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The congruence x1x2≡x3x4(modm) and mean values of character sums
Journal of Number Theory 130 (2010) 767–785
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Journal of Number Theory www.elsevier.com/locate/jnt
The congruence x1 x2 ≡ x3 x4 (mod m) and mean values of character sums Todd Cochrane a,∗ , Sanying Shi b a b
Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States Department of Mathematics, Tongji University, Shanghai 200092, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 May 2009 Communicated by K. Soundararajan MSC: 11A07 11D79 11L26 11L40 Keywords: Congruences Character sums Moments
For any positive integer m we obtain the asymptotic formula, B ∩ V = |B | + O 8ν (m) τ (m)(log m)3 (log log m)7 |B| , φ(m)
for the number of solutions of the congruence x1 x2 ≡ x3 x4 (mod m) with coordinates relatively prime to m in a box B of arbitrary size and position. We also obtain an upper bound for a fourth-order character sum moment,
+B 4 a χ ( x ) 8ν (m) τ (m)(log m)3 (log log m)7 B 2 . φ(m) 1
χ =χ0 x=a+1
© 2009 Elsevier Inc. All rights reserved.
1. Introduction The distribution of solutions of the congruence
x1 x2 ≡ x3 x4
(mod m),
(1)
where m is any natural number, arises naturally in the study of certain character sums. For any integers ai , B i , with 1 B i < m, 1 i 4 let B be the box of points
*
Corresponding author. E-mail addresses: [email protected] (T. Cochrane), [email protected] (S. Shi).
0022-314X/$ – see front matter doi:10.1016/j.jnt.2009.08.006
© 2009 Elsevier Inc.
All rights reserved.
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
B = x ∈ Z4 : a i x i < a i + B i , 1 i 4 ,
(2)
of cardinality |B | = B 1 B 2 B 3 B 4 , and B the subset of B of points with coordinates relatively prime to m,
B = x ∈ Z4 : a i x i < a i + B i , ( x i , m ) = 1 , 1 i 4 .
(3)
Let V ⊂ Z4 denote the set of all integer solutions of (1) and V ⊂ Z4 the solutions of (1) with (xi , m) = 1, 1 i 4. Thus |B ∩ V | denotes the number of solutions of (1) in B with coordinates relatively prime to m. Using the elementary relation
−1 −1
χ x1 x2 x3 x4
=
χ (mod m)
φ(m), if x ∈ V , 0,
if x ∈ B − V ,
1 where in the sum χ runs through the set of multiplicative characters (mod m) and x− denotes the i multiplicative inverse of xi (mod m), one readily obtains
1 −1 B ∩ V = |B | + 1 χ x1 x2 x− . 3 x4 φ(m) φ(m)
(4)
χ =χ0 x∈B
Here, χ0 denotes the principal character (mod m). In particular, taking all the ai = a + 1 and all of the B i = B we have
4 +B | a | B 1 B ∩ V = + χ (x) . φ(m) φ(m)
(5)
χ =χ0 x=a+1
Thus we see that the problem of estimating |B ∩ V | is equivalent to the problem of estimating the fourth-order moment of a character sum. Our main theorems are Theorem 1. For any integers a, B with B > 0 we have
4 +B a χ (x) 8ν (m) τ (m)(log m)3 (log log m)7 B 2 , φ(m) 1
χ =χ0 x=a+1
where τ (m) is the number of divisors of m and ν (m) is the number of distinct prime divisors of m. By an application of Hölder’s inequality we immediately deduce, Theorem 2. For any positive integer m and box B of type (2), we have B ∩ V = |B | + O 8ν (m) τ (m)(log m)3 (log log m)7 |B| . φ(m)
Indeed, by (4),
4 B ∩ V − |B | 1 φ(m) φ(m)
i =1 χ =χ0 ai xi
4 1/4 χ (xi ) .
(6)
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
769
Theorems 1 and 2 were proven for the case of prime moduli by Ayyad, Cochrane, and Zheng [1], in a slightly sharper form,
+B 4 a χ (x) B 2 log2 p . ( p − 1) 1
(7)
χ =χ0 x=a+1
Garaev and Garcia [5] improved this to B 2 (log p + log2 ( B 2 / p )). Estimates for a general modulus were obtained by Harman [6, Lemma 2], Vaughan [12, p. 184] and Friedlander and Iwaniec [4, Lemma 3], but only for the case where a = 0. The latter upper bound was B 2 log6 m, although Garaev and Garcia point out that it should be B 2 log8 m. This bound is sharper than the bound in Theorem 1 for values of m having lots of divisors, and suggests that the dependence on τ (m) and ν (m) in the theorem may be removable. As noted in [1], the best possible estimate one can hope for is B 2 log m. The proofs in [6,12,4] make use of L-functions associated with the characters, Perron’s formula and mean value estimates of fourth-order moments of L-functions. Unfortunately, they do not appear to generalize to intervals shifted away from the origin. The upper bound in Theorem 1 may be compared with the result of Montgomery and Vaughan [8, Theorem 1],
1
φ(m)
χ =χ0
4 B max χ (x) m2 , B x=1
and the result of Burgess [2, Lemma 1] for prime p,
4 p B χ (x + m) 6p 2 B 2 .
χ =χ0 x=1 m=1
Estimates for higher order moments of character sums were obtained by Cochrane and Zheng [3] for prime moduli, including
−1
( p − 1)
2k +B a χ (x) ε,k pk−1+ε + B k p ε .
χ =χ0 x=a+1
Of course, the goal here is to remove the factor pk−1+ε on the right-hand side, thus yielding a bound of the strength one gets upon assuming the Grand Riemann Hypothesis. Finally we note that Shparlinski has made applications of the bound in (7) for bounding incomplete multiple Kloosterman sums [9], determining the distribution of points on multidimensional modular hyperbolas [10] and addressing a multidimensional Lehmer problem [11]. 2. Background lemmas The conductor of a character character χ (mod m ), that is,
χ (mod m) is the smallest divisor m of m such that χ induces a
χ (x) = χ (x)χ0 (x), for x ∈ Z. In particular, the principal character χ0 has conductor 1 and χ0 (x) = 1 for all x ∈ Z. χ is called a primitive character (mod m) if cond(χ ) = m, and an improper character otherwise. For any integer y and character χ (mod m) we define the Gauss sum G m (χ , y ) by,
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
G m (χ , y ) =
m
χ (x)em (xy ),
x=1
where em (∗) = e 2π i ∗/m , and put G χ = G m (χ , 1). It is well known that if (mod m) then
G m (χ , y ) = χ ( y )G χ ,
|G χ | =
and
χ is a primitive character
√
m.
(8)
More generally, we have Lemma 1. For any multiplicative character χ (mod m) and integer y we have
G m (χ , y ) = G χ
y
dχ
d
d|( y ,m/m )
χ
m dm
μ
m dm
.
Proof. The formula seems to be well known and can be found for example in [7, Lemma 3.2] (with the need of a slight correction). For convenience, we give a proof here m
G m (χ , y ) =
χ (x)em (xy )
x=1 (x,m)=1
=
m x=1
=
m μ(d) χ (x)em (xy ) = μ(d) χ (x)em (xy )
d|(x,m)
μ(d)χ (d)
x=1 d|x
d|m m/d
χ (t )em/d (t y ).
t =1
d|m
Now, if (d, m ) > 1 then χ (d) = 0 and there is no contribution to the sum. If (d, m ) = 1 then d|(m/m ), m |(m/d) and we can write t = u + vm with u running through a complete set of residues (mod m ) and v a set of residues (mod m/(dm )), to obtain
G m (χ , y ) =
=
μ(d)χ (d)
d|(m/m )
u =1
m
μ(d)χ (d)
d|(m/m ) m |y dm
μ(d)χ
χ (u )em/d (u y )
v =1
m m (d) χ (u )em/d (u y ). dm
Next, substitute λ = m/(dm ) and use (8) to get
χ (u )em/d u + vm y m/(dm )
)
v =1
u =1
d|(m/m )
=
dm m m/(
u =1
em/(dm ) ( v y )
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
G m (χ , y ) =
μ
λ|( y ,m/m )
m
χ
m λ
m m λ
= Gχ
y
λχ
μ
λ
λ|( y ,m/m )
771
m λ χ (u )em (u y /λ) u =1
m
χ
m λ
m
m λ
2
.
Lemma 2. Let m be positive integer. Then for (x, m) = 1 we have
χ (x) =
μ(m/d)φ(d) =
d|(m,x−1)
χ (mod m) cond(χ )=m
μ(m/d)φ(d)δd (x),
(9)
d|m
where
δd (x) =
1 if x ≡ 1 (mod d), 0
otherwise.
Proof. This lemma is also well known and follows from the relation χ (mod m) χ (x) = φ(m) or 0 according as x ≡ 1 (mod m) or not, together with the Möbius inversion formula. 2 Finally, we shall make frequent appeal to elementary estimates such as
φ(m) m/ log log m, 1 d|m
d
=
σ (m)
σ (m) m log log m,
log log m,
m
1+
p |m
1
p
log log m.
3. Fundamental identity Let Zm denote the ring of integers (mod m) and U m the group of units in Zm . Let f be a complex4 with Fourier expansion valued function defined on Zm
f (x) =
a(y)em (x · y).
y (mod m)
Here, em (∗) = e
2π i m ∗
, x · y = x1 y 1 + x2 y 2 + x3 y 3 + x4 y 4 , and
y (mod m)
denotes the complete sum over
4 y ∈ Zm . Throughout the paper we shall use the abbreviations
x (mod m)
=
4 x∈Zm
=
m x1 =1
...
m x4 =1
and
4 x∈U m
=
x (mod m) (x,m)=1
denote the set of solutions of Let V m
x1 x2 ≡ x3 x4 with coordinates in U m .
(mod m),
=
m x1 =1 (x1 ,m)=1
...
m x4 =1 (x4 ,m)=1
.
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
4 Theorem 3 (Fundamental identity). For any complex-valued function f on Zm with Fourier coefficients a(y), we have
f (x) =
x∈ V m
m4 φ(α )
φ(m)
α2
α |m
μ(l) l2
l|m/α
...
λ1 |m/αl (λ1 ,αl)=1
4 μ(λi ) i =1
λ4 |m/αl (λ4 ,αl)=1
λi
S (α , l, λ),
where λ = (λ1 , λ2 , λ3 , λ4 ),
S (α , l, λ) =
a(d1 u 1 , d2 u 2 , d3 u 3 , d4 u 4 ),
u (u i ,αl)=1 u i (mod αlλi ) u 1 u 2 λ3 λ4 ≡u 3 u 4 λ1 λ2 (mod
(10)
α)
and di = αm , 1 i 4. l λi Proof. We have
f (x) =
x∈ V m
= =
=
1
φ(m)
f (x)
φ(m) χ
4 x∈U m
1
φ(m) χ
φ(m) χ
1 −1 a(y)em (x · y) χ x1 x2 x− 3 x4
4 y∈Zm
a(y)
4 y∈Zm
1
χ (mod m)
4 x∈U m
1
1 −1 χ x1 x2 x− 3 x4
1 −1 em (x · y)χ x1 x2 x− 3 x4
4 x∈U m
a(y)
m 2
em (xi y i )χ (xi )
i =1 x i =1
4 y∈Zm
m 4
em (xi y i )χ (xi ).
i =3 x i =1
Then by Lemma 1, the above
=
1
φ(m) χ
y
4
×
a(y)G 2χ G 2χ¯
2
di χ¯ ( y i /di )χ m/di m
μ m/di m
i =1 di |( y i ,m/m )
di χ ( y i /di )χ¯ m/di m
μ m/di m .
i =3 di |( y i ,m/m )
Let λi = m/di m , u i = y i /di . Note, we may assume (λi , m ) = 1 and (u i , m ) = 1, else the term is zero. Now G χ¯ = χ (−1)G χ , so G 2χ G 2χ¯ = m 2 . Then, letting du = (d1 u 1 , d2 u 2 , d3 u 3 , d4 u 4 ), we have that the above
=
1
φ(m) ×
m
2
m |m
u u i (mod m/di ) (u i ,m )=1
d1 |m/m
a(du)
...
d4 |m/m
χ
cond(χ )=m
d1 d2 d3 d4
4 i =1
μ(λi )
1 −1 −1 −1 . χ u3 u4 u− 1 u 2 λ1 λ2 λ3 λ4
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
773
We then use Lemma 2 to get this
=
1
φ(m)
×
m |m
m
2
di μ(λi )
d4 |m/m i =1
d1 |m/m
4
...
a(du)
u u i (mod m/di ) (u i ,m )=1
1 −1 −1 −1 . μ m /α φ(α )δα u 3 u 4 u − 1 u 2 λ1 λ2 λ3 λ4
α |m
Set l = m /α so that m = αl, di = λ mm = λmαl and i i
=
m4 φ(α )
φ(m)
α |m
α2
μ(l) l2
l|m/α
where S (α , l, λ) is as stated in the theorem.
...
λ1 |m/αl (λ1 ,αl)=1
4
di = αm4 l4
λ4 |m/αl (λ4 ,αl)=1
1
λ i . Then the above,
4 μ(λi ) i =1
λi
S (α , l, λ),
2
4. Initial upper bound on |B ∩ V | for special boxes Lemma 3. Let a2 , a4 ∈ Z, x1 , x3 , m, B 2 , B 4 be positive integers with (x1 , m) = 1 and d = (x1 , x3 ). Then
# (x2 , x4 ) ∈ Z2 : x1 x2 ≡ x3 x4 (mod m), a2 x2 < a2 + B 2 , a4 x4 < a4 + B 4
B 2 x1 dm
+
B 4 x3 dm
+ 1 min
B 2d x3
+ 1,
B 4d x1
+1 .
Proof. Let (x1 , m) = 1, d = (x1 , x3 ), so that in particular (d, m) = 1. We must solve the linear Diophantine equation
x1 x2 − x3 x4 = jm
(11)
with j ∈ Z. Now, since
x1 a2 − x3 (a4 + B 4 ) < x1 x2 − x3 x4 < x1 (a2 + B 2 ) − x3 a4 , it follows that jm runs through an interval of length B 2 x1 + B 4 x3 . Any solution of (11) must have d| jm and thus d| j and so there are at most
B 2 x1 dm
+
B 4 x3 dm
+1
choices for j. For any fixed j the solution set of (11) is given by
x2 = x20 + λx3 /d,
x4 = x40 + λx1 /d,
with λ ∈ Z. Since x2 and x4 run through intervals of lengths B 2 and B 4 respectively, there are at most
min choices for λ, and the lemma follows.
2
B 2d x3
+ 1,
B 4d x1
+1 ,
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
Lemma 4. Suppose that B is a box of the type
B = x ∈ Z4 : 1 x1 H , a2 x2 < a2 + K , 1 x3 K , a4 x4 < a4 + H , where a2 , a4 , H , K are integers with 0 < H , K < m. Then the number of solutions of (1) in B with (x1 , m) = 1 is less than
7 |B | m
+ 3 |B| log m.
In particular,
B ∩ V 7 |B| + 3 |B| log m.
(12)
m
Proof. For any fixed x1 , x3 let N (x1 , x3 ) denote the number of solutions (x2 , x4 ) of the congruence x1 x2 ≡ x3 x4 (mod m) in the desired interval. Then by Lemma 3 the number of solutions in B with (x1 , m) = 1 is
d H
H
K
N (x1 , x3 )
x1 =1 x3 =1 (x1 ,m)=1 (x1 ,x3 )=d
H
K
H
+
K
2K x1 dm
x1 =1 x3 =1 (x1 ,m)=1 K x1 H x3 (x1 ,x3 )=d
2H 2 K 2 m
+
dm
x1 =1 x3 =1 (x1 ,m)=1 K x1 H x3 (x1 ,x3 )=d
d H
2H x3
+ HK +
d H
x1 ,x3 K x1 H x3 (x1 ,x3 )=d
+1
d H
x1 ,x3 K x1 H x3 (x1 ,x3 )=d
2K x1 dm
+
+1
Hd x1
Kd x3
Hd x1
2H x3 dm
+1
+1
+
Kd
x3
.
Letting x1 = du 1 and x3 = du 3 the above is
2H 2 K 2 m 2H 2 K 2 m 7H 2 K 2 m
+ HK +
2H 2 u 2 3 d H u 3 K /d
+ HK +
mK
4H 2 K 2 1 m
+ 3H K log m.
d H
2
d3
2 2 2K u 1 +H + +K d H u 1 H /d
+ 2H K log(2m)
Hm
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
775
Lemma 5. Let m > 1 and B be a box of the type
B = x ∈ Z4 : | x i | ρ i , 1 i 4 , where ρ1 , ρ2 , ρ3 , ρ4 are real numbers with 0 < ρi < m. Let λ, β be any integers with (λβ, m) = 1. Then the number of solutions of the congruence
λx1 x2 ≡ β x3 x4 (mod m) with x ∈ B , (xi , m) = 1, 1 i 4, is less than
112
ρ12 ρ22 m
1/2 + ρ1 ρ2 log m
ρ32 ρ42 m
1/2 + ρ3 ρ4 log m
.
Note. The lemma is false if m = 1 and ρi < 1 for some i. This is what distinguishes the cases and α > 1 in Sections 5.3 and 5.2 below.
α=1
Proof of Lemma 5. Since m > 1 and (xi , m) = 1, we have xi = 0, 1 i 4. Thus B can be partitioned into 16 boxes cornered at the origin, each in one of the 16 hyperoctants and having points with all nonzero coordinates. It suffices to consider the box with all positive coordinates. Summing over those x with (xi , m) = 1, 1 i 4 we have
1=
m c =1
λx1 x2 ≡β x3 x4 (mod m) (xi ,m)=1 1xi ρi
1
2 1/2
x1 ,x2 λx1 x2 ≡c (mod m) 1xi ρi
=
x1 x2 ≡u 1 u 2 (mod m) 1x1 ,u 1 ρ1 1x2 ,u 2 ρ2
1
x3 ,x4 β x3 x4 ≡c (mod m) 1xi ρi
1
x1 ,x2 λx1 x2 ≡c (mod m) 1xi ρi
c
c
1/2
1
x3 x4
β x3 x4 ≡c (mod m) 1xi ρi
1
2 1/2
1/2 1
,
x3 u 4 ≡u 3 u 4 (mod m) 1x3 ,u 3 ρ3 1x4 ,u 4 ρ4
where throughout, the variables are coprime to m. The lemma now follows from Lemma 4.
2
5. Upper bound on |B ∩ V | for a general box Proposition 1. Suppose that B is any box of type (2). Put L = log log m. Then
B ∩ V mL τ (m) + m1/2 L 18 τ (m) log m( B 1 B 2 + B 3 B 4 ) 4 B i + m−1 4ν (m) τ (m) L 3 + 16ν (m) (log m) L 2 B 1 B 2 B 3 B 4 + L 10
+ m−1 2ν (m) L 5
1i < j
i =1
B i B j B k + m −1 L 5 B 1 B 2 B 3 B 4 .
Bi B j
1i < j 4
(13)
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
Proof. We may assume that B is a box of the type
B = x ∈ Z4 : | x i − c i | < B i / 2 , 1 i 4 , for some integers c i . Let V be the set of solutions of (1), B0 = {x ∈ Z4 : |xi | < B i /2, 1 i 4}. We apply the fundamental identity, Theorem 3, with
f (x) =
1
|B |
χB 0 ∗ χB ,
a normalized convolution of the characteristic functions of B0 and B . Since f (x) 1 on B we have
B ∩ V f (x).
(14)
x∈ V
The function f may be written in the manner
f (x) = f 1 (x1 ) f 2 (x2 ) f 3 (x3 ) f 4 (x4 ), and the Fourier coefficients a(y) of f (x) are given by,
a(y) = a1 ( y 1 )a2 ( y 2 )a3 ( y 3 )a4 ( y 4 ), where the Fourier coefficients ai ( y i ) of f i satisfy
2 ai ( y i ) = 1 sin (π B i y i /m) . mB sin2 (π y /m) i
i
From the relationship |sin(x)| π2 |x| for |x| π /2 we have,
ai ( y i ) 1 min B 2 , m2 /4 y 2 , i i mB i
for | y i | m/2.
(15)
5.1. Estimating S (α , l, λ) Let λ = (λ1 , λ2 , λ3 , λ4 ) be a 4-tuple of positive integers satisfying λi |m/αl, (λi , αl) = 1, 1 i 4. Write u i = v i + αt i with t i running through a residue system (mod lλi ) and v i through a system (mod α ). Then
S (α , l, λ) =
u (u i ,αl)=1 u i (mod αlλi ) u 1 u 2 λ3 λ4 ≡u 3 u 4 λ1 λ2 (mod
=
a(d1 u 1 , d2 u 2 , d3 u 3 , d4 u 4 )
α) 4
v (mod α ) ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
α)
ai di ( v i + αt i ) .
(16)
i =1 t i (mod lλi ) ( v i +αt i ,l)=1
To estimate the sum over t i we apply the upper bound in (15) with y i = di ( v i + αt i ). In order to assure | y i | m/2 we choose values for v i , t i as follows. If lλi is odd we let
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
−
l λi
< ti <
2
l λi 2
−α
,
2
α
< vi
777
,
2
while if lλi is even, we let
−
l λi 2
ti <
l λi 2
0 v i < α.
,
We shall suppose αl is odd, the proof for the even case being similar, and partition the interval for v i into “subintervals"
I ki := v i ∈ Z: δki
α l λi Bi
| v i | < 2ki
α l λi
Bi
,
where δki = 2ki −1 , for ki 1; δki = 0, for ki = 0. In particular, since | v i | < α /2 we may assume ki satisfies
l λi Bi
<
1 2ki
,
ki < log2 ( B i /lλi ).
(17)
Consider now the sum over t i for fixed v i ∈ I ki . If t i = 0 then |u i | = | v i | δki and by (15) (recalling di = m/(αlλi )),
ai (di v i )
Bi 22ki m
αd |t | m|t | If t i = 0 then u i = v i + αt i α2 |t i |, |di u i | 2i i = 2λ il i
ai di ( v i + αt i )
m B i (di u i )2
α l λi Bi
, |di u i | δki
m Bi
,
(18)
.
m|t i |2ki Bi
, by (17) and thus, by (15),
Bi 22ki mt i2
(19)
.
Summing over t i we see that
ai di ( v i + αt i )
|t i |lλi /2
Bi 22ki m
(20)
.
For any 4-tuple k = (k1 , k2 , k3 , k4 ) of nonnegative integers, let
R (k) = I k1 × I k2 × I k3 × I k4 and
B (k) = x ∈ Z: |xi | min
α /2, 2ki αlλi / B i , 1 i 4 .
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
5.2. The case α > 1 If
α > 1 then applying Lemma 5 to B (k) we get
1
v∈ B (k), ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
22k1 +2k2 α 4l4 λ21 λ22
α B 21 B 22
+
2k1 +k2 α 2l2 λ1 λ2 log α
12
B1 B2
α)
×
22k3 +2k4 α 4l4 λ23 λ24
α B 23 B 24
2k3 +k4 α 2l2 λ3 λ4 log α
+
B3 B4
2k1 +k2 +k3 +k4 α 3l4 λ1 . . . λ4 B1 B2 B3 B4
12
+
k1 k2 √ 2 2 + 2 +k3 +k4 α 5/2l3 λ1 λ2 λ3 λ4 log α
√
B1 B2 B3 B4
+ +
k3 k4 √ 2 2 + 2 +k1 +k2 α 5/2l3 λ1 λ2 λ3 λ4 log α
√
2
k1 2
k + 22
k + 23
k + 24
√
B3 B4 B1 B2
√
α 2l2 λ1 λ2 λ3 λ4 log α
B1 B2 B3 B4
.
Then letting the ki run from 0 to the bound in (17), we get from (16) and (20),
S (α , l, λ) ... k1
k4
|B | m4
...
k1
4
v∈ R (k), ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
k4
α)
i =1
Bi 22ki m
1 22k1 +2k2 +2k3 +2k4
1
5
1
v∈ B (k), ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
α)
1
α 3l4 λ1 . . . λ4 + α 2 l3 B 1 B 2 λ1 λ2 λ3 λ4 log 2 m
m4
1 5 + α 2 l3 B 3 B 4 λ3 λ4 λ1 λ2 log 2 m + α 2l2 B 1 B 2 B 3 B 4 λ1 λ2 λ3 λ4 log m
=:
1 m4
( T 1 + T 2 + T 3 + T 4 ),
(21)
say. By the fundamental identity, Theorem 3, the contribution to f (x) with x∈ V m above by
m4 φ(α ) 1
φ(m)
α |m
α
2
l|(m/α )
l2
λ1 |(m/αl) (λ1 ,αl)=1
...
λ4 |(m/αl) (λ4 ,αl)=1
1
λ1 λ2 λ3 λ4
α > 1 is bounded
S (α , l, λ),
where the λi run through the square-free divisors of m/αl. Inserting the bound for | S (α , l, λ)| in (21) yields the upper bound
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
φ(α ) 1
1
φ(m)
=
α |m
1
φ(m)
α
2
l|(m/α )
l2
...
λ1 |(m/αl) (λ1 ,αl)=1
779
(T 1 + T 2 + T 3 + T 4 ) λ1 λ2 λ3 λ4
λ4 |(m/αl) (λ4 ,αl)=1
( S 1 + S 2 + S 3 + S 4 ),
(22)
say, where S i is the contribution corresponding to the value T i . Let ν (m) denote the number of distinct prime factors of m and τ ∗ (m/αl) the number of square-free divisors of m/αl that are relatively prime to αl. Then
S1
φ(α )α
α |m
l2 τ ∗ (m/αl)4
φ(α )α (m/α )2 m2 τ (m).
(23)
α |m
l|m/α
Similarly,
S2
1
B 1 B 2 log 2 m
√
α φ(α )
α |m
1
B 1 B 2 log 2 m
lτ ∗ (m/αl)4
l|m/α
√
α φ(α )
α |m
m
α
1+
p |(m/α )
17 p
φ(α ) m(log log m)17 log m B 1 B 2 √
α
α |m
m
3/2
17
(log log m)
log m
B1 B2
1
1+ √
p |m
p
m3/2 log m(log log m)17 τ (m) B 1 B 2 ,
S3
1
B 3 B 4 log 2 m
√
α φ(α )
α |m
(24)
lτ ∗ (m/αl)4
l|m/α
m3/2 log m(log log m)17 τ (m) B 3 B 4 ,
(25)
and
S4
B 1 B 2 B 3 B 4 log m
α |m
B 1 B 2 B 3 B 4 log m
16
ν (m)
φ(α )
τ ∗ (m/αl)4
l|m/α
φ(α )16ν (m/α ) τ (m/α )
α |m
B 1 B 2 B 3 B 4 m(log log m) log m.
By (22), (23), (24), (25) and (26) we see that the contribution to four terms appearing on the right-hand side of (13).
x∈ V m
(26) f (x) with
α > 1 is the first
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
5.3. The case α = 1
α = 1. In this case, | B (k)|
Finally we consider the contribution with
S (α , l, λ) ... k1
k4
|B | m4
...
k1
m4
...
k1
1 22k1 +2k2 +2k3 +2k4
i =1 (
22k1 +2k2 +2k3 +2k4
+ 1) and
Bi 22ki m
1
v∈ B (k) 4 k i +1 2 l λi
1
2ki +1 lλi Bi
Bi
i =1
+1
4 1
m4
(lλi + B i ).
i =1
l|m
l2
f (x) with
x∈ V m
m4 1
α)
k4
Thus by Theorem 3 the contribution to
φ(m)
i =1
v∈ R (k) v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
k4
|B |
4
4
···
λ1 |m/l
φ(m)
l|m
l2
4 1
1
λ4 |m/l
4 1
1
α = 1 is bounded above by
λ1 λ2 λ3 λ4 m 4
(lλi + B i )
i =1
(l + B i /λi ),
i =1 λi |m/l
where the λi run through the square-free divisors of m/l. Letting τ ∗ (m/l), σ−∗ 1 (m/l) denote the number, and the sum of the reciprocals respectively, of the square-free divisors of m/l we see that the above is
1
4 1
φ(m)
l2
l|m
i =1
∗ lτ ∗ (m/l) + σ− 1 (m/l) B i .
The following estimates are readily obtained from the multiplicative properties of the sums
l2 τ ∗ (m/l)4 m2 ,
l|m
l|m
l|m
τ ∗ (m/l)2 σ−∗ 1 (m/l)2 4ν (m) τ (m)
1 l|m
l
∗ lτ ∗ (m/l)3 σ− 1 (m/l) m
τ
∗
(m/l)σ−∗ 1 (m/l)3
2ν (m)
p |m
1+
p |m
1+
p |m
1+
1 p
9
m(log log m)9 ,
p 1
2
p
3 1−
4ν (m) τ (m)(log log m)2 , 1 p
−1
2ν (m) (log log m)4 ,
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
1 l|m
l
σ ∗ (m/l)4 2 −1
1+
p |m
Using these estimates we see that the contribution to
1
φ(m)
m2 + m(log log m)9
1
4 (log log m)4 .
p
x∈ V m
f (x) with
α = 1 is bounded by
B i + 4ν (m) τ (m)(log log m)2
i
+ 2ν (m) (log log m)4
781
Bi B j
i< j
B i B j B k + (log log m)4 B 1 B 2 B 3 B 4 ,
i < j
the remaining terms on the right-hand side of (13). We observe that the first term m2 /φ(m) is dominated by mL τ (m) and so does not need to be included. 2 6. Proof of Theorem 1 We may assume 1 B < m. As noted in the Introduction, by the identity in (5), Theorem 1 is just a special case of Theorem 2 with B a box of the type I × I × I × I , and so our approach is to prove Theorem 2 for this case. Let I be the interval of points a x < a + B in Zm , and χ I the characteristic function for I , with Fourier expansion
χ I (x) =
m
a( y )em (xy ).
y =1
Here
a(0) =
B m
a( y ) = 1 sin π y B /m , m sin π y /m
,
for y = 0.
(27)
Let B denote the number of values in I relatively prime to m, so that
B =
ax
φ(m) B + θ τ (m), μ(d) = m
d|(x,m)
(28)
with |θ| 1. Let B = I × I × I × I and B the set of points in B with coordinates relatively prime to m, so that |B | = B 4 . Recalling that V is the set of integer solutions of x1 x2 ≡ x3 x4 (mod m) with (xi , m) = 1, 1 i 4, we have
B ∩ V =
1 χ I x1 x2 x− 3
axi
= a(0) B 3 +
a( y )
y =0
say. For the Main term, using a(0) =
1 em x1 x2 x− 3 y = Main + Error ,
axi
we deduce from (28) that
|B | B 3 3 B 3 τ (m) 2 − − a ( 0 ) B B = φ(m) φ(m) m B φ(m) B τ (m),
(29)
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
that is
Main =
|B | + O B 2 τ (m) . φ(m)
Thus we are left with proving that
|Error| 8ν (m) τ (m)(log m)3 (log log m)7 B 2 . If B < (log m)2 (log log m)5 τ (m) then we have trivially
|Error| B 3
a( y ) B 3 log m B 2 τ (m) log3 m(log log m)5 , y
and so we may assume that B > (log m)2 (log log m)5 τ (m). First observe that for any value of l, a+ B −1
em (lx) =
x=a (x,m)=1
μ(λ)
em (lλt ).
(30)
a+ B λ t < λ
λ|m
a
Letting B λ denote the number of integers that t runs through, we get
a + B −1 sin(π B lλ/m) λ em (lx) sin(π lλ/m) . x=a λ|m
(x,m)=1
Applying this estimate to the sum over x2 in the error term of (29) we have
|Error|
a( y ) λ|m y =0
1 sin(π B λ λ yx1 x− 3 /m) . −1 sin(π λ yx1 x3 /m) ax
(31)
i
(xi ,m)=1
We break the sum into two pieces E 1 + E 2 where E 1 is the sum over values of λ > B and E 2 the sum over λ B. For E 1 we use the trivial upper bound
1 sin(π B λ λ yx1 x− 3 /m) B λ 1, −1 sin(π λ yx1 x3 /m) to get
|E1| B2
a( y ) τ (m) B 2 log m. λ|m λ> B
y
−1 For E 2 , λ B, and so B λ λB + 1 2B λ . Putting l ≡ yx1 x3 (mod m/λ) we have
|E2|
λ|m |l| 2mλ λ B
0<| y |m/2
a( y ) sin(π B λ λl/m) sin(π λl/m)
axi
1.
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
783
We split the latter sum into E 2,0 + E 2,1 where E 2,0 is the contribution with l = 0 and E 2,1 the contribution with l = 0, and make use of the inequalities,
a( y ) = 1 sin(π B y /m) 1 min B , m , m sin(π y /m) m y
for 0 < | y | m/2,
and
sin(π B λ λl/m) min B , m , sin(π λl/m) λ lλ
for |λl| m/2.
If l = 0 the sum over x1 , x3 is zero unless (m/λ)| y, in which case it equals B 2 . Thus, putting y= m λ t we have
| E 2,0 |
m B 2 a λt λ B λ|m 0<|t | λ 2
B3
1 λ|m
λ
λ λ
0<|t | 2
tm
B3 m
τ (m) log m B 2 τ (m) log m.
(32)
Assume now that 0 < l 2mλ . Then any solution of the congruence x1 y ≡ x3l (mod m/λ) with (x1 , m) = (x3 , m) = 1 must have mλ y. Put d = ( y , l, m/λ), y = y /d, l = l/d. Since mλ y we have m d< m λ , that is, λ < d . Note that the congruence x1 y ≡ x3 l (mod m/λ) is equivalent to x1 y ≡ x3 l (mod m/λd) and that any solution of the latter congruence with (x1 , m) = (x3 , m) = 1, must also have ( y , m/λd) = (l , m/λd) = 1. Thus the contribution to E 2 with l = 0 is bounded by
1
| E 2,1 |
m
min B ,
m d|m λ|(m/d) 0<|l | m 0<| y | 2d 2λd λ
×
m
min
d| y |
B
,
m
λ l λd
1.
axi
Let y = u + λmd t, with |u | m/2λd, and |t | λ2 . Next, for any nonnegative i , j let
Ri j =
x1 , u , x3 , l : a x1 , x3 < a + B , δi
m dB
|u | < 2
i
m dB
, δj
m dB
|l | < 2
j
m
dB
and
Bi j =
x1 , u , x3 , l : a x1 , x3 < a + B , |u | < min 2
i
m
,
m
dB 2λd
, |l | < min 2
j
m
,
m
dB 2λd
,
i where δi = 2i −1 for i > 0, δ0 = 0. Since |u | 2m λd we may assume λ < B /2 . If t = 0 then | y | = |u |, m while if t = 0 then | y | 2λd |t |. Thus, for any point (x1 , u , x3 , l ) ∈ Ri j satisfying the congruence
x1 u ≡ x3l
(mod m/λd),
(33)
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
we have,
min B ,
|t |λ/2
m
min B ,
d| y |
m
+λ
du
1
0<|t |λ/2
t
B 2i
log m.
Let N i j denote the number of points in Ri j satisfying (33) with coordinates relatively prime to m/λd. Then
| E 2,1 |
1 B m
d|m λ|(m/d) λ
B2
=
m
log m
i
log m
2i
j
B 2 jλ
Ni j
1 2− i − j N i j . λ d|m λ|(m/d) λ
i
(34)
j
The dimensions B i of the box Bi j , satisfy
B1 = B,
B2
2i m dB
B3 = B,
,
B4
2 jm dB
.
Put L = log log(m). Suppose first that B < m/λd. Then we can apply Proposition 1 to get
Ni j
m dλ
+ BL
10
+
m
τ (m/dλ) L +
dλ
mL 10 dB
i
2 +2
j
m
d
i
+ 2ν (m/dλ) L 5 B2i λ + B2 j λ +
m
2i + j λ + λ
dB
m d
dB 2
m
j
16ν (m/dλ) (log m) L 2 2 2 + 2
λd 2i + j mλ + 4ν (m/λd) τ (m/λd) L 3 2i λ + 2 j λ + B 2 +
Putting
L 18 log mτ (m/dλ) 2i /2 + 2 j /2 + 1/2
L 5 2i + j .
ν = ν (m) and τ = τ (m) and noting that λ < B < m/λd gives Ni j
m dλ
+
τL + m
λd
L
dλ
10
+ 2ν L 5
m
+ m d
m
L 18 log mτ 2i /2 + 2 j /2 + 1/2 mL 10 dB 2i +
m d
i
2 +2 2j +
j
d
j
i
16ν (log m) L 2 2 2 + 2
m m m m 1 + 4ν τ L 3 2 i + 2j + + 2i + j
m i+ j 2 d
λd
+λ
m d
λd
λd
L 5 2i + j .
d
B
(35)
If B > λmd then we have a trivial bound for N i j obtained by letting x1 , u , l be arbitrary values, thus B + 1 choices for x3 , and so determining at most m/λ d
Ni j B i
j
2i m dB
+1
2 jm dB
+1
B m/λd
+1 .
If either 2dBm < 1 or 2dBm < 1 then N i j = 0 for this would require u = 0 or l = 0 by the definition of Bi j , but then u or l would not be relatively prime to m/λd since in the sum for E 2,1 , m/λd > 1.
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
785
Thus it follows that N i j 2i + j mdλ , which is less than the final term in (35). Summing over i , j and noting that i , j run through intervals of length at most log2 m, we get
i
2− i − j N i j
j
m d
τL λ
L 18
+√
+ 4ν τ L 3
λ
log mτ + 16ν (log m) L 2 +
log m
λ
+
log2 m B
L 10
λ
+
L 10 B
log m
+ 2ν L 5 log2 m + λ L 5 log2 m .
Summing over λ, d in (34) then yields
| E 2,1 | B 2 log m τ L 2 + L 19 τ log m + 16ν L 4 log m + L 11 + L 12 log m/ B + 4ν L 4 τ log m + 4ν τ L 5 log2 m/ B + 2ν L 7 log2 m + L 6 τ log2 m . Finally, using the assumption B > τ L 5 log2 m we have
| E 2,1 | B 2 log m 16ν L 4 log m + 4ν L 4 τ log m + 2ν L 7 log2 m + L 6 τ log2 m , completing the proof. References [1] A. Ayyad, T. Cochrane, Z. Zheng, The congruence x1 x2 ≡ x3 x4 (mod p), the equation x1 x2 = x3 x4 , and mean values of character sums, J. Number Theory 59 (2) (1996) 398–413. [2] D.A. Burgess, Mean values of character sums, Mathematika 33 (1) (1986) 1–5. [3] T. Cochrane, Z. Zheng, High order moments of character sums, Proc. Amer. Math. Soc. 126 (4) (1998) 951–956. [4] J.B. Friedlander, H. Iwaniec, The divisor problem for arithmetic progressions, Acta Arith. 45 (3) (1985) 273–277. [5] M.Z. Garaev, V.C. Garcia, The equation x1 x2 = x3 x4 + λ in fields of prime order and applications, J. Number Theory 128 (9) (2008) 2520—2537 (English summary). [6] G. Harman, Diophantine approximation with square-free integers, Math. Proc. Cambridge Philos. Soc. 95 (1984) 383–388. [7] H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53, Amer. Math. Soc., Providence, RI, 2004. [8] H.L. Montgomery, R.C. Vaughan, Mean values of character sums, Canad. J. Math. 31 (3) (1979) 476–587. [9] I.E. Shparlinski, Bounds of incomplete multiple Kloosterman sums, J. Number Theory 126 (1) (2007) 68–73. [10] I.E. Shparlinski, On the distribution of points on multidimensional modular hyperbolas, Proc. Japan Acad. Ser. A Math. Sci. 83 (2) (2007) 5–9. [11] I.E. Shparlinski, On a generalised Lehmer problem for arbitrary powers, East–West J. Math. (2008) 197–204 (special volume). [12] R.C. Vaughan, Diophantine approximation by prime numbers, III, Proc. London Math. Soc. 33 (1976) 177–192.
Contents lists available at ScienceDirect
Journal of Number Theory www.elsevier.com/locate/jnt
The congruence x1 x2 ≡ x3 x4 (mod m) and mean values of character sums Todd Cochrane a,∗ , Sanying Shi b a b
Department of Mathematics, Kansas State University, Manhattan, KS 66506, United States Department of Mathematics, Tongji University, Shanghai 200092, China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 14 May 2009 Communicated by K. Soundararajan MSC: 11A07 11D79 11L26 11L40 Keywords: Congruences Character sums Moments
For any positive integer m we obtain the asymptotic formula, B ∩ V = |B | + O 8ν (m) τ (m)(log m)3 (log log m)7 |B| , φ(m)
for the number of solutions of the congruence x1 x2 ≡ x3 x4 (mod m) with coordinates relatively prime to m in a box B of arbitrary size and position. We also obtain an upper bound for a fourth-order character sum moment,
+B 4 a χ ( x ) 8ν (m) τ (m)(log m)3 (log log m)7 B 2 . φ(m) 1
χ =χ0 x=a+1
© 2009 Elsevier Inc. All rights reserved.
1. Introduction The distribution of solutions of the congruence
x1 x2 ≡ x3 x4
(mod m),
(1)
where m is any natural number, arises naturally in the study of certain character sums. For any integers ai , B i , with 1 B i < m, 1 i 4 let B be the box of points
*
Corresponding author. E-mail addresses: [email protected] (T. Cochrane), [email protected] (S. Shi).
0022-314X/$ – see front matter doi:10.1016/j.jnt.2009.08.006
© 2009 Elsevier Inc.
All rights reserved.
768
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
B = x ∈ Z4 : a i x i < a i + B i , 1 i 4 ,
(2)
of cardinality |B | = B 1 B 2 B 3 B 4 , and B the subset of B of points with coordinates relatively prime to m,
B = x ∈ Z4 : a i x i < a i + B i , ( x i , m ) = 1 , 1 i 4 .
(3)
Let V ⊂ Z4 denote the set of all integer solutions of (1) and V ⊂ Z4 the solutions of (1) with (xi , m) = 1, 1 i 4. Thus |B ∩ V | denotes the number of solutions of (1) in B with coordinates relatively prime to m. Using the elementary relation
−1 −1
χ x1 x2 x3 x4
=
χ (mod m)
φ(m), if x ∈ V , 0,
if x ∈ B − V ,
1 where in the sum χ runs through the set of multiplicative characters (mod m) and x− denotes the i multiplicative inverse of xi (mod m), one readily obtains
1 −1 B ∩ V = |B | + 1 χ x1 x2 x− . 3 x4 φ(m) φ(m)
(4)
χ =χ0 x∈B
Here, χ0 denotes the principal character (mod m). In particular, taking all the ai = a + 1 and all of the B i = B we have
4 +B | a | B 1 B ∩ V = + χ (x) . φ(m) φ(m)
(5)
χ =χ0 x=a+1
Thus we see that the problem of estimating |B ∩ V | is equivalent to the problem of estimating the fourth-order moment of a character sum. Our main theorems are Theorem 1. For any integers a, B with B > 0 we have
4 +B a χ (x) 8ν (m) τ (m)(log m)3 (log log m)7 B 2 , φ(m) 1
χ =χ0 x=a+1
where τ (m) is the number of divisors of m and ν (m) is the number of distinct prime divisors of m. By an application of Hölder’s inequality we immediately deduce, Theorem 2. For any positive integer m and box B of type (2), we have B ∩ V = |B | + O 8ν (m) τ (m)(log m)3 (log log m)7 |B| . φ(m)
Indeed, by (4),
4 B ∩ V − |B | 1 φ(m) φ(m)
i =1 χ =χ0 ai xi
4 1/4 χ (xi ) .
(6)
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
769
Theorems 1 and 2 were proven for the case of prime moduli by Ayyad, Cochrane, and Zheng [1], in a slightly sharper form,
+B 4 a χ (x) B 2 log2 p . ( p − 1) 1
(7)
χ =χ0 x=a+1
Garaev and Garcia [5] improved this to B 2 (log p + log2 ( B 2 / p )). Estimates for a general modulus were obtained by Harman [6, Lemma 2], Vaughan [12, p. 184] and Friedlander and Iwaniec [4, Lemma 3], but only for the case where a = 0. The latter upper bound was B 2 log6 m, although Garaev and Garcia point out that it should be B 2 log8 m. This bound is sharper than the bound in Theorem 1 for values of m having lots of divisors, and suggests that the dependence on τ (m) and ν (m) in the theorem may be removable. As noted in [1], the best possible estimate one can hope for is B 2 log m. The proofs in [6,12,4] make use of L-functions associated with the characters, Perron’s formula and mean value estimates of fourth-order moments of L-functions. Unfortunately, they do not appear to generalize to intervals shifted away from the origin. The upper bound in Theorem 1 may be compared with the result of Montgomery and Vaughan [8, Theorem 1],
1
φ(m)
χ =χ0
4 B max χ (x) m2 , B x=1
and the result of Burgess [2, Lemma 1] for prime p,
4 p B χ (x + m) 6p 2 B 2 .
χ =χ0 x=1 m=1
Estimates for higher order moments of character sums were obtained by Cochrane and Zheng [3] for prime moduli, including
−1
( p − 1)
2k +B a χ (x) ε,k pk−1+ε + B k p ε .
χ =χ0 x=a+1
Of course, the goal here is to remove the factor pk−1+ε on the right-hand side, thus yielding a bound of the strength one gets upon assuming the Grand Riemann Hypothesis. Finally we note that Shparlinski has made applications of the bound in (7) for bounding incomplete multiple Kloosterman sums [9], determining the distribution of points on multidimensional modular hyperbolas [10] and addressing a multidimensional Lehmer problem [11]. 2. Background lemmas The conductor of a character character χ (mod m ), that is,
χ (mod m) is the smallest divisor m of m such that χ induces a
χ (x) = χ (x)χ0 (x), for x ∈ Z. In particular, the principal character χ0 has conductor 1 and χ0 (x) = 1 for all x ∈ Z. χ is called a primitive character (mod m) if cond(χ ) = m, and an improper character otherwise. For any integer y and character χ (mod m) we define the Gauss sum G m (χ , y ) by,
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
G m (χ , y ) =
m
χ (x)em (xy ),
x=1
where em (∗) = e 2π i ∗/m , and put G χ = G m (χ , 1). It is well known that if (mod m) then
G m (χ , y ) = χ ( y )G χ ,
|G χ | =
and
χ is a primitive character
√
m.
(8)
More generally, we have Lemma 1. For any multiplicative character χ (mod m) and integer y we have
G m (χ , y ) = G χ
y
dχ
d
d|( y ,m/m )
χ
m dm
μ
m dm
.
Proof. The formula seems to be well known and can be found for example in [7, Lemma 3.2] (with the need of a slight correction). For convenience, we give a proof here m
G m (χ , y ) =
χ (x)em (xy )
x=1 (x,m)=1
=
m x=1
=
m μ(d) χ (x)em (xy ) = μ(d) χ (x)em (xy )
d|(x,m)
μ(d)χ (d)
x=1 d|x
d|m m/d
χ (t )em/d (t y ).
t =1
d|m
Now, if (d, m ) > 1 then χ (d) = 0 and there is no contribution to the sum. If (d, m ) = 1 then d|(m/m ), m |(m/d) and we can write t = u + vm with u running through a complete set of residues (mod m ) and v a set of residues (mod m/(dm )), to obtain
G m (χ , y ) =
=
μ(d)χ (d)
d|(m/m )
u =1
m
μ(d)χ (d)
d|(m/m ) m |y dm
μ(d)χ
χ (u )em/d (u y )
v =1
m m (d) χ (u )em/d (u y ). dm
Next, substitute λ = m/(dm ) and use (8) to get
χ (u )em/d u + vm y m/(dm )
)
v =1
u =1
d|(m/m )
=
dm m m/(
u =1
em/(dm ) ( v y )
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
G m (χ , y ) =
μ
λ|( y ,m/m )
m
χ
m λ
m m λ
= Gχ
y
λχ
μ
λ
λ|( y ,m/m )
771
m λ χ (u )em (u y /λ) u =1
m
χ
m λ
m
m λ
2
.
Lemma 2. Let m be positive integer. Then for (x, m) = 1 we have
χ (x) =
μ(m/d)φ(d) =
d|(m,x−1)
χ (mod m) cond(χ )=m
μ(m/d)φ(d)δd (x),
(9)
d|m
where
δd (x) =
1 if x ≡ 1 (mod d), 0
otherwise.
Proof. This lemma is also well known and follows from the relation χ (mod m) χ (x) = φ(m) or 0 according as x ≡ 1 (mod m) or not, together with the Möbius inversion formula. 2 Finally, we shall make frequent appeal to elementary estimates such as
φ(m) m/ log log m, 1 d|m
d
=
σ (m)
σ (m) m log log m,
log log m,
m
1+
p |m
1
p
log log m.
3. Fundamental identity Let Zm denote the ring of integers (mod m) and U m the group of units in Zm . Let f be a complex4 with Fourier expansion valued function defined on Zm
f (x) =
a(y)em (x · y).
y (mod m)
Here, em (∗) = e
2π i m ∗
, x · y = x1 y 1 + x2 y 2 + x3 y 3 + x4 y 4 , and
y (mod m)
denotes the complete sum over
4 y ∈ Zm . Throughout the paper we shall use the abbreviations
x (mod m)
=
4 x∈Zm
=
m x1 =1
...
m x4 =1
and
4 x∈U m
=
x (mod m) (x,m)=1
denote the set of solutions of Let V m
x1 x2 ≡ x3 x4 with coordinates in U m .
(mod m),
=
m x1 =1 (x1 ,m)=1
...
m x4 =1 (x4 ,m)=1
.
772
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
4 Theorem 3 (Fundamental identity). For any complex-valued function f on Zm with Fourier coefficients a(y), we have
f (x) =
x∈ V m
m4 φ(α )
φ(m)
α2
α |m
μ(l) l2
l|m/α
...
λ1 |m/αl (λ1 ,αl)=1
4 μ(λi ) i =1
λ4 |m/αl (λ4 ,αl)=1
λi
S (α , l, λ),
where λ = (λ1 , λ2 , λ3 , λ4 ),
S (α , l, λ) =
a(d1 u 1 , d2 u 2 , d3 u 3 , d4 u 4 ),
u (u i ,αl)=1 u i (mod αlλi ) u 1 u 2 λ3 λ4 ≡u 3 u 4 λ1 λ2 (mod
(10)
α)
and di = αm , 1 i 4. l λi Proof. We have
f (x) =
x∈ V m
= =
=
1
φ(m)
f (x)
φ(m) χ
4 x∈U m
1
φ(m) χ
φ(m) χ
1 −1 a(y)em (x · y) χ x1 x2 x− 3 x4
4 y∈Zm
a(y)
4 y∈Zm
1
χ (mod m)
4 x∈U m
1
1 −1 χ x1 x2 x− 3 x4
1 −1 em (x · y)χ x1 x2 x− 3 x4
4 x∈U m
a(y)
m 2
em (xi y i )χ (xi )
i =1 x i =1
4 y∈Zm
m 4
em (xi y i )χ (xi ).
i =3 x i =1
Then by Lemma 1, the above
=
1
φ(m) χ
y
4
×
a(y)G 2χ G 2χ¯
2
di χ¯ ( y i /di )χ m/di m
μ m/di m
i =1 di |( y i ,m/m )
di χ ( y i /di )χ¯ m/di m
μ m/di m .
i =3 di |( y i ,m/m )
Let λi = m/di m , u i = y i /di . Note, we may assume (λi , m ) = 1 and (u i , m ) = 1, else the term is zero. Now G χ¯ = χ (−1)G χ , so G 2χ G 2χ¯ = m 2 . Then, letting du = (d1 u 1 , d2 u 2 , d3 u 3 , d4 u 4 ), we have that the above
=
1
φ(m) ×
m
2
m |m
u u i (mod m/di ) (u i ,m )=1
d1 |m/m
a(du)
...
d4 |m/m
χ
cond(χ )=m
d1 d2 d3 d4
4 i =1
μ(λi )
1 −1 −1 −1 . χ u3 u4 u− 1 u 2 λ1 λ2 λ3 λ4
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
773
We then use Lemma 2 to get this
=
1
φ(m)
×
m |m
m
2
di μ(λi )
d4 |m/m i =1
d1 |m/m
4
...
a(du)
u u i (mod m/di ) (u i ,m )=1
1 −1 −1 −1 . μ m /α φ(α )δα u 3 u 4 u − 1 u 2 λ1 λ2 λ3 λ4
α |m
Set l = m /α so that m = αl, di = λ mm = λmαl and i i
=
m4 φ(α )
φ(m)
α |m
α2
μ(l) l2
l|m/α
where S (α , l, λ) is as stated in the theorem.
...
λ1 |m/αl (λ1 ,αl)=1
4
di = αm4 l4
λ4 |m/αl (λ4 ,αl)=1
1
λ i . Then the above,
4 μ(λi ) i =1
λi
S (α , l, λ),
2
4. Initial upper bound on |B ∩ V | for special boxes Lemma 3. Let a2 , a4 ∈ Z, x1 , x3 , m, B 2 , B 4 be positive integers with (x1 , m) = 1 and d = (x1 , x3 ). Then
# (x2 , x4 ) ∈ Z2 : x1 x2 ≡ x3 x4 (mod m), a2 x2 < a2 + B 2 , a4 x4 < a4 + B 4
B 2 x1 dm
+
B 4 x3 dm
+ 1 min
B 2d x3
+ 1,
B 4d x1
+1 .
Proof. Let (x1 , m) = 1, d = (x1 , x3 ), so that in particular (d, m) = 1. We must solve the linear Diophantine equation
x1 x2 − x3 x4 = jm
(11)
with j ∈ Z. Now, since
x1 a2 − x3 (a4 + B 4 ) < x1 x2 − x3 x4 < x1 (a2 + B 2 ) − x3 a4 , it follows that jm runs through an interval of length B 2 x1 + B 4 x3 . Any solution of (11) must have d| jm and thus d| j and so there are at most
B 2 x1 dm
+
B 4 x3 dm
+1
choices for j. For any fixed j the solution set of (11) is given by
x2 = x20 + λx3 /d,
x4 = x40 + λx1 /d,
with λ ∈ Z. Since x2 and x4 run through intervals of lengths B 2 and B 4 respectively, there are at most
min choices for λ, and the lemma follows.
2
B 2d x3
+ 1,
B 4d x1
+1 ,
774
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
Lemma 4. Suppose that B is a box of the type
B = x ∈ Z4 : 1 x1 H , a2 x2 < a2 + K , 1 x3 K , a4 x4 < a4 + H , where a2 , a4 , H , K are integers with 0 < H , K < m. Then the number of solutions of (1) in B with (x1 , m) = 1 is less than
7 |B | m
+ 3 |B| log m.
In particular,
B ∩ V 7 |B| + 3 |B| log m.
(12)
m
Proof. For any fixed x1 , x3 let N (x1 , x3 ) denote the number of solutions (x2 , x4 ) of the congruence x1 x2 ≡ x3 x4 (mod m) in the desired interval. Then by Lemma 3 the number of solutions in B with (x1 , m) = 1 is
d H
H
K
N (x1 , x3 )
x1 =1 x3 =1 (x1 ,m)=1 (x1 ,x3 )=d
H
K
H
+
K
2K x1 dm
x1 =1 x3 =1 (x1 ,m)=1 K x1 H x3 (x1 ,x3 )=d
2H 2 K 2 m
+
dm
x1 =1 x3 =1 (x1 ,m)=1 K x1 H x3 (x1 ,x3 )=d
d H
2H x3
+ HK +
d H
x1 ,x3 K x1 H x3 (x1 ,x3 )=d
+1
d H
x1 ,x3 K x1 H x3 (x1 ,x3 )=d
2K x1 dm
+
+1
Hd x1
Kd x3
Hd x1
2H x3 dm
+1
+1
+
Kd
x3
.
Letting x1 = du 1 and x3 = du 3 the above is
2H 2 K 2 m 2H 2 K 2 m 7H 2 K 2 m
+ HK +
2H 2 u 2 3 d H u 3 K /d
+ HK +
mK
4H 2 K 2 1 m
+ 3H K log m.
d H
2
d3
2 2 2K u 1 +H + +K d H u 1 H /d
+ 2H K log(2m)
Hm
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
775
Lemma 5. Let m > 1 and B be a box of the type
B = x ∈ Z4 : | x i | ρ i , 1 i 4 , where ρ1 , ρ2 , ρ3 , ρ4 are real numbers with 0 < ρi < m. Let λ, β be any integers with (λβ, m) = 1. Then the number of solutions of the congruence
λx1 x2 ≡ β x3 x4 (mod m) with x ∈ B , (xi , m) = 1, 1 i 4, is less than
112
ρ12 ρ22 m
1/2 + ρ1 ρ2 log m
ρ32 ρ42 m
1/2 + ρ3 ρ4 log m
.
Note. The lemma is false if m = 1 and ρi < 1 for some i. This is what distinguishes the cases and α > 1 in Sections 5.3 and 5.2 below.
α=1
Proof of Lemma 5. Since m > 1 and (xi , m) = 1, we have xi = 0, 1 i 4. Thus B can be partitioned into 16 boxes cornered at the origin, each in one of the 16 hyperoctants and having points with all nonzero coordinates. It suffices to consider the box with all positive coordinates. Summing over those x with (xi , m) = 1, 1 i 4 we have
1=
m c =1
λx1 x2 ≡β x3 x4 (mod m) (xi ,m)=1 1xi ρi
1
2 1/2
x1 ,x2 λx1 x2 ≡c (mod m) 1xi ρi
=
x1 x2 ≡u 1 u 2 (mod m) 1x1 ,u 1 ρ1 1x2 ,u 2 ρ2
1
x3 ,x4 β x3 x4 ≡c (mod m) 1xi ρi
1
x1 ,x2 λx1 x2 ≡c (mod m) 1xi ρi
c
c
1/2
1
x3 x4
β x3 x4 ≡c (mod m) 1xi ρi
1
2 1/2
1/2 1
,
x3 u 4 ≡u 3 u 4 (mod m) 1x3 ,u 3 ρ3 1x4 ,u 4 ρ4
where throughout, the variables are coprime to m. The lemma now follows from Lemma 4.
2
5. Upper bound on |B ∩ V | for a general box Proposition 1. Suppose that B is any box of type (2). Put L = log log m. Then
B ∩ V mL τ (m) + m1/2 L 18 τ (m) log m( B 1 B 2 + B 3 B 4 ) 4 B i + m−1 4ν (m) τ (m) L 3 + 16ν (m) (log m) L 2 B 1 B 2 B 3 B 4 + L 10
+ m−1 2ν (m) L 5
1i < j
i =1
B i B j B k + m −1 L 5 B 1 B 2 B 3 B 4 .
Bi B j
1i < j 4
(13)
776
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
Proof. We may assume that B is a box of the type
B = x ∈ Z4 : | x i − c i | < B i / 2 , 1 i 4 , for some integers c i . Let V be the set of solutions of (1), B0 = {x ∈ Z4 : |xi | < B i /2, 1 i 4}. We apply the fundamental identity, Theorem 3, with
f (x) =
1
|B |
χB 0 ∗ χB ,
a normalized convolution of the characteristic functions of B0 and B . Since f (x) 1 on B we have
B ∩ V f (x).
(14)
x∈ V
The function f may be written in the manner
f (x) = f 1 (x1 ) f 2 (x2 ) f 3 (x3 ) f 4 (x4 ), and the Fourier coefficients a(y) of f (x) are given by,
a(y) = a1 ( y 1 )a2 ( y 2 )a3 ( y 3 )a4 ( y 4 ), where the Fourier coefficients ai ( y i ) of f i satisfy
2 ai ( y i ) = 1 sin (π B i y i /m) . mB sin2 (π y /m) i
i
From the relationship |sin(x)| π2 |x| for |x| π /2 we have,
ai ( y i ) 1 min B 2 , m2 /4 y 2 , i i mB i
for | y i | m/2.
(15)
5.1. Estimating S (α , l, λ) Let λ = (λ1 , λ2 , λ3 , λ4 ) be a 4-tuple of positive integers satisfying λi |m/αl, (λi , αl) = 1, 1 i 4. Write u i = v i + αt i with t i running through a residue system (mod lλi ) and v i through a system (mod α ). Then
S (α , l, λ) =
u (u i ,αl)=1 u i (mod αlλi ) u 1 u 2 λ3 λ4 ≡u 3 u 4 λ1 λ2 (mod
=
a(d1 u 1 , d2 u 2 , d3 u 3 , d4 u 4 )
α) 4
v (mod α ) ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
α)
ai di ( v i + αt i ) .
(16)
i =1 t i (mod lλi ) ( v i +αt i ,l)=1
To estimate the sum over t i we apply the upper bound in (15) with y i = di ( v i + αt i ). In order to assure | y i | m/2 we choose values for v i , t i as follows. If lλi is odd we let
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
−
l λi
< ti <
2
l λi 2
−α
,
2
α
< vi
777
,
2
while if lλi is even, we let
−
l λi 2
ti <
l λi 2
0 v i < α.
,
We shall suppose αl is odd, the proof for the even case being similar, and partition the interval for v i into “subintervals"
I ki := v i ∈ Z: δki
α l λi Bi
| v i | < 2ki
α l λi
Bi
,
where δki = 2ki −1 , for ki 1; δki = 0, for ki = 0. In particular, since | v i | < α /2 we may assume ki satisfies
l λi Bi
<
1 2ki
,
ki < log2 ( B i /lλi ).
(17)
Consider now the sum over t i for fixed v i ∈ I ki . If t i = 0 then |u i | = | v i | δki and by (15) (recalling di = m/(αlλi )),
ai (di v i )
Bi 22ki m
αd |t | m|t | If t i = 0 then u i = v i + αt i α2 |t i |, |di u i | 2i i = 2λ il i
ai di ( v i + αt i )
m B i (di u i )2
α l λi Bi
, |di u i | δki
m Bi
,
(18)
.
m|t i |2ki Bi
, by (17) and thus, by (15),
Bi 22ki mt i2
(19)
.
Summing over t i we see that
ai di ( v i + αt i )
|t i |lλi /2
Bi 22ki m
(20)
.
For any 4-tuple k = (k1 , k2 , k3 , k4 ) of nonnegative integers, let
R (k) = I k1 × I k2 × I k3 × I k4 and
B (k) = x ∈ Z: |xi | min
α /2, 2ki αlλi / B i , 1 i 4 .
778
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
5.2. The case α > 1 If
α > 1 then applying Lemma 5 to B (k) we get
1
v∈ B (k), ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
22k1 +2k2 α 4l4 λ21 λ22
α B 21 B 22
+
2k1 +k2 α 2l2 λ1 λ2 log α
12
B1 B2
α)
×
22k3 +2k4 α 4l4 λ23 λ24
α B 23 B 24
2k3 +k4 α 2l2 λ3 λ4 log α
+
B3 B4
2k1 +k2 +k3 +k4 α 3l4 λ1 . . . λ4 B1 B2 B3 B4
12
+
k1 k2 √ 2 2 + 2 +k3 +k4 α 5/2l3 λ1 λ2 λ3 λ4 log α
√
B1 B2 B3 B4
+ +
k3 k4 √ 2 2 + 2 +k1 +k2 α 5/2l3 λ1 λ2 λ3 λ4 log α
√
2
k1 2
k + 22
k + 23
k + 24
√
B3 B4 B1 B2
√
α 2l2 λ1 λ2 λ3 λ4 log α
B1 B2 B3 B4
.
Then letting the ki run from 0 to the bound in (17), we get from (16) and (20),
S (α , l, λ) ... k1
k4
|B | m4
...
k1
4
v∈ R (k), ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
k4
α)
i =1
Bi 22ki m
1 22k1 +2k2 +2k3 +2k4
1
5
1
v∈ B (k), ( v i ,α )=1 v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
α)
1
α 3l4 λ1 . . . λ4 + α 2 l3 B 1 B 2 λ1 λ2 λ3 λ4 log 2 m
m4
1 5 + α 2 l3 B 3 B 4 λ3 λ4 λ1 λ2 log 2 m + α 2l2 B 1 B 2 B 3 B 4 λ1 λ2 λ3 λ4 log m
=:
1 m4
( T 1 + T 2 + T 3 + T 4 ),
(21)
say. By the fundamental identity, Theorem 3, the contribution to f (x) with x∈ V m above by
m4 φ(α ) 1
φ(m)
α |m
α
2
l|(m/α )
l2
λ1 |(m/αl) (λ1 ,αl)=1
...
λ4 |(m/αl) (λ4 ,αl)=1
1
λ1 λ2 λ3 λ4
α > 1 is bounded
S (α , l, λ),
where the λi run through the square-free divisors of m/αl. Inserting the bound for | S (α , l, λ)| in (21) yields the upper bound
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
φ(α ) 1
1
φ(m)
=
α |m
1
φ(m)
α
2
l|(m/α )
l2
...
λ1 |(m/αl) (λ1 ,αl)=1
779
(T 1 + T 2 + T 3 + T 4 ) λ1 λ2 λ3 λ4
λ4 |(m/αl) (λ4 ,αl)=1
( S 1 + S 2 + S 3 + S 4 ),
(22)
say, where S i is the contribution corresponding to the value T i . Let ν (m) denote the number of distinct prime factors of m and τ ∗ (m/αl) the number of square-free divisors of m/αl that are relatively prime to αl. Then
S1
φ(α )α
α |m
l2 τ ∗ (m/αl)4
φ(α )α (m/α )2 m2 τ (m).
(23)
α |m
l|m/α
Similarly,
S2
1
B 1 B 2 log 2 m
√
α φ(α )
α |m
1
B 1 B 2 log 2 m
lτ ∗ (m/αl)4
l|m/α
√
α φ(α )
α |m
m
α
1+
p |(m/α )
17 p
φ(α ) m(log log m)17 log m B 1 B 2 √
α
α |m
m
3/2
17
(log log m)
log m
B1 B2
1
1+ √
p |m
p
m3/2 log m(log log m)17 τ (m) B 1 B 2 ,
S3
1
B 3 B 4 log 2 m
√
α φ(α )
α |m
(24)
lτ ∗ (m/αl)4
l|m/α
m3/2 log m(log log m)17 τ (m) B 3 B 4 ,
(25)
and
S4
B 1 B 2 B 3 B 4 log m
α |m
B 1 B 2 B 3 B 4 log m
16
ν (m)
φ(α )
τ ∗ (m/αl)4
l|m/α
φ(α )16ν (m/α ) τ (m/α )
α |m
B 1 B 2 B 3 B 4 m(log log m) log m.
By (22), (23), (24), (25) and (26) we see that the contribution to four terms appearing on the right-hand side of (13).
x∈ V m
(26) f (x) with
α > 1 is the first
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
5.3. The case α = 1
α = 1. In this case, | B (k)|
Finally we consider the contribution with
S (α , l, λ) ... k1
k4
|B | m4
...
k1
m4
...
k1
1 22k1 +2k2 +2k3 +2k4
i =1 (
22k1 +2k2 +2k3 +2k4
+ 1) and
Bi 22ki m
1
v∈ B (k) 4 k i +1 2 l λi
1
2ki +1 lλi Bi
Bi
i =1
+1
4 1
m4
(lλi + B i ).
i =1
l|m
l2
f (x) with
x∈ V m
m4 1
α)
k4
Thus by Theorem 3 the contribution to
φ(m)
i =1
v∈ R (k) v 1 v 2 λ3 λ4 ≡ v 3 v 4 λ1 λ2 (mod
k4
|B |
4
4
···
λ1 |m/l
φ(m)
l|m
l2
4 1
1
λ4 |m/l
4 1
1
α = 1 is bounded above by
λ1 λ2 λ3 λ4 m 4
(lλi + B i )
i =1
(l + B i /λi ),
i =1 λi |m/l
where the λi run through the square-free divisors of m/l. Letting τ ∗ (m/l), σ−∗ 1 (m/l) denote the number, and the sum of the reciprocals respectively, of the square-free divisors of m/l we see that the above is
1
4 1
φ(m)
l2
l|m
i =1
∗ lτ ∗ (m/l) + σ− 1 (m/l) B i .
The following estimates are readily obtained from the multiplicative properties of the sums
l2 τ ∗ (m/l)4 m2 ,
l|m
l|m
l|m
τ ∗ (m/l)2 σ−∗ 1 (m/l)2 4ν (m) τ (m)
1 l|m
l
∗ lτ ∗ (m/l)3 σ− 1 (m/l) m
τ
∗
(m/l)σ−∗ 1 (m/l)3
2ν (m)
p |m
1+
p |m
1+
p |m
1+
1 p
9
m(log log m)9 ,
p 1
2
p
3 1−
4ν (m) τ (m)(log log m)2 , 1 p
−1
2ν (m) (log log m)4 ,
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
1 l|m
l
σ ∗ (m/l)4 2 −1
1+
p |m
Using these estimates we see that the contribution to
1
φ(m)
m2 + m(log log m)9
1
4 (log log m)4 .
p
x∈ V m
f (x) with
α = 1 is bounded by
B i + 4ν (m) τ (m)(log log m)2
i
+ 2ν (m) (log log m)4
781
Bi B j
i< j
B i B j B k + (log log m)4 B 1 B 2 B 3 B 4 ,
i < j
the remaining terms on the right-hand side of (13). We observe that the first term m2 /φ(m) is dominated by mL τ (m) and so does not need to be included. 2 6. Proof of Theorem 1 We may assume 1 B < m. As noted in the Introduction, by the identity in (5), Theorem 1 is just a special case of Theorem 2 with B a box of the type I × I × I × I , and so our approach is to prove Theorem 2 for this case. Let I be the interval of points a x < a + B in Zm , and χ I the characteristic function for I , with Fourier expansion
χ I (x) =
m
a( y )em (xy ).
y =1
Here
a(0) =
B m
a( y ) = 1 sin π y B /m , m sin π y /m
,
for y = 0.
(27)
Let B denote the number of values in I relatively prime to m, so that
B =
ax
φ(m) B + θ τ (m), μ(d) = m
d|(x,m)
(28)
with |θ| 1. Let B = I × I × I × I and B the set of points in B with coordinates relatively prime to m, so that |B | = B 4 . Recalling that V is the set of integer solutions of x1 x2 ≡ x3 x4 (mod m) with (xi , m) = 1, 1 i 4, we have
B ∩ V =
1 χ I x1 x2 x− 3
axi
= a(0) B 3 +
a( y )
y =0
say. For the Main term, using a(0) =
1 em x1 x2 x− 3 y = Main + Error ,
axi
we deduce from (28) that
|B | B 3 3 B 3 τ (m) 2 − − a ( 0 ) B B = φ(m) φ(m) m B φ(m) B τ (m),
(29)
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
that is
Main =
|B | + O B 2 τ (m) . φ(m)
Thus we are left with proving that
|Error| 8ν (m) τ (m)(log m)3 (log log m)7 B 2 . If B < (log m)2 (log log m)5 τ (m) then we have trivially
|Error| B 3
a( y ) B 3 log m B 2 τ (m) log3 m(log log m)5 , y
and so we may assume that B > (log m)2 (log log m)5 τ (m). First observe that for any value of l, a+ B −1
em (lx) =
x=a (x,m)=1
μ(λ)
em (lλt ).
(30)
a+ B λ t < λ
λ|m
a
Letting B λ denote the number of integers that t runs through, we get
a + B −1 sin(π B lλ/m) λ em (lx) sin(π lλ/m) . x=a λ|m
(x,m)=1
Applying this estimate to the sum over x2 in the error term of (29) we have
|Error|
a( y ) λ|m y =0
1 sin(π B λ λ yx1 x− 3 /m) . −1 sin(π λ yx1 x3 /m) ax
(31)
i
(xi ,m)=1
We break the sum into two pieces E 1 + E 2 where E 1 is the sum over values of λ > B and E 2 the sum over λ B. For E 1 we use the trivial upper bound
1 sin(π B λ λ yx1 x− 3 /m) B λ 1, −1 sin(π λ yx1 x3 /m) to get
|E1| B2
a( y ) τ (m) B 2 log m. λ|m λ> B
y
−1 For E 2 , λ B, and so B λ λB + 1 2B λ . Putting l ≡ yx1 x3 (mod m/λ) we have
|E2|
λ|m |l| 2mλ λ B
0<| y |m/2
a( y ) sin(π B λ λl/m) sin(π λl/m)
axi
1.
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
783
We split the latter sum into E 2,0 + E 2,1 where E 2,0 is the contribution with l = 0 and E 2,1 the contribution with l = 0, and make use of the inequalities,
a( y ) = 1 sin(π B y /m) 1 min B , m , m sin(π y /m) m y
for 0 < | y | m/2,
and
sin(π B λ λl/m) min B , m , sin(π λl/m) λ lλ
for |λl| m/2.
If l = 0 the sum over x1 , x3 is zero unless (m/λ)| y, in which case it equals B 2 . Thus, putting y= m λ t we have
| E 2,0 |
m B 2 a λt λ B λ|m 0<|t | λ 2
B3
1 λ|m
λ
λ λ
0<|t | 2
tm
B3 m
τ (m) log m B 2 τ (m) log m.
(32)
Assume now that 0 < l 2mλ . Then any solution of the congruence x1 y ≡ x3l (mod m/λ) with (x1 , m) = (x3 , m) = 1 must have mλ y. Put d = ( y , l, m/λ), y = y /d, l = l/d. Since mλ y we have m d< m λ , that is, λ < d . Note that the congruence x1 y ≡ x3 l (mod m/λ) is equivalent to x1 y ≡ x3 l (mod m/λd) and that any solution of the latter congruence with (x1 , m) = (x3 , m) = 1, must also have ( y , m/λd) = (l , m/λd) = 1. Thus the contribution to E 2 with l = 0 is bounded by
1
| E 2,1 |
m
min B ,
m d|m λ|(m/d) 0<|l | m 0<| y | 2d 2λd λ
×
m
min
d| y |
B
,
m
λ l λd
1.
axi
Let y = u + λmd t, with |u | m/2λd, and |t | λ2 . Next, for any nonnegative i , j let
Ri j =
x1 , u , x3 , l : a x1 , x3 < a + B , δi
m dB
|u | < 2
i
m dB
, δj
m dB
|l | < 2
j
m
dB
and
Bi j =
x1 , u , x3 , l : a x1 , x3 < a + B , |u | < min 2
i
m
,
m
dB 2λd
, |l | < min 2
j
m
,
m
dB 2λd
,
i where δi = 2i −1 for i > 0, δ0 = 0. Since |u | 2m λd we may assume λ < B /2 . If t = 0 then | y | = |u |, m while if t = 0 then | y | 2λd |t |. Thus, for any point (x1 , u , x3 , l ) ∈ Ri j satisfying the congruence
x1 u ≡ x3l
(mod m/λd),
(33)
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T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
we have,
min B ,
|t |λ/2
m
min B ,
d| y |
m
+λ
du
1
0<|t |λ/2
t
B 2i
log m.
Let N i j denote the number of points in Ri j satisfying (33) with coordinates relatively prime to m/λd. Then
| E 2,1 |
1 B m
d|m λ|(m/d) λ
B2
=
m
log m
i
log m
2i
j
B 2 jλ
Ni j
1 2− i − j N i j . λ d|m λ|(m/d) λ
i
(34)
j
The dimensions B i of the box Bi j , satisfy
B1 = B,
B2
2i m dB
B3 = B,
,
B4
2 jm dB
.
Put L = log log(m). Suppose first that B < m/λd. Then we can apply Proposition 1 to get
Ni j
m dλ
+ BL
10
+
m
τ (m/dλ) L +
dλ
mL 10 dB
i
2 +2
j
m
d
i
+ 2ν (m/dλ) L 5 B2i λ + B2 j λ +
m
2i + j λ + λ
dB
m d
dB 2
m
j
16ν (m/dλ) (log m) L 2 2 2 + 2
λd 2i + j mλ + 4ν (m/λd) τ (m/λd) L 3 2i λ + 2 j λ + B 2 +
Putting
L 18 log mτ (m/dλ) 2i /2 + 2 j /2 + 1/2
L 5 2i + j .
ν = ν (m) and τ = τ (m) and noting that λ < B < m/λd gives Ni j
m dλ
+
τL + m
λd
L
dλ
10
+ 2ν L 5
m
+ m d
m
L 18 log mτ 2i /2 + 2 j /2 + 1/2 mL 10 dB 2i +
m d
i
2 +2 2j +
j
d
j
i
16ν (log m) L 2 2 2 + 2
m m m m 1 + 4ν τ L 3 2 i + 2j + + 2i + j
m i+ j 2 d
λd
+λ
m d
λd
λd
L 5 2i + j .
d
B
(35)
If B > λmd then we have a trivial bound for N i j obtained by letting x1 , u , l be arbitrary values, thus B + 1 choices for x3 , and so determining at most m/λ d
Ni j B i
j
2i m dB
+1
2 jm dB
+1
B m/λd
+1 .
If either 2dBm < 1 or 2dBm < 1 then N i j = 0 for this would require u = 0 or l = 0 by the definition of Bi j , but then u or l would not be relatively prime to m/λd since in the sum for E 2,1 , m/λd > 1.
T. Cochrane, S. Shi / Journal of Number Theory 130 (2010) 767–785
785
Thus it follows that N i j 2i + j mdλ , which is less than the final term in (35). Summing over i , j and noting that i , j run through intervals of length at most log2 m, we get
i
2− i − j N i j
j
m d
τL λ
L 18
+√
+ 4ν τ L 3
λ
log mτ + 16ν (log m) L 2 +
log m
λ
+
log2 m B
L 10
λ
+
L 10 B
log m
+ 2ν L 5 log2 m + λ L 5 log2 m .
Summing over λ, d in (34) then yields
| E 2,1 | B 2 log m τ L 2 + L 19 τ log m + 16ν L 4 log m + L 11 + L 12 log m/ B + 4ν L 4 τ log m + 4ν τ L 5 log2 m/ B + 2ν L 7 log2 m + L 6 τ log2 m . Finally, using the assumption B > τ L 5 log2 m we have
| E 2,1 | B 2 log m 16ν L 4 log m + 4ν L 4 τ log m + 2ν L 7 log2 m + L 6 τ log2 m , completing the proof. References [1] A. Ayyad, T. Cochrane, Z. Zheng, The congruence x1 x2 ≡ x3 x4 (mod p), the equation x1 x2 = x3 x4 , and mean values of character sums, J. Number Theory 59 (2) (1996) 398–413. [2] D.A. Burgess, Mean values of character sums, Mathematika 33 (1) (1986) 1–5. [3] T. Cochrane, Z. Zheng, High order moments of character sums, Proc. Amer. Math. Soc. 126 (4) (1998) 951–956. [4] J.B. Friedlander, H. Iwaniec, The divisor problem for arithmetic progressions, Acta Arith. 45 (3) (1985) 273–277. [5] M.Z. Garaev, V.C. Garcia, The equation x1 x2 = x3 x4 + λ in fields of prime order and applications, J. Number Theory 128 (9) (2008) 2520—2537 (English summary). [6] G. Harman, Diophantine approximation with square-free integers, Math. Proc. Cambridge Philos. Soc. 95 (1984) 383–388. [7] H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53, Amer. Math. Soc., Providence, RI, 2004. [8] H.L. Montgomery, R.C. Vaughan, Mean values of character sums, Canad. J. Math. 31 (3) (1979) 476–587. [9] I.E. Shparlinski, Bounds of incomplete multiple Kloosterman sums, J. Number Theory 126 (1) (2007) 68–73. [10] I.E. Shparlinski, On the distribution of points on multidimensional modular hyperbolas, Proc. Japan Acad. Ser. A Math. Sci. 83 (2) (2007) 5–9. [11] I.E. Shparlinski, On a generalised Lehmer problem for arbitrary powers, East–West J. Math. (2008) 197–204 (special volume). [12] R.C. Vaughan, Diophantine approximation by prime numbers, III, Proc. London Math. Soc. 33 (1976) 177–192.