Average Bounds for Kloosterman Sums Over Primes

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by  A. J. Irving

Average Bounds for Kloosterman Sums Over Primes
3
A.J. Irving
1
0
Mathematical Institute, Oxford
2

n
a
J
1 Introduction

9
2
In this paper we consider bounds for sums of the form

]
T
ap
N S (a;x) = e( )
q
q
. p∼x
h (pX,q)=1
t
a
m where p runs over primes and p x denotes the inequality x p < 2x. These sums may be
∼ ≤
[ bounded trivially by x. If (a,q) = 1 then we conjecture that for any ǫ > 0 a bound of

2 1
v Sq(a;x) ǫ x2+ǫqǫ
≪
2
7 should be true. This conjecture, however, seems to be far out of reach of current methods.
3
A bound for S (a;x) is given by Garaev, [5], in the case that q is prime. He shows that
6 q
. for x < q we have, for any ǫ > 0,
1
0 15 2 1
3 max Sq(a;x) ǫ (x16 +x3q4)qǫ.
1 (a,q)=1| | ≪
:
v 3
This gives us a nontrivial estimate for the sum provided that x q4+δ for some δ > 0.
i
X For q x q1167+δ Garaev uses this bound to prove an asymptotic≥for the number of prime
r ≥ ≥
a solutions, p1,p2,p3 with pi [0,x], to the congruence
∈
p (p +p ) λ (mod q).
1 2 3
≡
Fouvry andShparlinski, [4], generalise Garaev’sboundto arbitraryq andthelarger range
3 4
q4 x q3. They also show that if we average over q then a sharper bound is possible.
≤ ≤ 3
Specifically, their Theorem 5 states that if Q2 x 2 then for every ǫ > 0 we have
≥ ≥
13 3 13 5
max S (a;x) (Q10x5 +Q12x6)Qǫ. (1)
q ǫ
(a,q)=1| | ≪
q∼Q
X
3
This bound is nontrivial when x Q4+δ. Fouvry and Shparlinski use their estimates to
≥
study multiplicative properties of the set
p p +p p +p p : p x,p prime .
1 2 1 3 2 3 i i
{ ∼ }
1
They show, for example, that for x sufficiently large this set contains numbers with a prime
factor exceeding x1.10028. Baker, [1, Theorem 2], has recently improved the bound (1) in the
1
range Q2 x 2Q. His result is
≤ ≤
11 4 11
max S (a;x) (Q10x5 +Qx12)Qǫ. (2)
q ǫ
(a,q)=1| | ≪
q∼Q
X
This is nontrivial for Q x12+δ and sharper than (1) when x Q1−δ. Baker applies this
≥ ≤
bound to the same ternary form problem as Fouvry and Shparlinski; combining it with a
variant on the sieve argument he improves 1.10028 to 1.1673.
We are motivated by a new application of these sums to Diophantine approximation. For
this application we need only consider average bounds. We will show that by generalising
the arguments from [4] it is possible to obtain a sharper estimate than that given in (1).
Theorem 1.1. For any ǫ > 0 we have
5 5 9 7 13
max S (a;x) (Q4x8 +Qx10 +Q6x18)Qǫ
q ǫ
(a,q)=1| | ≪
q∼Q
X
3 2
for Q2 x Q3.
≥ ≥
2
This gives us a nontrivial result for x Q3+δ. The proof uses similar methods to those of
≥
Fouvry and Shparlinski. However we introduce higher moments into one of their estimates.
This results in a problem of counting solutions to a congruence with a larger number of
variables; one which we can solve with a sharp bound when we average over q.
Using this theorem we give a further improvement of the exponent in the ternary form
problem. Let P+(n) denote the largest prime factor of n.
Theorem 1.2. Let θ = 1.188... be the unique root of the equation
1
21θ 19
42θ 65+38log − = 0.
− 4
(cid:18) (cid:19)
Then, for any θ < θ ,
1
x3
# p ,p ,p : p x,p prime,P+(p p +p p +p p ) > xθ .
1 2 3 i i 1 2 1 3 2 3 θ
{ ∼ } ≫ (logx)3
In some applications of Theorem 1.1 the maximum over a is not necessary. We therefore
prove a bound when a is constant, which is stronger provided that a is not too large.
Theorem 1.3. For any integer a > 0 and any ǫ > 0 we have
a 1 1 11 7 2
S (a;x) (1+ )2(Q2x 8 +Q6x3)(aQ)ǫ
q ǫ
| | ≪ xQ
q∼Q
X
4 1
for Q3 x Q2.
≥ ≥
2
This is nontrivial for Q21+δ x Q34−δ. The proof exploits the fact that, since there is
≤ ≤
no maximum over a, we can reorder summations to give an inner sum over q Q. This is a
∼
longer range than those arising in the proof of Theorem 1.1. After inverting the Kloosterman
fractions in such sums we reach a situation in which the Weil estimate for short Kloosterman
sums can be used.
The sums in this last theorem are essentially bilinear forms with Kloosterman fractions,
which were studied for arbitrary coefficients by Duke, Friedlander and Iwaniec in [3]. In the
case that one of the coefficients is the indicator function of the primes then our theorem does
better than the general result of [3], provided that x and Q are sufficiently close in size.
We are most interested in the situation when x Q, that is Q x Q, as this is the
≍ ≪ ≪
case in our application to Diophantine approximation. For this reason we have given theo-
rems which, given our current ideas, are as sharp as possible in this case. For x sufficiently
different in size to Q it is possible to improve the above theorems. In order to compare the
various results we note that when x Q we have the following bounds, valid for any ǫ > 0.
≍
1. Using Fouvry and Shparlinski’s bound (1) or Baker’s (2), we get
23
max S (a;x) Q12+ǫ.
q ǫ
(a,q)=1| | ≪
q∼Q
X
2. Theorem 1.1 improves this to
19
max S (a;x) Q10+ǫ.
q ǫ
(a,q)=1| | ≪
q∼Q
X
3. If 0 < a Q2 then using Theorem 2 from Duke, Friedlander and Iwaniec, [3], we get
≪
a bound
95
S (a;x) Q48+ǫ.
q ǫ
| | ≪
q∼Q
X
4. If 0 < a Q2 then Theorem 1.3 gives a bound
≪
15
S (a;x) Q8 +ǫ.
q ǫ
| | ≪
q∼Q
X
These bounds should be compared with the trivial bound of Q2 and the conjectured best
3
bound of Q2+ǫ.
Acknowledgements
This work was completed as part of my DPhil, for which I was funded by EPSRC grant
EP/P505666/1. I am very grateful to the EPSRC for funding me and to my supervisor,
Roger Heath-Brown, for all his help and advice.
3
2 Lemmas
We require the following estimate for short Kloosterman sums coming from the Weil bound.
Lemma 2.1. For integers a and q with q > 1, and reals Y < Z we have, for any ǫ > 0, that
an Z Y 1
e( ) ((a,q)( − +1)+q2)qǫ.
ǫ
q ≪ q
Y<n≤Z
X
(n,q)=1
Proof. This is a slight weakening of Lemma 1 from Fouvry and Shparlinski, [4]. It follows
immediately on inserting the estimate
n1−ǫ φ(n) n
≪ ≪
as well as the standard bound for the divisor function, τ, into that lemma.
WealsorequirethefollowingestimateforthenumberofsolutionstoacertainDiophantine
equation.
Lemma 2.2. Let k N and ǫ > 0 be fixed. For any N 0 we have
∈ ≥
1 1 1 1
# (n ,...,n ) Z2k : 0 < n N, +...+ = +...+ Nk+ǫ.
1 2k i k,ǫ
{ ∈ ≤ n n n n } ≪
1 k k+1 2k
Proof. Suppose that
1 1 1 1
+...+ = +...+
n n n n
1 k k+1 2k
with 0 < n N for all i. Let P = n . We clearly have 0 < P < N2k.
i i
≤
Suppose that a prime p divides n for some i. When we multiply the above equation by
i
Q
P, to remove the denominators, all the terms except one will contain a factor n and so will
i
be divisible by p. In order to have equality the exceptional term must also be divisible by p.
Hence, since p is prime, there must be an index j = i with p n . It follows that if p P for a
j
6 | |
prime p then p2 P, whence P is square-full.
|
The number of square-full integers up to x is O(√x), see Golomb [6] for example. Thus
there are O(Nk) possible values for P. For each such value of P the number of solutions
to the equation is bounded by the divisor function τ (P). The result follows on using the
2k
standard estimate for these divisor functions.
(k)
Now let J (q) denote the number of solutions to the congruence
M
m +...+m m +...+m (mod q) (3)
1 k k+1 2k
≡
with 1 m M and (m ,q) = 1. The following generalises Fouvry and Shparlinski’s result,
i i
≤ ≤
[4, Lemma 3].
4
Lemma 2.3. Fix some k N. For any ǫ > 0 and any M 1 we have
∈ ≥
J(k)(q) (QMk +M2k)Mǫ.
M ≪k,ǫ
q∼Q
X
Proof. For each (m ,...,m ) with 1 m M we count the number of q Q with
1 2k i
≤ ≤ ∼
(q,m ) = 1 for which the congruence (3) holds. If
i
1 1 1 1
+...+ = +...+
m m m m
1 k k+1 2k
then the congruence is satisfied for every q Q for which q is coprime to m . Using
i
∼
Lemma 2.2 it follows that the contribution from such 2k-tuples (m ,...,m ) is
1 2k
Q
QMk+ǫ.
ǫ,k
≪
In the alternative case we define
2k k 2k
F = m ( m−1 m−1)
i i − i
i=1 i=1 i=k+1
Y X X
so that F is a non-zero integer with F M2k. Since q F there are O(Mǫ) possible values
| | ≪ |
for q. Thus the contribution from such 2k-tuples (m ,...,m ) is
1 2k
M2k+ǫ
ǫ,k
≪
so the result follows.
3 Estimates for Bilinear Sums
Throughout this section let α ,β be arbitrary complex numbers bounded by 1. We will
l m
prove estimates for sums
alm
W = W = α β e( ),
a,q l m
q
l∼L,m∼M
X
(ml,q)=1
either individually or on average over q Q. If β = 1 then we call W a Type I sum, if not
m
∼
then it is Type II.
Firstly we use Lemma 2.3 to estimate Type II sums on average. This is a generalisation
of a bound of Fouvry and Shparlinski, [4, Corollary 5].
Lemma 3.1. Suppose that 1 L,M Q. For any integer k 1 and any ǫ > 0 we have
≤ ≤ ≥
1 2k−1 1 2k−1
max W Q(Q2kL 2k M2 +L 2k M)Qǫ.
a,q ǫ,k
(a,q)=1| | ≪
q∼Q
X
5
Proof. By H¨older’s inequality we have
2k
alm
W2k L2k−1 β e( ) .
(cid:12) m (cid:12)
≤ q
(cid:12) (cid:12)
Xl∼L (cid:12) mX∼M (cid:12)
(l,q)=1(cid:12)(m,q)=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since L < Q we may bound this by extend(cid:12)ing the sum over(cid:12)l to a sum over all residues
modulo q:
2k
q
alm
W2k L2k−1 β e( ) .
(cid:12) m (cid:12)
≤ q
(cid:12) (cid:12)
Xl=1 (cid:12) mX∼M (cid:12)
(cid:12)(m,q)=1 (cid:12)
(cid:12) (cid:12)
Expanding, reordering the summation and(cid:12)using the orthog(cid:12)onality of additive characters
(cid:12) (cid:12)
then results in
W2k L2k−1QJ(k)(q).
≪ M
Using H¨older’s inequality and Lemma 2.3 we then get
2k−1 1 (k) 1
max W L 2k Q2k J (q)2k
(a,q)=1| | ≪ M
q∼Q q∼Q
X X
1
2k
2k−1 (k)
L 2k Q J (q)
≤ M
!
q∼Q
X
2k−1 1
L 2k Q(QMk +M2k)2kMǫ
ǫ,k
≪
1 2k−1 1 2k−1
Q(Q2kL 2k M2 +L 2k M)Qǫ.
≪
If we remove the maximum over a then we can obtain a sharper estimate by exploiting
the sum over q.
Lemma 3.2. For any integer a > 0, any L,M,Q 1, and any ǫ > 0, we have
≥
a 1 1 1 5 3
W (1+ )2(QLM2 +Q2L4M2)(aQ)ǫ.
a,q ǫ
| | ≪ LMQ
q∼Q
X
Proof. We first consider the case when α ,β are supported on integers coprime to a. We
l m
trivially have
W W (l)
a,q 1
| | ≤
q∼Q l∼L
X X
(l,a)=1
6
where
alm
W (l) = β e( ) .
1 (cid:12) m (cid:12)
q
(cid:12) (cid:12)
Xq∼Q (cid:12) mX∼M (cid:12)
(q,l)=1(cid:12)(m,aq)=1 (cid:12)
(cid:12) (cid:12)
By Cauchy’s inequality we get (cid:12) (cid:12)
(cid:12) (cid:12)
2
alm
W2 Q β e( ) .
1 ≤ (cid:12) m q (cid:12)
(cid:12) (cid:12)
Xq∼Q (cid:12) mX∼M (cid:12)
(q,l)=1(cid:12)(m,aq)=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Expanding and reordering the summation(cid:12)then gives us the b(cid:12)ound
a(m m )lm m
W2 Q (cid:12) e( 1 − 2 1 2)(cid:12).
1 ≤ (cid:12) q (cid:12)
m1,Xm2∼M (cid:12)(cid:12) Xq∼Q (cid:12)(cid:12)
(m1m2,a)=1(cid:12)(q,lm1m2)=1 (cid:12)
(cid:12) (cid:12)
We can write (cid:12) (cid:12)
(cid:12) (cid:12)
1 qq
lm m = −
1 2
lm m
1 2
where q is an inverse of q modulo lm m . Therefore
1 2
a(m m ) a(m m )q
W2 Q (cid:12) e 1 − 2 e 1 − 2 (cid:12).
1 ≤ (cid:12) qlm m − lm m (cid:12)
m1,Xm2∼M (cid:12)(cid:12) Xq∼Q (cid:18) 1 2 (cid:19) (cid:18) 1 2 (cid:19)(cid:12)(cid:12)
(m1m2,a)=1(cid:12)(q,lm1m2)=1 (cid:12)
(cid:12) (cid:12)
If we let (cid:12) (cid:12)
(cid:12) (cid:12)
a(m m )
1 2
f(t) = e( − )
lm m t
1 2
then the factor f(q) can be removed using summation by parts. For t Q we have
∼
a
f′(t)
≪ LMQ2
and thus
a a(m m )q
W2 Q(1+ ) max(cid:12) e( 1 − 2 )(cid:12).
1 ≪ LMQ m1,Xm2∼M Q′∼Q(cid:12)(cid:12)(cid:12) Q≤Xq<Q′ lm1m2 (cid:12)(cid:12)(cid:12)
(m1m2,a)=1 (cid:12)(q,lm1m2)=1 (cid:12)
(cid:12) (cid:12)
We get a contribution to this from pairs m = m(cid:12) which is bounded by (cid:12)
1 2
(cid:12) (cid:12)
a
Q(1+ )MQ.
LMQ
7
For the remaining terms let
b = a(m m )
1 2
−
and
c = lm m
1 2
so that the inner sum is
bq
e( ).
c
Q≤q<Q′
X
(q,c)=1
We may bound this using Lemma 2.1 by
Q 1
((b,c)( +1)+(LM2)2)(LM2)ǫ.
≪ǫ LM2
The result would be trivial if LM2 Q2. We thus assume that LM2 Q2, which allows us
≥ ≤
to replace (LM2)ǫ by Qǫ in our bound.
1
The contribution to our estimate for W2 from the term (LM2)2 is then
1
a 1
Q(1+ )L2M3Qǫ
ǫ
≪ LMQ
and that from the remaining terms is
a Q
Q(1+ )( +1)Qǫ (m m ,lm m ),
≪ǫ LMQ LM2 1 − 2 1 2
m1,Xm2∼M
m16=m2
where we have used that (m m l,a) = 1.
1 2
If we write h = m m = 0 then
1 2
− 6
(m m ,lm m ) (h,lm (m +h))
1 2 1 2 1 1
− ≪
m1,Xm2∼M mX1∼M0<Xh≪M
m16=m2
= (h,lm2)
1
mX1∼M0<Xh≪M
M2Qǫ,
ǫ
≪
since one has in general that
H
(h,n) Hτ(n) Hnǫ
ǫ
≪ ≪
h=1
X
for any n N.
∈
8
We conclude that
a 1 Q
W2 Q(1+ )(QM +L2M3 + +M2)Qǫ.
1 ≪ǫ LMQ L
Since L,M 1 this simplifies to
≥
a 1
W2 Q(1+ )(QM +L2M3)Qǫ
1 ≪ǫ LMQ
and therefore
a 1 1 1 5 3
W (1+ )2(QLM2 +Q2L4M2)Qǫ.
ǫ
≪ LMQ
Thiscompletes theproofundertheassumptionthatthecoefficients aresupportedonintegers
coprime to a. To remove this assumption we begin by writing (l,a) = u, a = bu and l = ku
to get
alm
W = α β e( )
a,q l m
q
l∼L,m∼M
X
(ml,q)=1
bkm
= α β e( ).
uk m
q
a=ub k∼L/u,m∼M
X X
(u,q)=1(mk,q)=1,(k,b)=1
Next we set (m,b) = v, m = vj and b = cv to rewrite this as
ckj
α β e( ).
uk vj
q
a=uvc k∼L/u,j∼M/v
(uXv,q)=1 X
(jk,q)=1,(k,vc)=1,(j,c)=1
It follows that
ckj
W (cid:12) α β e( )(cid:12)
a,q uk vj
| | ≤ (cid:12) q (cid:12)
q∼Q a=uvc q∼Q (cid:12) k∼L/u,j∼M/v (cid:12)
X X X (cid:12) X (cid:12)
(uv,q)=1(cid:12)(jk,q)=1,(k,vc)=1,(j,c)=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
ckj
(cid:12) α β e( )(cid:12).
≤ (cid:12) uk vj q (cid:12)
a=uvcq∼Q(cid:12) k∼L/u,j∼M/v (cid:12)
X X(cid:12) X (cid:12)
(cid:12)(jk,q)=1,(k,vc)=1,(j,c)=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
For each factorisation a = uvc the inner sum now has coefficients supported on integers
coprime to c so the above bound applies. The number of factorisations is O(aǫ) so the bound
for the general sum is the same as that for the sum with coprimality conditions except for
an additional factor aǫ.
9
Finally we use Lemma 2.1 directly, to estimate Type I sums when L is small.
Lemma 3.3. Suppose that β = 1. Then, for any L,M 1 and any ǫ > 0 we have
m
≥
LM 1
W ((a,q)( +L)+Q2L)Qǫ.
ǫ
≪ Q
Proof. We have
alm
W e( ) .
(cid:12) (cid:12)
≤ q
(cid:12) (cid:12)
Xl∼L (cid:12) mX∼M (cid:12)
(l,q)=1(cid:12)(m,q)=1 (cid:12)
(cid:12) (cid:12)
The result follows on applying Lemma 2.1 t(cid:12)o the inner su(cid:12)m.
(cid:12) (cid:12)
Summing this result over q Q we immediately get the following.
∼
Lemma 3.4. Suppose that L,M,Q 1 and that β = 1. For any ǫ > 0 we have
m
≥
3
max W (LM +Q2L)Qǫ.
a,q ǫ
(a,q)=1| | ≪
q∼Q
X
In addition if a > 0 then we have
3
W (LM +Q2L)(aQ)ǫ.
a,q ǫ
| | ≪
q∼Q
X
4 Proof of the Theorems
In the sums S (a;x) we replace the indicator function of the primes with the von Mangoldt
q
1
function Λ(n). The contribution of prime powers pα with α > 1 is ǫ x2+ǫ. This is smaller
≪
than any of the bounds we will establish so it may be ignored. In addition the factor logp
may be removed using partial summation with the cost of a factor xǫ Qǫ. It is thus
≪
sufficient to establish the theorems for the sums containing Λ.
We decompose Λ(n) using Vaughan’s Identity, as described by Davenport in [2, Chapter
1
24]. We will use U = V x3; the precise choice of U for each theorem will be given later.
≤
The sum S (a;x) is decomposed into Type I and II sums with LM x. The precise forms
q
≍
of the sums are given by Fouvry and Shparlinski in [4]. The coefficients are not all bounded
by 1 but they are bounded by a divisor function. This divisor function may be absorbed into
the Qǫ term. We must estimate Type I sums for L U2 and Type II sums for U L x/U.
≤ ≤ ≤
Since U2 x/U any Type I sum with U L L2 may be regarded as a Type II sum. Hence
≤ ≤ ≤
it will be enough to consider Type I sums with L U and Type II sums with U L x/U.
≤ ≤ ≤
The variables of summation are restricted by the condition lm x. In the Type II sums
∼
this may be removed by Fourier analysis, see for example the start of Garaev’s proof, [5,
Lemma 2.4]. For the Type I sums, if we are simply treating them as Type II sums then the
same argument applies, whereas if we are using Lemma 3.4 then it is clear that a condition
lm x can be introduced by modifying the proof.
∼
10