≺ N^-1/2 q^3/4 - δ_0.
(By comparison, we note
the trivial bounds
ℱ≺ q^1/2 and
𝒪≺ H.)
We achieve this
by applying the above estimates with

    R :=
    T N /q S,
      
    S := q^2/18,
      
    T := q^5/18.

Then (<ref>)
is clear, while (<ref>)
follows from (<ref>).
The required bound for 𝒪
follows readily from (<ref>).
We now deduce the required bound for ℱ.
Note 
that the first term on the RHS
of (<ref>)
is acceptable thanks to our choice of R.
Note also
from (<ref>)
that q S T ≤ N ≺ q^3/2;
from our choice of R, it follows that
1/S T ≫ (1 + N /R^2 S)/ R^1/2 S.
The bound

for |ℱ|^2


then readily simplifies to
|ℱ|^2


≺ q^-2 δ_0≺ N^-1  q^3/2 - 2 δ_0.
(By solving a linear programming problem,
we see moreover that these choices give the optimal bound
for L(π⊗χ, 1/2)
derivable from the above propositions.)







§ ESTIMATES FOR ℱ



We now prove Proposition <ref>.



 §.§ Reciprocity


Our assumption (<ref>) implies that
for
all r,s,t with α_r β_s γ_t ≠ 0,
the function
V_r,s,t' (x)
:=
V(x)
e(t N x/q r s)
is inert.
By the Chinese remainder theorem,
we have
e_q (t n /r s)
=
e_q r s
(t n)
e_r s(- t n/q) for (r s, q) = 1.
We may thus rewrite

    ℱ = 
    ∑ _r ,s,tα_r β_s γ_t
    𝒮(r,s,t),

where

    𝒮(r,s,t)
    :=
    q^1/2∑_n
    V_r,s,t' (n/N)/Nλ(1,n)
    e_r s(- t n/q).



 §.§ Voronoi

We introduce the notation

    c := c(r s, t) := r s/(r s, t),
       
    a := a(r s, t) :=
    -t/(r s, t),

so that
e_r s (- t n/q)
=
e_c (a n/q) and (a,c) = 1.
Applying Voronoi summation
(<ref>),
we obtain

    𝒮(r,s,t)
    =
     q^1/2 c^3/2/N∑ _±, n 
     d | c
    V_±, r,s,t”( n d^2/c^3/N)
    λ(n,d)/n d^3/2
    K_c/d(± q n/a)

for some smooth functions
V_±, r,s,t” 
satisfying
(x ∂_x)^j V_±, r,s,t”(x)
≺min(x^1-θ, x^-A)
for fixed j, A ∈ℤ_≥ 0.



 §.§ Cleaning up

The Weil bound,
the Rankin–Selberg bound
(<ref>)
and the condition N ≺ q^3/2 give

    𝒮(r,s,t) ≺ q^1/2 c^3/2/N ≺ N^-1/2 q^1/2 (q c / N)^3/2.

If (r s, t) ≠ 1,
then (because t is prime)
c = r s / t,
hence by (<ref>),

    ∑_r,s,t: (r s, t) ≠ 1α_r β_s γ_t
    𝒮(r,s,t)
    ≺
    N^-1/2 q^1/2( q R S /T N)^3/2.

Since
the square of the latter is the first term
on the RHS of (<ref>),
the proof of Proposition <ref>
reduces to
that of an adequate bound for
the sum

    ℱ_1 := ∑_r,s,t : (r s, t) = 1α_r
    β_s γ_t 𝒮(r,s,t).

If (rs , t) = 1,
then c = r s and a = -t,
hence

    ℱ_1 = 
      ∑_±, n, r, d 
    α_r
      λ(n,d)/√(n d)∑ _
          s, t : 
    
          d | r s, (r s, t) = 1
        β_s γ_t
      Φ(n,d,r,s,t)

with

    Φ(n,d,r,s,t)
    :=
    q^1/2 (r s)^3/2/N √(n) d
    V_±, r,s,t”( n d^2/(r s)^3/N)
    K_ r s / d( ∓ q n / t).



  With slightly more case-by-case analysis in the arguments to follow,
  one can verify that the reduction performed here to the case
  (rs , t) = 1 is unnecessary,
  hence that the bound (<ref>)
  remains valid in the stated generality
  even after deleting the first term on its RHS.




 §.§ Cauchy–Schwarz

Let >0 be fixed and small.
The rapid decay of V_±,r,s,t”
implies that truncating (<ref>)
to n d^2 ≤ q^ (R S)^3/N
introduces the negligible error Ø(q^-∞).
By the Rankin–Selberg bound (<ref>),
we have

    ∑_±, n, r, d: 
    
        n d^2 ≤ q^ (R S)^3/N
      
    |α_r|
    |λ(n,d)|^2/n d≺ 1.

It follows by Cauchy–Schwarz
that

    |ℱ_1|^2
    ≺∑_±, n, r, d
      
    |α_r|
    |∑ _
          s, t : 
    
          d | r s, (r s, t) = 1
        β_s γ_t
      Φ(n,d,r,s,t)
    |^2 + Ø(q^-∞).




 §.§ Application of exponential sum bounds

Opening the square,
expanding the definition of Φ
and wastefully discarding some summation conditions,
we obtain

    |ℱ_1|^2
      ≺q (R S)^3/N^2∑_±,r,d,s_1,s_2,t_1,t_2: 
    
          d|(r s_1,r s_2)
        |α_r β_s_1β_s_2γ_t_1γ_t_2|/d^2
      |𝒞| + Ø(q^-∞),

where 𝒞 is defined
for (r,s_1,s_2,t_1,t_2) in the support of
α_r β_s_1β_s_2γ_t_1γ_t_2
by

    𝒞
    :=
    1/X∑_n
    U (n/X)
    K_r s_1/d
    (∓ q n/t_1)
    
      K_r s_2/d
      (∓ q n/t_2)

with

    X := (r s_1)^3/2 (r s_2)^3/2/d^2 N≍(R S)^3/d^2 N

and

    U(x) :=
    1/x
    V_±,r,s_1,t_1”( X x d^2/(r s_1)^3/N)
    
      V_±,r,s_2,t_2”( X x d^2/(r s_2)^3/N)
    .

We have
(x ∂_x)^j U(x)
≺min(x^1 - 2 θ, x^-A) for fixed j, A ∈ℤ_≥ 0.



By a smooth dyadic partition of unity,
we may write

    U(x)
      = ∑_Y ∈exp(ℤ)min(Y^1-2θ, Y^-10) U_Y(x/Y),

where each function U_Y is inert.
Substituting (<ref>)
into (<ref>)
and applying the incomplete exponential sum estimates
recorded in Appendix <ref>,
we obtain
with

    Δ :=
    q
    (r s_2/d)^2 t_2 - (r s_1/d)^2 t_1/(r s_1/d,r s_2/d)^2
    =
    q
    s_2^2 t_2 - s_1^2 t_1/(s_1,s_2)^2

that

    𝒞≺1/X∑_Y ∈exp(ℤ)min(Y^1-2 θ, Y^-10)
    (
      X Y
      (Δ, r s_1d, r s_2d)^1/2/[r s_1d,r s_2d]^1/2
      + [r s_1d,r s_2d]^1/2).

Since θ < 1/2, the above sum is dominated
by the contribution from Y = 1;
estimating that contribution a bit crudely with respect to d,
we obtain

    𝒞≺ d^1/2(Δ, r s_1, r s_2)^1/2/r^1/2 [s_1,s_2]^1/2
    +
    r^1/2 [s_1,s_2]^1/2/d^1/2 X.



 §.§ Diagonal and off-diagonal

To state the
estimates to be obtained
shortly,
we introduce
the notation

    𝔼_r,s_1,s_2,t_1,t_2 :=
    1/R S^2 T^2∑_
        r :
        
    
        R ≤ s ≤ 2 R
      ∑_
        s_1, s_2: 
    
        S ≤ s_1,s_2 ≤ 2 S
      ∑_
        t_1, t_2: 
    
        T ≤ t_1,t_2 ≤ 2 T
      .

We estimate separately the contribution of each term on the RHS
of (<ref>) to ℱ_1 via
(<ref>), splitting off the contribution to
the first from terms with Δ = 0.
We obtain in this way that

    |ℱ_1|^2 ≺q (R S)^3/N^2∑_i=0,1,2ℬ_i +
    Ø(q^-∞),

where

    ℬ_0    :=
                      𝔼_r,s_1,s_2,t_1,t_2
                      1_Δ = 0(s_1, s_2)^1/2/[s_1,s_2]^1/2,
      
    ℬ_1    :=
                      𝔼_r,s_1,s_2,t_1,t_2
                      1_Δ≠ 0(Δ , r s_1, r s_2)^1/2/r^1/2 [s_1,s_2]^1/2,
      
    ℬ_2    :=
                      N/(R S)^3𝔼_r,s_1,s_2,t_1,t_2
                      r^1/2 [s_1,s_2]^1/2.

(In deriving
the estimate involving ℬ_2,
we used the slightly wasteful
bound
1/d^21/d^1/2 X≪N/ (R S)^3.)
Noting that Δ = 0 iff s_2^2 t_2 = s_1^2 t_1,
we verify
using the divisor bound
that

    ℬ_0
         ≺1/ S T, 
    ℬ_1    ≺1/R^1/2 S, 
    ℬ_2
         ≺N/(R S)^3
        R^1/2 S.

These estimates combine to give an adequate estimate
for ℱ_1.




§ ESTIMATES FOR 𝒪



We now prove Proposition <ref>.


 §.§ Cauchy–Schwarz

Using again the Rankin–Selberg bound (<ref>),
we obtain

    |𝒪|^2
    ≺∑_n|V(n/N)|^2/N|∑_s, t, h : h ≠ 0β_s γ_t Ŵ (h/H)
      S_χ(h, t n/s; q)/√(q)|^2.



 §.§ Elementary exponential sum bounds

Let >0 be fixed but sufficiently small.
Since q is prime
and R satisfies the  lower bound in
(<ref>),
we know that the integers h and q
are coprime whenever
0 ≠ |h| ≤ q^ H.
By the rapid decay of Ŵ,
we may truncate the h-sum
to |h| ≤ q^ H with negligible
error Ø(q^-∞).
We then open the square and apply Cauchy–Schwarz,
leading us to  consider
for s_1,t_1,h_1,s_2,t_2,h_2
with

    S ≤ s_i ≤ 2 S,
         
      T ≤ t_i ≤ 2 T,
         
      0 ≠ |h_i| ≤ q^ H

the sums 

    Π :=
      ∑_n
      |V(n/N)|^2/NS_χ(h_1, t_1 n/s_1; q)/√(q)S_χ(h_2, t_2 n/s_2; q)/√(q).

We apply Poisson summation.
By the lower bound on N in (<ref>)
and the assumption δ_0  = 1/36 < 1/4,
we have N ≫ q^1+ for some fixed > 0.
Thus only the zero frequency ξ = 0
after Poisson
contributes non-negligibly,
and so
Π≺
q^-1Π_0 + Ø(q^-∞)
with

    Π_0
    :=
    ∑_n (q)S_χ(h_1, t_1 n/s_1; q)/√(q)S_χ(h_2, t_2 n/s_2; q)/√(q).

Opening the Kloosterman sums
and executing the n-sum
gives

    Π_0 = ∑_x,y(q)^*
    1 _t_1 / s_1 x = t_2/s_2 yχ(x/y)
    e_q(h_1 x - h_2 y).

Our assumptions imply that
the quantities s_i, t_i, h_i
are all coprime to q,
so after a change of variables
we arrive
at

    |Π_0|
    =
    |∑_x(q)^* e_q((s_1 t_2 h_1 - s_2 t_1 h_2) x)|
    ≤
    (t_1 s_2 h_2 - t_2 s_1 h_1, q).



 §.§ Diagonal vs. off-diagonal

We have shown thus far that

    |𝒪|^2 ≺
    H^2
    1/(S T H)^2∑_s_1,t_1,h_1,s_2,t_2,h_2
    q^-1
    (t_1 s_2 h_2 - t_2 s_1 h_1, q)
    +
    Ø(q^-∞),

where the sum is restricted
by the condition (<ref>).
By our assumption (<ref>),
the quantities t_1 s_2 h_2 and t_2 s_1 h_1
are congruent modulo q precisely when they are equal.
By the divisor bound, the number of tuples
for which t_1 s_2 h_2 = t_2 s_1 h_1
is
≺ q^2  S T H.
Since > 0 was arbitrary,
we obtain

    |𝒪|^2
      ≺
      H^2
      ( 1/S T H
        + 1/q).

By another application of our assumption
(<ref>),
the first term in the latter bound dominates,
giving the required bound for 𝒪.

The proof of our main result (Theorem <ref>)
is now complete.



§ CORRELATIONS OF KLOOSTERMAN SUMS


The estimates recorded here are unsurprising, but we were unable
to find references containing all cases that we require
(compare with e.g. <cit.>).

  Let s be a natural number.
  Let a,b,c,d ∈ℤ/s
  be congruence classes
  for which
  (d,s) = 1.
  For each prime p | s,
  let 𝒳_0(p) ⊆ℤ/p
  be a subset of cardinality p - Ø(1).
  Let 𝒳 denote the set of elements
  x ∈ℤ/s
  for which
  
  
  * the class of x modulo p belongs
    to 𝒳_0(p) for each p | s, and
  
  * (c x + d,s) = 1.
  
  
  
  
  
  
  Define
  ϕ : 𝒳→ℤ/s
  by
  
    ϕ(x) := x a x + b/c x + d.

  Then the exponential sum
  Σ := s^-1∑_x ∈𝒳 e_s(ϕ(x))
  
  
  
  
  
  
  
  
  
  
  
  
  
  satisfies
  
    |Σ|
        ≤
        2^Ø(ω(s))(a,b,s)/s^1/2 (a,s)^1/2,

    where ω(s) denotes the number of prime divisors
    of s, without multiplicity.


  We may assume that s = p^n for some prime p.  For n=0,
  there is nothing to show.  For n=1, we appeal either to the
  Weil bound, to bounds for Ramanujan sums, or to the trivial
  bound according as (a,p) = 1, or (a,p) = p and
  (a,b,p) = 1, or (a,b,p) = p.  We treat the remaining cases
  by induction on n ≥ 2.
  
  
  
  
  
  If (a,b,p) > 1, then
  the conclusion follows by our inductive hypothesis applied to
  s/p, a/p, b/p, c, d.  We may thus assume that (a,b,p) = 1.
  
  
  
  
  
  
  
  
  A short calculation gives the identities
  of rational functions
  
    ϕ '(x)
        = a c x^2 + 2 a d x + b d/(c x + d)^2,
          ϕ”(x)
        =  2 a + 2 c ϕ '(x)/c x + d.

  
  
  
  
  
  Write n = 2 α or 2 α + 1,
  and set
  ℛ := {x ∈𝒳/p^α : ϕ '(x)
  ≡ 0 p^α}.
  Then by p-adic stationary phase <cit.>,
  
    Σ≪
        s^-1/2∑_x ∈ℛ
        (ϕ”(x), p)^1/2.

  If (a,p) > 1, then
  (b,p) = 1
  and
  ϕ '(x) ≡ b d / (c x + d)^2 p,
  so (ϕ'(x), p) = 1.
  Thus
  ℛ = ∅ and Σ = 0.  
  Assume otherwise
  that (a,p) = 1.
  For x ∈ℛ,
  we have
  ϕ”(x) ≡ 2 a / (c x + d) p^α,
  so that
  
    x ∈ℛ
        (ϕ”(x), p) = (2 a, p^α) = (2,p^α) ≪ 1.

  Thus Σ≪ s^-1/2#ℛ
  and, by Hensel's lemma,
  #ℛ≪ 1.
  The proof of the required bound is then complete.




  Let s_1,s_2 be natural numbers.
  Let
  a_1,a_2,b_1,b_2 be integers with (b_1,s_1) = (b_2,s_2) =
  1.
  Set ℓ_i := a_i/b_i ∈ℤ/s_i.
  Set
  
    Δ :=
        s_2^2 b_2 a_1 - s_1^2 b_1 a_2/(s_1,s_2)^2.

  
  
  * 
    Let ξ be an integer.
    Set
    
    Σ  :=
          1/[s_1,s_2]∑ _x([s_1,s_2])
          K_s_1(ℓ_1 x)
          K_s_2(ℓ_2 x)
          e _[s_1,s_2](ξ x)

    Then
    
    |Σ|
          ≤ 2^Ø(ω([s_1,s_2]))(Δ,ξ,s_1,s_2)/ [s_1,s_2]^1/2 (ξ, s_1, s_2)^1/2.

    
    In particular,
    
    |Σ|
          ≤  2^Ø(ω([s_1,s_2]))(Δ,ξ,s_1,s_2)^1/2/ [s_1,s_2]^1/2.

  
  * 
    Let V : ℝ→ℂ be a smooth
    function
    satisfying
    x^m ∂_x^n V(x) ≺ 1
    for all fixed m,n ∈ℤ_≥ 0.
    Let X > 0.
    
    Assume that s_1, s_2 = Ø(q^Ø(1)).
    Then
    
    ∑_n
          V (n/X)
          K_s_1(ℓ_1 n)
          K_s_2(ℓ_2 n)≺
          X (Δ, s_1, s_2)^1/2/[s_1,s_2]^1/2
          +
          [s_1,s_2]^1/2.

  


  These estimates are not sharp if either (a_1,s_1) or
  (a_2,s_2) is large, but that case is unimportant for us.
  In fact, we have recorded (<ref>) only for
  completeness; the slightly weaker bound
  (<ref>) is the relevant one for our
  applications.
  We note finally that K_s is real-valued.



  We begin with (<ref>).
  Each side of (<ref>)
  factors naturally as a product over primes,
  so we may assume that s_i = p^n_i for some prime p.
  By the change of variables x ↦ b_1 b_2 x,
  we may reduce further to the case b_1 = b_2 = 1,
  so that ℓ_i = a_i.
  
  In the case that some ℓ_i is divisible by p, the
  quantity K_s_i(ℓ_i x) is independent of x, has
  magnitude at most s_i^-1/2, and vanishes if n_i > 1.
  The required estimate then follows in the stronger form
  Σ≪ (s_1 s_2)^-1/2 by opening the other
  Kloosterman sum and executing the sum over x.  We will thus
  assume henceforth that ℓ_1 and ℓ_2 are coprime to
  p.

  Write w_i := s_i / (s_1,s_2),
  so that w_1 s_2 = s_1 w_2 = [s_1,s_2]
  and
  Δ = w_2^2 b_2 a_1 - w_1^2 b_1 a_2.
  By opening the Kloosterman sums
  and summing over x,
  we obtain
  
    Σ = 
        1/√(s_1 s_2)∑ _x_1(s_1)^* ∑ _x_2(s_2)^* _w_1 ℓ_2 x_2^-1≡ w_2 ℓ_1 x_1^-1 + ξ 
          ([s_1,s_2])
        e_[s_1,s_2]
        (w_2 x_1 - w_1 x_2).

  Consider first the case s_1 = s_2 =: s,
  so that w_1 = w_2 = 1 and Δ = ℓ_1 - ℓ_2 and [s_1,s_2] = (s_1,s_2) = s.
  The subscripted identity in (<ref>) then shows
  that x_2 is determined uniquely by x_1 =: x
  and, after a short calculation, that
  
    Σ = 1/s∑_x(s)^*
        e_s(x ξ x + Δ x/ξ x + ℓ_1).

  By the previous lemma,
  it follows that
  
    Σ≪(Δ,ξ,s)/s^1/2 (ξ,s)^1/2,

  as required.

  Suppose now that s_1 ≠ s_2.
  Without loss of generality,
  s_1 < s_2.
  Then w_1 = 1 and w_2 = s_2/s_1;
  in particular, w_2 is divisible by p.
  The summation condition
  in (<ref>)
  shows
  
  
  that
  Σ = 0 unless
  (ξ,p) = 1, as we henceforth assume.
  Since (ℓ_1 ℓ_2, p) = 1,
  we have (Δ, p) = 1,
  so our goal is to show that
  Σ≪ s_2^-1/2.
  We introduce the variable
  
    y := ξ x_1 + w_2 ℓ_1.

  Then
  
    x_1 = y - w_2 ℓ_1/ξ,
           
        x_2 =
        w_1 ℓ_2 /yy - w_2 ℓ_1/ξ,

  and as y runs over (ℤ/s_1)^*,
  the pair (x_1, x_2)
  traverses
  the set indicated in (<ref>).
  A short calculation
  gives
  
    w_2 x_1 -
        w_1 x_2
        =
        - Δ/ξ
        +
        w_2/ξ
        ( y + ℓ_2 ℓ_1/y),

  hence
  
    Σ = 
        1/√(s_1 s_2)
        e_s_2 (  - Δ/ξ )
        ∑ _y(s_1)^* 
          e_s_1
          (  1/ξ ( y + ℓ_2 ℓ_1/y) )
        _√(s_1)
          K_s_1(ℓ_2 ℓ_1/ξ^2)
        .

  The required conclusion then follows
  from the Weil bound.

  To prove (<ref>),
  we first apply Poisson summation
  to write
  the LHS of (<ref>)
  as
  
    X
        ∑_ξV̂
        (ξ/[s_1,s_2]/X)
        1/[s_1,s_2]∑_x([s_1,s_2])
        K_s_1(ℓ_1 x) K_s_2(ℓ_2 x)
        e_[s_1,s_2](ξ x),

  where V̂ satisfies
  estimates analogous to those assumed for V.
  We then apply (<ref>).
  The ξ = 0
  term in (<ref>)
  then contributes the first
  term on the RHS of (<ref>),
  while an adequate
  estimate for the remaining terms
  follows from the consequence
  
    ∑_ξ≠ 0
        |V̂|
        (ξ/[s_1,s_2]/X)
        (Δ,ξ,s_1,s_2)^1/2≺ [s_1,s_2]/X

  of the divisor bound.





§ COMPARISON WITH MUNSHI'S APPROACH

We outline Munshi's approach <cit.> to the sums Σ
arising as in 
<ref>
after a standard application of the approximate functional
equation, and compare with our own treatment.
For simplicity we focus on the most difficult range N ≈  q^3/2.



 §.§ Averaged Petersson formula

Munshi employs the following decomposition of the diagonal symbol:

    δ(m,n)    =  1/B^⋆∑_b∈ℬ∑_ψ(b) (1-ψ(-1)) ∑_f ∈ S_k(b,ψ)w_f^-1λ_f(m)λ_f(n) 
         - 2 π i^-k1/B^⋆∑_b∈ℬ∑_ψ(b) (1-ψ(-1)) ∑_c
                    ≡ 0 (b)S_ψ(m,n,c)/c J_k-1 (4 π√(m n)/c).

Here ℬ is
a suitable set
of natural numbers, ψ runs over a suitable collection of odd Dirichlet
characters modulo b ∈ℬ,
and B^⋆ denotes the appropriate
normalizing factor.




 §.§ Munshi's initial transformations

Set A(n) := λ(1,n).
Munshi writes[
  For the sake of comparison,
  we note that Munshi used the notation
  R,L,P,M
  corresponding to our
  R,S,T,q.
]

    ∑_n ∼ N A(n) χ(n) ≈1/S∑_s ∼ S∑_n ∼ N A(n) ∑_r ∼ NSχ(r/s) δ(r,ns)

where s runs over primes of size S. Munshi applies (<ref>) to
δ(r,nℓ)
with ℬ = {t q : t ∼ T}, where t runs over
primes of size T,
and the characters ψ are taken to be trivial modulo q.
The use of (<ref>) produces two main
contributing terms,
ℱ^M
from the sum of Fourier coefficients
and 𝒪^M
from the sum of Kloosterman sums,
given roughly by

    ℱ^M ≈1/T^2S∑_s∑_t∑_ψ(t)∑_n ∼ N∑_r ∼ NS A(n) χ(r/s) ∑_f ∈ S_k(t q, ψ)ω_f^-1λ_f(r)λ_f(n s)

and 

    𝒪^M ≈1/T^2 S∑_s∑_t∑_ψ(t)∑_n ∼ N∑_r ∼ NS A(n) χ(r/s) ∑_c ≪√(q) S/T1/c t qS_ψ (r, n s;c t q)

which Munshi then works to balance with the appropriate choices
of S and T.
(The superscripted M
has been included to disambiguate from the closely related expressions
defined in <ref> of this paper.)
In (<ref>) we
sum over moduli c up to the transition range of
the resulting J-Bessel function, which we
do not display for notational simplicity.
(For the analogous problem in spectral or t-aspects,
the J-Bessel function
plays an important analytic role;
cf. forthcoming work of Yongxiao Lin.)




 §.§ Outline of Munshi's method

We now present a brief outline of Munshi's treatment of ℱ^M and 𝒪^M (see <cit.> for details).


  §.§.§ Treatment of  ℱ^M



  * Dualize the n-sum via the _3 ×_2 functional equation.

  * Dualize the r-sum via the _2 ×_1 functional equation.

  * Sum over f via the Petersson trace formula.
  The diagonal contribution is negligible.  The off-diagonal contribution is a c-sum over Kloosterman
  sums of the form S_ψ(t^2qn,rs;ctq) with c ≪√(q) T^2.

  * Factor the Kloosterman sums
  modulo t and modulo cq.
  This yields Gauss sums modulo t; evaluate them.
  Sum over ψ modulo t.
  Factor the remaining Kloosterman sum modulo c and modulo
  q.
  The mod q contribution gives a Ramanujan sum
  equal to -1.

  * The n-sum now oscillates only modulo c.  Apply _3
  Voronoi and reciprocity.

  * Dualize
  the c-sum modulo r via Poisson.
  Only the zero dual frequency contributes.
  It remains to estimate sums of the form
  
    √(q)/T^4∑_t∼ T∑_s ∼ S∑_r ∼√(q)T/S∑_n ∼ T^3 A(n) χ(r s/t)  S( - n q/t, 1; r s).


  * Pull the n,r sums outside and apply Cauchy-Schwarz.

  * Conclude via Poisson in n.
  
Such a treatment produces the following bound

    ℱ≪ N [T/q^1/4S^1/2+(T S/q^1/2)^1/4+noise_ℱ],

where noise_ℱ comes from all of the other technical aspects resulting from working outside of the transition ranges and appropriately setting up the remaining object for each step of the above proof.


  §.§.§ Treatment of 𝒪^M



  * Factor the Kloosterman sums
  modulo t and cq.
  Evaluate
  the sum over ψ; this simplifies the Kloosterman
  sums modulo t to additive characters.
  Apply reciprocity.
  One now has oscillations only modulo cq.

  * Apply Poisson to the r sum.
  Only the zero frequency contributes non-negligibly
  to the dual sum.
  One is now left with estimating sums of the form
  
    1/TS √(q)∑_t∼ T∑_s ∼ S∑_c ≪√(q) S/T∑_n∼ N A(n)  χ(t c/s) 𝒟(n s/ t c;q),

  where
  
    𝒟(u;q):=∑_b(q) 
     (b(b-1),q)=1χ(b-1) e_q((b^-1 - 1) u)


  * Apply Cauchy–Schwarz with the n-sum outside.

  * Conclude via Poisson in n.

Such a treatment produces the following bound

    𝒪≪ N [q^1/4/T+S/T+noise_𝒪]

where noise_𝒪 comes from all of the other technical aspects resulting from working outside of the transition ranges and appropriately setting up the remaining object for each step of the above proof.


  §.§.§ Optimization

Ignoring the contributions from noise_ℱ and  noise_𝒪 in (<ref>) and (<ref>), one first restricts S<q^1/4, sets 

    T/q^1/4S^1/2=(TS/q^1/2)^1/4

to get that S=Tq^-1/6, and then sets 

    T/q^1/4S^1/2=q^1/4/T

to get that T=q^5/18 and S=q^2/18 which would produce a combined bound of 

    ∑_n ∼ N A(n)χ(n) ≪ N [q^-1/36+noise_ℱ+𝒪].

Therefore, the best possible bound that one could hope to
achieve is a saving over the convexity bound of size
q^-1/36. However, due to all of the technical obstacles that
present themselves in the course of the proof, Munshi's
original approach <cit.>
produced a saving of q^-1/1612,
improved in the preprint <cit.>
to q^-1/308.



 §.§ Discovering the key identity (<ref>)


After a topics course taught by the first author in the Fall of 2016
and subsequent discussions with the second author in June 2017, the key identity in this paper was discovered hidden within Munshi's work.  Indeed, starting from (<ref>) in the treatment of ℱ^M, if one were to now apply Voronoi summation in the n sum followed by an application of reciprocity for the resulting additive characters, then one would need to instead analyze sums of the form

    1/T^2∑_t∼ T∑_s ∼ S∑_r ∼√(q)T/S∑_n ∼ q^3/2A(n) χ(r s/t)  e_q(-n t/r s).

Viewing -t/rs as the u in (<ref>), we see that an application of Poisson summation in r returns us to the dual of our original object of interest (from the h=0 frequency of the dual) plus a sum which is the “_3 dual” of 𝒪^M (from the dual non-zero h frequencies) as expressed in (<ref>)

    1/TS√(q)∑_t∼ T∑_s ∼ S∑_h ≪√(q)S/T∑_n ∼ q^3/2A(n) S_χ(h t/s,n,q).

By “_3 dual,” we mean that Voronoi summation in n
applied to (<ref>) returns one to objects of the form
(<ref>). This observation led to the simplification presented in this
paper whereby many of the initial steps of Munshi's argument, as outlined above, are eliminated.




 §.§ Acknowledgements

This work was initiated during a visit of PN to RH at The Ohio
State University in June 2017. RH thanks PN for taking the time
to schedule that visit to Columbus on his return to ETH from
MSRI.  RH also thanks the Department of Mathematics at The Ohio
State University for giving him the opportunity to teach a
topics course on R. Munshi's delta method during the Fall 2016
semester. PN thanks The Ohio State University, the STEAM
Factory, and the Erdős Institute for their hospitality.
We thank Ritabrata Munshi for encouragement
and Yongxiao Lin for helpful corrections and feedback on an earlier draft.
We thank the referee
for helpful feedback
which has led
to corrections and
improvements to the exposition.




plain

