Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

0.54 MB·

by  Khang Tran

Pair correlation of roots of rational functions with rational
generating functions and quadratic denominators
Khang Tran
6 Department of Mathematics
1
University of Illinois at Urbana-Champaign
0
2
Alexandru Zaharescu

n
Department of Mathematics
a
J University of Illinois at Urbana-Champaign

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1

Dedicated in honor of Mourad Ismail and Dennis Stanton
]
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Abstract
C
. ForanyrationalfunctionswithcomplexcoefficientsA(z),B(z)andC(z),whereA(z),C(z)
h
are not identically zero, we consider the sequence of rational functions H (z) with generating
m
at function(cid:80)Hm(z)tm =1/(A(z)t2+B(z)t+C(z)). Weprovideanexplicitformulaforthelim-
m itingpaircorrelationfunctionoftherootsof(cid:81)n H (z),asn→∞,countingmultiplicities,
m=0 m
on certain closed subarcs J of a curve C where the roots lie. We give an example where the
[
limiting pair correlation function does not exist if J contains the endpoints of C.

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v
1
8
1 Introduction
3
4
0 The root distribution of certain sequences of rational functions with complex coefficients has some
.
1 peculiar connections with the discriminant of the denominator of its generating function. In par-
0 ticular, one of the authors [22] showed that if the generating function of a sequence of polynomials
6
is the reciprocal of a quadratic or cubic polynomial, then the roots of the corresponding sequence
1
of polynomials form a dense set on an explicit algebraic curve. Moreover, in several cases, the
:
v endpoints of the curve are the roots of the discriminant of the denominator. For the root distribu-
i
X tion of various sequences of polynomials and their generating functions, see Figure 1, Figure 2 and
Figure 3 below. For other connections between discriminants and the root distribution, see [23].
r
a Our goal here is to study the pair correlation of the roots of a sequence of rational functions when
the denominator of the generating function is quadratic. Pair correlation is one of the important
statistics concerned with the local spacing distribution of a sequence, and there are many results
MathematicsSubjectClassification2010: 11B05,11K99.
Keywordsandphrases: paircorrelation,rootdistribution,generatingfunctions.
The first author acknowledges support from NSF grant DMS-0838434 ”EMSW21MCTP: Research Experience
forGraduateStudents”.
Thesecondauthor’sresearchwaspartiallysupportedbyNSFgrantDMS-0901621.
1
in the literature on the local spacing distribution of a variety of sequences of interest in number
theory (see [1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21]).
Let T denote the set of triples (A(z),B(z),C(z)) with A(z),B(z),C(z) rational functions of z
with complex coefficients, A(z) and C(z) not identically zero. For each (A(z),B(z),C(z)) in T we
consider the sequence of rational functions H (z) with generating function given by
m
∞
(cid:88)
H (z)tm =1/(A(z)t2+B(z)t+C(z)). (1)
m
m=0
It is easy to see that there exists a finite set of points in the complex plane which contains all the
poles of all the rational functions H (z), m = 0,1,... (the multiplicity of the poles may increase
m
as m increases). By contrast, if we consider all the zeros of all H (z), we obtain an infinite set of
m
complex numbers which (except in some degenerate cases) is dense on a certain curve C which will
be described below. This naturally raises the question of how these points are distributed along C.
Our goal is to study the pair correlation of the roots of (cid:81)n H (z), counting multiplicities.
m=0 m
The rational function B2(z)/A(z)C(z) plays an important role in our investigation. In this con-
nection, let us consider the equivalence relation ∼ on T defined such that two elements of T are
equivalent if and only if the corresponding rational function B2(z)/A(z)C(z) is the same. We will
say that an element (A(z),B(z),C(z)) of T is in standard form provided C(z)=1, A(z) and B(z)
arepolynomials,andthereisnorootofB(z)whichissimultaneouslyalsoadouble(orhigherorder)
root of A(z). The last condition above can be restated as saying that the greatest common divisor
(B2(z),A(z)) of B2(z) and A(z) is square free. As examples, the triples (A(z),B(z),C(z)) used in
Figures1and2belowareinstandardform, whiletheoneusedinFigure3isnotinstandardform,
since in that example (B2(z),A(z)) = z2. It is easy to see that in every equivalence class there is
exactlyoneelementofT instandardform. Indeed,noticefirstthatforany(A(z),B(z),C(z))inT,
(A(z),B(z),C(z)) and (A(z)/C(z),B(z)/C(z),1) belong to the same equivalence class. Also, for
any nonzero rational function E(z), the triple (E2(z)A(z)/C(z),E(z)B(z)/C(z),1) belongs to the
same equivalence class. Here one can choose E(z) such that (E2(z)A(z)/C(z),E(z)B(z)/C(z),1)
is in standard form. For instance, choose first a polynomial E(z) such that both E2(z)A(z)/C(z)
and E(z)B(z)/C(z) are polynomials. If E(z)B(z)/C(z) has a root α which is also a multiple root
of E2(z)A(z)/C(z), divide E(z) by z −α. Then continue this procedure until all such roots are
eliminated. It follows that any equivalence class contains at least one triple in standard form.
Next, if (A (z),B (z),C (z)) ∼ (A (z),B (z),C (z)) and both triples are in standard form, then
1 1 1 2 2 2
C (z) = C (z) = 1 and B2(z)/A (z) = B2(z)/A (z). Therefore B2(z) divides B2(z)A (z), and
1 2 1 1 2 2 1 2 1
since (B2(z),A(z)) is square free, this forces B (z) to divide B (z). Similarly, B (z) divides B (z),
1 2 2 1
so B (z) = B (z) and then A (z) = A (z). In conclusion, each equivalence class contains exactly
1 2 1 2
onetripleinstandardform. Letusalsoremarkthat, givenanarbitrarytriple(A(z),B(z),C(z))in
T,theprocessoffindingtheuniquetripleinitsequivalenceclasswhichisinstandardformdescribed
above also provides us with a clear understanding of how the roots of the corresponding rational
functionsH (z)change. Moreprecisely,replacing(A(z),B(z),C(z))by(A(z)/C(z),B(z)/C(z),1)
m
has the effect of multiplying each H (z) by the fixed rational function C(z). Moreover, replacing
m
(A(z)/C(z),B(z)/C(z),1)by(E2(z)A(z)/C(z),E(z)B(z)/C(z),1)hasthesameeffectasreplacing
t by E(z)t, that is, has the effect of multiplying each H (z) by E(z)m. In conclusion, the distribu-
m
tion of zeros of the rational functions H (z) corresponding to a given triple (A(z),B(z),C(z)) in
m
T is exactly the same as the one obtained by replacing (A(z),B(z),C(z)) by the unique triple in
standardforminitsequivalenceclass, exceptatthelocationoffinitelymanypointsinthecomplex
plane (where the zeros and poles of C(z) and E(z) above lie).
2
Taking into account the above discussion, we restrict ourselves in what follows to study the
pair correlation of zeros of (cid:81)m H (z) for triples (A(z),B(z),C(z)) which are in standard form.
k=0 k
Note that in this case all the H (z) are polynomials, and H (z)=1, so we may restrict the above
m 0
producttok ≥1. WeconsiderthepaircorrelationproblemonanarbitrarysubarcJ ofthecurveC
onwhichtherootsofH (z)lie. Thisrestrictionsometimesimpliesanexclusionoftheendpointsof
m
C from J. We will see an explicit example in Section 3 where the limiting pair correlation function
exists on any proper subinterval which does not contain these endpoints. This function does not
existiftheseendpointsbelongtoJ. Thepaircorrelationfunctionisexplicitlygiveninthefollowing
theorem.
Theorem 1 Let A(z) and B(z) be polynomials in z with complex coefficients, A(z) not identically
zero, such that the greatest common divisor of B2(z) and A(z) is square free, and consider the
polynomials H (z) with generating function
m
∞
(cid:88) 1
H (z)tm = .
m A(z)t2+B(z)t+1
m=0
Then the set of roots of all the H (z) is dense on a fixed curve C. Also, let
m
B2(z)
h(z)=
A(z)
and let J be a closed subarc of C such that h(z) is piecewise continuously differentiable on J and
the inverse function exists. The function
f(t):=h−1(4cos2πt)
maps a subinterval I ⊂ [0,4] onto J. If f(cid:48)(z) (cid:54)= 0 on I then the limiting pair correlation function
of the roots of (cid:81)m H (z), counting multiplicities, exists on J, as m→∞, and is given by
k=1 k
ˆ
(cid:18) (cid:19)
l(J) l(J) dt
g (x)= g x , (2)
J |I|2 I |I||f(cid:48)(t)| |f(cid:48)(t)|
I
where
6 (cid:88) 2x
g (x)= σ(k)log (3)
I π2x2 k
1≤k≤2x
and σ is the sum of divisors function.
Thefirstpartofthetheorem, statingthatthesetofrootsofallH (z)isdenseonafixedcurve
m
C has been established in [22]. For the sake of completeness, we will present a proof in Section
2. The proof of the main part of the theorem involving the study of pair correlation is given in
Section 3. The methodology of the proof is similar to that used in [2], and with the one employed
in [1], where a more elaborate argument is needed since it addresses the effect of addition on pair
correlation functions concerning fractions of bounded height. Section 4 provides an example of the
pair correlation function for a specific sequence of polynomials.
3
2 Distribution of roots on a fixed curve
In this section we consider the root distribution of the sequence of polynomials H (z). We recall
m
the q-analogue of the discriminant, a very useful concept introduced by Mourad Ismail [13]. The
q-discriminant of a polynomial P(x) of degree n with the leading coefficient p is
(cid:89)
Disc (P(x);q)=p2n−2qn(n−1)/2 (q−1/2x −q1/2x )(q1/2x −q−1/2x )
x i j i j
1≤i<j≤n
where x ,...,x are roots of P(x). This q-discriminant equals 0 if and only x /x = q for some
1 n i j
roots x ,x . In the special case when q →1, this q-discriminant gives the ordinary discriminant of
i j
a polynomial. The following theorem was established in [22].
Theorem 2 Let
1 (cid:88)
= H (z)tm,
A(z)t2+B(z)t+1 m
where A(z)(cid:54)=0. The roots of H (z) are dense on the curve C defined by the conditions
m
B2(z)
(cid:61) = 0
A(z)
and
B2(z)
0≤(cid:60) ≤4
A(z)
in the complex plane.
Proof Let z be a root of H (z) with A(z) (cid:54)= 0. Let t = t (z) and t = t (z) be the roots of
m 1 1 2 2
A(z)t2+B(z)t+1. If t =t then z ∈C since B2(z)=4A(z). We consider t (cid:54)=t . Using partial
1 2 1 2
fractions, we have
1 1
=
A(z)t2+B(z)t+1 A(z)(t−t )(t−t )
1 2
1 (cid:88) tm+1−tm+1
= 1 2 tm. (4)
A(z) (t −t )tm+1tm+1
1 2 1 2
So the roots of H (z) are the roots of t =qt , where q is an (m+1)-th root of unity and q (cid:54)=1,
m 1 2
A(z)(cid:54)=0. These roots are the roots of the q-analogue of discriminant
Disc (A(z)t2+B(z)t+1;q)=q(cid:0)B2(z)−(q+q−1+2)A(z)(cid:1).
t
Hence
B2(z)
= q+q−1+2
A(z)
= 2(cid:60)q+2
Thus z ∈C since q is an (m+1)-th root of unity.
To show the density of the roots of H (z), we let ζ ∈ C and U be an open neighborhood of ζ
m
such that the rational function B2(z)/A(z) is analytic on U. Since the set of (m+1)-th roots of
4
unity is dense on the unit circle as m→∞, the set of values 2(cid:60)q+2 is dense on the interval [0,4]
where q is an (m+1)-th root of unity. By the open mapping theorem, the map B2(z)/A(z) maps
U to an open set containing some point 2(cid:60)q+2 for large m. Thus there is some point z ∈U such
that
B2(z)
=q+q−1+2
A(z)
or
Disc (A(z)t2+B(z)t+1;q)=0.
t
This implies that q is the quotient of the two roots of A(z)t2 +B(z)t+1. Hence z is a root of
H (z) by (4).
m
The root distribution of various sequences of polynomials with their generating functions are
given by figures below.
1.5
1.0
0.5
0.5 1.0 1.5 2.0 2.5 3.0
(cid:45)0.5
(cid:45)1.0
(cid:45)1.5
Figure 1: Roots of H (z) with (cid:80)H (z)tm =1/(z2t2+(z2−2z+3)t+1)
30 m
2
1
0.5 1.0 1.5 2.0
(cid:45)1
(cid:45)2
Figure 2: Roots of H (z) with (cid:80)H (z)tm =1/(z2t2+(z2−2z+5)t+1)
30 m
5
1.0
0.5
1 2 3 4
(cid:45)0.5
(cid:45)1.0
Figure 3: Roots of H (z) with (cid:80)H (z)tm =1/(z2t2+(z2−2z)t+1)
30 m
In the next section, we will analyze the pair correlation of the roots of (cid:81)n H (z), counting
m=1 m
multiplicities.
3 Pair correlation on the corresponding curve
In this section, we will prove Theorem 1. In the proof of Theorem 2, the map h(z) maps a
root of H (z) to q +q−1 +2 where q is a (m+1)-th root of unity. So it is essential to con-
m
sider the pair correlation function of the sequence of finite sequences M(Q) = {x ,...,x } =
1 N
{1/2,1/3,2/3,1/4,2/4,3/4,1/5,2/5,3/5,4/5,...} on a subinterval I = [a,b] of (0,1). Each finite
sequence is a sequence of N = Q(Q − 1)/2 numbers p/q with 1 ≤ p < q and q ≤ Q. Let
M (Q)=M(Q)∩I. The number of points N (Q) in the interval I is
I I
(cid:88) (cid:88)
N (Q) = 1
I
q≤Qqa≤p≤qb
(cid:88)
= q(b−a)+O(1)
q≤Q
(b−a)Q2
= +O(Q)
2
= N(b−a)+O(Q).
We consider the quantity
(cid:26) (cid:27)
1 λ
R (λ) = # (x ,x )∈M (Q):i(cid:54)=j,0<|x −x |≤ .
MI(Q) 2N(b−a) i j I i j N
Let the limiting pair correlation measure R (λ) = lim R (λ) if the limit exists. If
MI Q→∞ MI(Q)
moreover R (λ) can be written in the form
MI(Q)
ˆ
λ
R (λ)= g (x)dx,
MI I
0
the function g (λ) is called the pair correlation function associated to the given sequence of finite
I
sets (M (Q)) . We will need the following lemma. The proof of this lemma is similar to that of
I Q∈N
Lemma 1 in [2].
6
Lemma 3 For any subinterval I ⊂(0,1), the pair correlation function of the sequence of finite sets
(M (Q)) exists and is given by
I Q∈N
6 (cid:88) 2x
g (x)= σ(k)log
I π2x2 k
1≤k≤2x
for any x>0.
Proof Let G and H be smooth functions defined on R with SuppG⊂(0,1) and SuppH ⊂(0,Λ).
Here SuppG denotes the closure in R of the set of points where G is nonzero, and similarly for
SuppH. Let
(cid:88)
h(y) = H(N(y+n)),
n∈Z
(cid:88)
g(y) = G(y+n)
n∈Z
and
(cid:88)
S = h(x−y)g(x)g(y).
x,y∈M(Q)
Wenotethatthesummationsinthedefinitionsofh(y)andg(y)containonlyfinitelymanynonzero
terms. This allows us to interchange summations in the subsequent computations. The functions
h(y), g(y) have Fourier series expansions,
(cid:88)
h(y)= c e(ny)
n
n∈Z
and
(cid:88)
g(y)= a e(ny).
n
n∈Z
By substituting these Fourier expansions into the formula of S, we obtain
(cid:88) (cid:88) (cid:88) (cid:88)
S = c e(m(x−y)) a e(nx) a e(ry)
m n r
x,y∈M(Q) m n r
(cid:88) (cid:88) (cid:88)
= c a a e((m+n)x) e((r−m)y)
m n r
m,n,r x∈M(Q) y∈M(Q)
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) (cid:88) (m+n)p (cid:88) (cid:88) (r−m)p
= c a a e e
m n r q q
m,n,r q≤Q1≤p<q q≤Q1≤p<q
 
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:88)  (cid:88) (cid:88) (m+n)p (cid:88) (cid:88) (m+n)p 
= cmanar e q + e q 
m,n,r q≤Q 1≤p<q q≤Q 1≤p<q
q|m+n q(cid:45)m+n
 
(cid:18) (cid:19) (cid:18) (cid:19)
 (cid:88) (cid:88) (r−m)p (cid:88) (cid:88) (r−m)p 
× e + e .
 q q 
q≤Q 1≤p<q q≤Q 1≤a<q
q|r−m q(cid:45)r−m
7
Using the fact that for any integer l,
(cid:88) (cid:18)lp(cid:19) (cid:40)0 if q (cid:45)p
e = ,
q q if q|p
1≤p≤q
we obtain
  
(cid:88)  (cid:88) (cid:88)  (cid:88) (cid:88) 
S = cmanar (q−1)− 1 (q−1)− 1
  
m,n,r q≤Q q≤Q q≤Q q≤Q
q|m+n q(cid:45)m+n q|r−m q(cid:45)r−m
  
(cid:88)  (cid:88)  (cid:88) 
= cmanar q−Q q−Q
  
m,n,r q≤Q q≤Q
q|m+n q|r−m
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
= c a a q q−Q c a a q
m n r m n r
m,n,r q≤Q q≤Q m,n,r q≤Q
q|m+n q|r−m q|m+n
(cid:88) (cid:88) (cid:88)
−Q c a a q+Q2 c a a .
m n r m n r
m,n,r q≤Q m,n,r
q|r−m
By the Poisson summation formula (see [7, Theorem 2.2, page 37] or [10] for a generalized
version), we have
(cid:88) (cid:88)
a = g(r)=0
r
r∈Z r∈Z
since SuppG⊂(0,1). With this fact, S is reduced to
(cid:88) (cid:88) (cid:88)
S = c a a q q.
m n r
m,n,r q≤Q q≤Q
q|m+n q|r−m
With the change in the index of summation given by m(cid:48) = m+n, n(cid:48) = r−m and m = r(cid:48), we
obtain
(cid:88) (cid:88) (cid:88)
S = c a a q q
r(cid:48) m(cid:48)−r(cid:48) n(cid:48)+r(cid:48)
m(cid:48),n(cid:48),r(cid:48) q≤Q q≤Q
q|m(cid:48) q|n(cid:48)
(cid:88) (cid:88)
= q q c a a
1 2 r(cid:48) m(cid:48)−r(cid:48) n(cid:48)+r(cid:48)
q1,q2≤Q m(cid:48),n(cid:48),r(cid:48)
q1|m(cid:48),q2|n(cid:48)
(cid:88) (cid:88) (cid:88) (cid:88)
= q q c a a .
1 2 r(cid:48) mq1−r(cid:48) nq2+r(cid:48)
q1,q2≤Q r(cid:48)∈Z m n
8
The formula of the coefficients of the Fourier series of G(t) gives
ˆ
a = G(t)e(−(mq −r)t)dt
mq1−r 1
ˆR
= G(t)e(rt)e(−mq t)dt
1
ˆR
(cid:18) (cid:19) (cid:18) (cid:19)
1 t rt
= G e e(−mt)dt.
q q q
R 1 1 1
By the Poisson summation formula,
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) 1 m rm
a = G e .
mq1−r q q q
m m∈Z 1 1 1
Following a similar procedure, one obtains
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) 1 n −rn
a = G e .
nq2+r q q q
m n∈Z 2 2 2
Applying these two equations to the formula of S, we have
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) (cid:88) 1 m rm (cid:88) 1 n −rn
S = q q c G e G e
1 2 r q q q q q q
q1,q2≤Q r∈Z m∈Z 1 1 1 n∈Z 2 2 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18)(cid:18) (cid:19) (cid:19)
(cid:88) (cid:88) m n (cid:88) m n
= G G c e − r
q q r q q
q1,q2≤Qm,n∈Z 1 2 r 1 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
(cid:88) (cid:88) m n (cid:88) m n
= G G H N r+ − .
q q q q
q1,q2≤Qm,n∈Z 1 2 r 1 2
SinceSuppG⊂(0,1),SuppH ⊂(0,Λ)andN ∼Q2/2,wehavethatifr (cid:54)=0thenG(m/q )G(n/q )H(N(r+
1 2
m/q − n/q )) = 0 when N is large. Let δ = (q ,q ), q = d δ and q = d δ. Let a,b sat-
1 2 1 2 1 1 2 2
isfy ad +bd = 1 with 0 < b < d . We consider a change in the index of summation given by
1 2 1
m=bm(cid:48)+d n(cid:48) and n=−am(cid:48)+d n(cid:48). This yields
1 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) bm n −am n Nm
S = G + G + H
d δ δ d δ δ d d δ
d1δ,d2δ≤Qm,n∈Z 1 2 1 2
(d1,d2)=1
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) 1 bm 1 bm m Nm
= G +n G +n− H .
δ d δ d d d d d δ
d1δ,d2δ≤Qm,n∈Z 1 1 1 2 1 2
(d1,d2)=1
Since SuppH ⊂(0,Λ) and N ≥(1−(cid:15))Q2/2 for large Q, the summand is nonzero only if
mδ ≤2Λ/(1−(cid:15)).
By choosing (cid:15) sufficiently small, we have
m≤2Λ/δ.
9
This inequality and the fact that SuppG ⊂ (0,1) imply that the summand is nonzero for finitely
many values of n, which satisfy
(cid:18) (cid:19)
1 bm
0< +n <1.
δ d
1
From the fact that
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
1 bm m 1 bm m
G +n− =G +n +O ,
δ d d d δ d δd d
1 1 2 1 1 2
we have
(cid:88) (cid:88) (cid:18)1(cid:18)bm (cid:19)(cid:19)2 (cid:18) Nm (cid:19)
S = G +n H +O(log2Q)
δ d d d δ
1 1 2
d1δ,d2δ≤Qmδ≤2Λ,n
(d1,d2)=1
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) b Nm
= f H
n d d d δ
1 1 2
mδ≤2Λ,nd1,d2≤Q/δ
(d1,d2)=1
where
(cid:18)1 (cid:19)2
f (x):=G (mx+n) .
n δ
Let K be a large positive integer whose value (as a function of Q) will be given later. The formula
of S above becomes
K (cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) (cid:88) b Nm
S = f H +O(log2Q)
n d d d δ
1 1 2
mnδ∈≤Z2Λi=1 d(1d,1d,2d≤2)Q=/1δ
b∈[d1i/K,d1(i+1)/K]
K (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
(cid:88) (cid:88) (cid:88) i 1 Nm
= f +O H +O(log2Q)
n K K d d δ
1 2
mnδ∈≤Z2Λi=1 d(1d,1d,2d≤2)Q=/1δ
b∈[d1i/K,d1(i+1)/K]
(cid:88) (cid:88)K (cid:18) i (cid:19) (cid:88) (cid:18) Nm (cid:19) (cid:18)Q2(cid:19)
= f H +O +O(log2Q).
n K d d δ K
1 2
mnδ∈≤Z2Λi=1 d(1d,1d,2d≤2)Q=/1δ
b∈[d1i/K,d1(i+1)/K]
Wenotethatifmin(d ,d )≤Q1−(cid:15) thenthemaintermis0forlargeQ. Ifmin(d ,d )>Q1−(cid:15) then
1 2 1 2
the chain rule implies that (cid:107)DH(cid:107) (cid:28)1/Q1−3(cid:15). Then Lemma 8 in [4] gives
∞
¨
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:88) Nm 6 Nm
H = H dxdy+O(Q3/2+(cid:15))
d d δ π2K xyδ
1 2 Q1−(cid:15)≤x,y≤Q/δ
d1,d2≤Q/δ
(d1,d2)=1
b∈[d1i/K,d1(i+1)/K] ¨
6 (cid:18)Q2m(cid:19)
= H dxdy+O(Q3/2+(cid:15))
π2K 2xyδ
¨Q1−(cid:15)≤x,y≤Q/δ
6Q2 (cid:18) m (cid:19)
= H dxdy+O(Q3/2+(cid:15)).
π2K 2xyδ
Q−(cid:15)≤x,y≤1/δ
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