Extreme values of class numbers of real quadratic fields

0.17 MB·English

by  Youness Lamzouri

EXTREME VALUES OF CLASS NUMBERS OF REAL QUADRATIC
FIELDS
YOUNESS LAMZOURI
5
Abstract. WeimprovearesultofH.L.MontgomeryandJ.P.Weinbergerbyestab-
1
lishingtheexistenceofinfinitelymanyfundamentaldiscriminantsd>0forwhichthe
0
2 classnumberoftherealquadraticfieldQ(√d)exeeds(2eγ+o(1))√d(loglogd)/logd.
We believe this bound to be best possible. We also obtain upper and lower bounds
b
of nearly the same order of magnitude, for the number of real quadratic fields with
e
F discriminant d x which have such an extreme class number.
≤

6

]
T
N 1. Introduction
.
h An important problem in number theory is to understand the size of the class
t
a
number of an algebraic number field. The case of a quadratic field has a long history
m
goingback to Gauss. Let d beafundamental discriminant andh(d) betheclass number
[

of the field Q(√d). When d < 0, in which case Q(√d) is imaginary quadratic, J. E.
2
v
Littlewood [7] established, assuming the Generalized Riemann Hypothesis GRH, that
3
0 2eγ
0 (1.1) h(d) +o(1) d loglog d ,
1 ≤ π | | | |
0 (cid:18) (cid:19)
p
. where γ is the Euler-Mascheroni constant. Littlewood used Dirichlet’s class number
1
0 formula, which for d < 4, asserts that
5 −
1 d
: (1.2) h(d) = | | L(1,χ ),
v d
π ·
i p
X
where χ = d is the Kronecker symbol. He then deduced (1.1) from the bound
d
r
a ·
(1.3) (cid:0) (cid:1) L(1,χ ) (2eγ +o(1))loglog d ,
d
≤ | |
which he obtained under GRHfor all fundamental discriminants d. Furthermore, under
thesamehypothesis, Littlewood[7]provedthatthereexist infinitelymanyfundamental
discriminants d (both positive and negative) for which
(1.4) L(1,χ ) (eγ +o(1))loglog d ,
d
≥ | |
and hence for those d < 0, one has
eγ
(1.5) h(d) +o(1) d loglog d ,
≥ π | | | |
(cid:18) (cid:19)
p
2010 Mathematics Subject Classification. Primary 11R11, 11M20.
1
2 YOUNESSLAMZOURI
by the class number formula (1.2). The omega result (1.4) was later established uncon-
ditionally by S. Chowla [1].
The case of a real quadratic field is notoriously difficult, due to the presence of
non-trivial units in Q(√d). For example, Gauss’s conjecture that there are infinitely
many positive discriminants d for which h(d) = 1 is still open. When d is positive, the
class number h(d) is heavily affected by the size of the regulator R of Q(√d). In this
d
case, Dirichlet’s class number formula asserts that
√d
(1.6) h(d) = L(1,χ ).
d
R ·
d
Recall that R = logε where ε is the fundamental unit of the quadratic field Q(√d),
d d d
defined as ε = (a+ b√d)/2, where b > 0 and a is the smallest positive integer such
d
that (a,b) is a solution to the Pell equations m2 dn2 = 4. Since ε > √d/2, it
d
− ±
follows that when d is large we have
1
(1.7) R +o(1) logd,
d
≥ 2
(cid:18) (cid:19)
and hence by Littlewood’s bound (1.3) we deduce that
loglogd
(1.8) h(d) 4eγ +o(1) √d .
≤ · logd
(cid:0) (cid:1)
for all positive fundamental discriminants d, under the assumption of GRH. In 1977, H.
L. Montgomery and J. P. Weinberger [8] showed that this bound cannot be improved,
apart from the value of the constant. Indeed, they proved that there exist infinitely
many real quadratic fields Q(√d) such that
loglogd
(1.9) h(d) √d .
≫ · logd
Recently, W. Duke investigated generalizations of this result to higher degree number
fields. In[3],heobtainedthecorresponding omegaresult fortheclassnumber ofabelian
cubic fields, while in [2], assuming certain hypotheses (including GRH), he obtained
similar results in the case of totally real number fields of a fixed degree whose normal
closure has the symmetric group as Galois group.
It is widely believed that the true nature of extreme values of L(1,χ ) is given
d
by the omega result (1.4) rather than the GRH bound (1.3). A. Granville and K.
Soundararajan [5] investigated the distribution of large values of L(1,χ ) and their
d
results give strong support to this conjecture. In view of (1.7) and the class number
formula (1.6), this leads to the following conjecture
Conjecture 1.1. For all large positive fundamental discriminants d we have
loglogd
(1.10) h(d) 2eγ +o(1) √d .
≤ · logd
(cid:0) (cid:1)
EXTREME VALUES OF CLASS NUMBERS OF REAL QUADRATIC FIELDS 3
In this paper, we prove the existence of infinitely many real quadratic fields Q(√d)
for which the class number h(d) is as large as the conjectured upper bound (1.10). We
also obtain upper and lower bounds of nearly the same order of magnitude, for the
number of real quadratic fields with discriminant d x for which the class number is
≤
that large.
Theorem 1.2. Let x be large.
(a) There are at least x1/2 1/loglogx real quadratic fields Q(√d) with discriminant
−
d x, such that
≤
loglogd
(1.11) h(d) (2eγ +o(1))√d .
≥ · logd
(b) Furthermore, there are at most x1/2+o(1) real quadratic fields Q(√d) with dis-
criminant d x, for which (1.11) holds.
≤
To prove the omega result (1.9), Montgomery and Weinberger worked over the
following special family of fundamental discriminants, first studied by Chowla
:= d square-free of the form d = 4n2 +1 for some n 1 .
D { ≥ }
This family has the advantage that the regulator R is as small as possible. Indeed, if
d
d = 4n2 + 1 , then the fundamental unit of Q(√d) is ǫ = 2n+ √d 2√d, and
d
∈ D ≤
hence R = (1/2+o(1))logd. Therefore, the class number formula (1.6) implies that
d
√d
(1.12) h(d) = (2+o(1)) L(1,χ )
d
logd ·
for d . Montgomery and Weinberger showed that there exist infinitely many d
∈ D ∈ D
for which L(1,χ ) loglogd, from which they deduced (1.9).
d
≫
Let (x) denote the set of d such that d x. To establish the first part of
D ∈ D ≤
Theorem 1.2, we prove that there exist many fundamental discriminants d (x) for
∈ D
which L(1,χ ) is as large as one could hope for, namely (eγ + o(1))loglogd. Our
d
≥
argument uses elements from the work of Montgomery and Weinberger as well as some
new ideas.
The proof of the second part of Theorem 1.2 relies on two main ingredients. First,
using elementary methods we bound the number of real quadratic fields Q(√d) with
discriminant d x which have small regulator. Then, we combine Heath-Brown’s
≤
quadratic large sieve [6] with zero-density estimates to show that Littlewood’s GRH
boundL(1,χ ) (2eγ+o(1))loglogdholdsunconditionallyforallbut atmost x1/2+o(1)
d
≤
fundamental discriminants 0 < d < x.
4 YOUNESSLAMZOURI
2. Real quadratic fields with extreme class number: proof of
Theorem 1.2, part (a)
To obtain large values of L(1,χ ), a general strategy is to construct fundamental
d
discriminants d for which χ (p) = 1 for all the small primes p, typically up to y = logd.
d
Montgomery and Weinberger [8] noticed that for square-free d of the form d = 4n2+1,
one has χ (p) = 1 for all the primes p dividing n. Hence, this reduces the problem to
d
estimating the number of fundamental discriminants d (x) for which (d 1)/4 is
∈ D −
divisible by all the small primes. To this end they established the following lemma.
Lemma 2.1 (Lemma 1 of [8]). The number of integers d x such that d is square-free
≤
and d = 4n2 +1 where q n equals
|
√x 2
1 +O x1/3logx .
2q − p2
p∤q (cid:18) (cid:19)
Y (cid:0) (cid:1)
Taking q = p, where √logx y (logx)/8 is a real number, and noting that
p y ≤ ≤
≤
q = ey(1+o(1)) by the prime number theorem, yields
Q
Corollary 2.2. Let √logx y (logx)/8 be a real number. The number of funda-
≤ ≤
mental discriminants d (x) such that χ (p) = 1 for all primes p y is at least
d
∈ D ≤
x1/2e y(1+o(1)).
−
Montgomery and Weinberger then used zero density estimates to prove that for any
0 < δ < 1, all but at most xδ fundamental discriminants 1 d x satisfy
≤ ≤
χ (p)
d
(2.1) logL(1,χ ) = +O (1),
d δ
p
p y
X≤
where (logx)δ y logx is a real number. Taking y = (logx)/9 and δ = 1/4 in (2.1),
≤ ≤
and using Corollary 2.2 produces more than x3/8 fundamental discriminants d x in
≤
for which L(1,χ ) loglogd.
d
D ≫
In order to improve this estimate, we first replace (2.1) with a better approximation
toL(1,χ ),duetoGranvilleandSoundararajan[5],whichisobtainedusingzerodensity
d
estimates together with the large sieve.
Proposition 2.3 (Proposition 2.2 of [5]). Let A > 2 be fixed. Then, for all but at most
Q2/A+o(1) primitive characters χ (mod q) with q Q we have
≤
1
χ(p) − 1
L(1,χ) = 1 1+O .
− p loglogQ
p (logQ)A(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
≤Y
Let √logx y (logx)/8 be a real number. Then, note that
≤ ≤
(2.2)
1 1
χd(p) − χd(p) − χd(p) 1
1 = 1 exp +O .
− p − p  p √ylogy 
p (logx)A(cid:18) (cid:19) p y(cid:18) (cid:19) y<p<(logx)A (cid:18) (cid:19)
≤Y Y≤ X
 
EXTREME VALUES OF CLASS NUMBERS OF REAL QUADRATIC FIELDS 5
By Corollary 2.2, there are many fundamental discriminants d (x) for which the
∈ D
product (1 χ (p)/p) 1 is as large as possible. The key ingredient in the proof of
p y − d −
≤
the first part of Theorem 1.2 is the following proposition which gives an upper bound
Q
for the 2k-th moment of χ (p)/p as d varies in (x), uniformly for k in a
y<p<(logx)A d D
large range. In particular, we shall later deduce that with very few exceptions in (x),
P D
the prime sum χ (p)/p is small.
y<p<(logx)A d
P
Proposition 2.4. Let √logx < y < logx be a real number, and z = (logx)A where
A > 2 is a constant. Then, for every positive integer k logx/(8Aloglogx) we have
≤
2k
k
χ (p) ck
d √x ,
p ≪ ylogy
!
d (x) y<p<z (cid:18) (cid:19)
∈XD X
for some absolute constant c > 0.
To prove this result we first need the following lemma, which gives a non-trivial
bound for a certain character sum.
Lemma 2.5. Let q be an odd positive integer, and write q = q2q where q is square-
1 0 0
free. If x q2 then
≥
4n2 +1 x
,
q ≪ q
n x(cid:18) (cid:19) 0
X≤
where ( ) is the Jacobi symbol modulo q.
·
q
Proof. Let f(n) = 4n2 +1. First we have
(2.3)
q q q
f(n) f(n) f(a) x f(a)
= = 1 = +O(q).
q q q q q
n x(cid:18) (cid:19) a=1 n x (cid:18) (cid:19) a=1 (cid:18) (cid:19) n x a=1(cid:18) (cid:19)
X≤ Xn aX≤modq X n aX≤modq X
≡ ≡
Note that q f(a) is a complete character sum, and hence if q = pα1 pαk is the
a=1 q 1 ··· k
prime factorizati(cid:16)on of(cid:17)q, then
P
q f(a) k pαjj f(a ) k pαjj f(a ) αj
j j
(2.4) = = ,
q  pαj   p 
Xa=1 (cid:18) (cid:19) Yj=1 aXj=1 j ! Yj=1 aXj=1(cid:18) j (cid:19)
   
   
by multiplicativity and the Chinese Remainder theorem. Now, if α = 2β is even we
j j
use the trivial bound
p2βj
j f(a) 2βj
(2.5) p2βj.
(cid:12)(cid:12)(cid:12)Xa=1(cid:18) pj (cid:19) (cid:12)(cid:12)(cid:12) ≤ j
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6 YOUNESSLAMZOURI
On the other hand, if α = 2β +1 is odd, then f(aj) αj = f(aj) , and hence
j j pj pj
(cid:16) (cid:17) (cid:16) (cid:17)
pj2βj+1 f(a) 2βj+1 pj2βj−1 pj f(bp +c) pj f(c)
(2.6) = j = p2βj .
p p j p
a=1 (cid:18) j (cid:19) b=0 c=1 (cid:18) j (cid:19) c=1 (cid:18) j (cid:19)
X X X X
If p is a prime number, then the sum p n2+b is a Jacobsthal sum, and it is known
n=1 p
that (see for example Storer [9]) (cid:16) (cid:17)
P
p
n2 +b
= 1 if p ∤ b.
p −
n=1(cid:18) (cid:19)
X
Therefore, we deduce
pj 4c2 +1 4
= = 1.
p − p −
c=1 (cid:18) j (cid:19) (cid:18) j(cid:19)
X
Inserting this estimate in (2.6) yields
p2βj+1
j f(a) 2βj+1
(2.7) = p2βj.
(cid:12) p (cid:12) j
(cid:12)(cid:12) Xa=1 (cid:18) j (cid:19) (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Combining the bounds (2.5) a(cid:12)nd (2.7) in (2.4) yi(cid:12)elds
(cid:12) (cid:12)
q
f(a) q
.
(cid:12)(cid:12)Xa=1(cid:18) q (cid:19)(cid:12)(cid:12) ≤ q0
(cid:12) (cid:12)
(cid:3)
The result follows upon inserting(cid:12)this bound i(cid:12)n (2.3).
(cid:12) (cid:12)
Proof of Proposition 2.4. As before, we let f(n) = 4n2 + 1. By positivity of the sum-
mand we have
2k
2k f(n)
χd(p) p
.
p ≤  (cid:16) p (cid:17)
!
d∈XD(x) yX<p<z nX≤√x yX<p<z
 
Expending the inner sum we obtain
2k
f(n) b (m;y,z) f(n)
p 2k m
(2.8) = ,
 (cid:16) p (cid:17) m (cid:16) (cid:17)
y<p<z m
X X
 
where
(2.9) b (m;y,z) := 1.
r
y<p1p1X,p..r.,=prm<z
···
Note that 0 b (m;y,z) r!. Moreover, b (m;y,z) = 0 unless m = pα1 pαs where
≤ r ≤ r 1 ··· s
y < p < p < < p < z are distinct primes and Ω(m) = α + +α = r (where
1 2 s 1 s
··· ···
EXTREME VALUES OF CLASS NUMBERS OF REAL QUADRATIC FIELDS 7
Ω(m) is the number of prime divisors of m counting multiplicities). In this case, we
have
r
(2.10) b (m;y,z) = .
r
α ,...,α
(cid:18) 1 s(cid:19)
Using this formula, one can easily deduce that if n and m are positive integers with
Ω(n) = ℓ and Ω(m) = r then
ℓ+r
(2.11) b (mn;y,z) b (n;y,z)b (m;y,z).
ℓ+r ℓ r
≤ ℓ
(cid:18) (cid:19)
Since z2k x1/4, then it follows from Lemma 2.5 that
≤
(2.12)
2k
f(n)
p = b2k(m;y,z) f(n) √x b2k(m;y,z),
 (cid:16) p (cid:17) m m ≪ mm
n √x y<p<z m n √x(cid:18) (cid:19) m 0
X≤ X X X≤ X
 
where m is the square-free part of m. Let m be such that Ω(m) = 2k and p m =
0
| ⇒
y < p < z, and write m = m2m , where m is square-free. Put Ω(m ) = ℓ. Then by
1 0 0 1
(2.11) we have
2k
b (m;y,z) b (m2;y,z)b (m ;y,z)
2k ≤ 2ℓ 2ℓ 1 2k−2ℓ 0
(cid:18) (cid:19)
2k 2ℓ
(b (m ;y,z))2b (m ;y,z)
ℓ 1 2k 2ℓ 0
≤ 2ℓ ℓ −
(cid:18) (cid:19)(cid:18) (cid:19)
(2k)! k
b (m ;y,z)b (m ;y,z),
ℓ 1 2k 2ℓ 0
≤ k! ℓ −
(cid:18) (cid:19)
since b (m ;y,z) ℓ!. Therefore, we deduce that
ℓ 1
≤
k
b (m;y,z) (2k)! k b (m ;y,z) b (m ;y,z)
2k ℓ 1 2k 2ℓ 0
−
mm ≤ k! ℓ m2 m2
Xm 0 Xℓ=0 (cid:18) (cid:19)Xm1 1 Xm0 0
k
2k(2k)! 1 ck k
,
≤ k! p2 ≤ ylogy
!
y<p<z (cid:18) (cid:19)
X
for some positive constant c > 0 if y is large enough, since
r
b (n;y,z) 1
r
(2.13) =
n2 p2
!
n y<p<z
X X
and 1/p2 1/(ylogy) by the prime number theorem. Inserting this bound in
y<p<z ≪
(cid:3)
(2.12) completes the proof.
P
We are now ready to prove the first part of Theorem 1.2.
Proof of Theorem 1.2, part (a). Let z = (logx)6, and √logx y (logx)/8 be a real
≤ ≤
number to be chosen later. Then, by Proposition 2.3 and equation (2.2) it follows that
8 YOUNESSLAMZOURI
for all but at most x2/5 fundamental discriminants d (x), we have
∈ D
1
χd(p) − χd(p) 1
(2.14) L(1,χ ) = 1 exp +O ..
d
− p p √ylogy !
p y(cid:18) (cid:19) y<p<z (cid:18) (cid:19)
Y≤ X
Furthermore, taking k = [logx/(50loglogx)] in Proposition 2.4 implies that the num-
ber of fundamental discriminants d (x) such that
∈ D
χ (p) 1
d
>
p (loglogx)1/4
(cid:12) (cid:12)
(cid:12)y<Xp<z (cid:12)
(cid:12) (cid:12)
is (cid:12) (cid:12)
(cid:12) (cid:12)
k
logx
(2.15) √x .
≪ ylogy(loglogx)1/3
(cid:18) (cid:19)
On the other hand, it follows from Corollary 2.2 that there are at least √xe y(1+o(1))
−
fundamental discriminants d (x) for which χ (p) = 1 for all primes p y. There-
d
∈ D ≤
fore, choosing y = logx/(2loglogx) we deduce from (2.14) and (2.15) that there are
at least x1/2 1/loglogx fundamental discriminants d (x) such that χ (p) = 1 for all
− d
∈ D
primes p y, (2.14) holds and
≤
χ (p) 1
d
.
p ≤ (loglogx)1/4
(cid:12) (cid:12)
(cid:12)y<Xp<z (cid:12)
(cid:12) (cid:12)
For these d, we have by (2.14(cid:12)) that (cid:12)
(cid:12) (cid:12)
1
L(1,χ ) = eγloglogx 1+O .
d (loglogx)1/4
(cid:18) (cid:18) (cid:19)(cid:19)
Inserting this estimate in (1.12) completes the proof.
(cid:3)
3. An upper bound for the number of real quadratic fields with
extreme class number: proof of Theorem 1.2, part (b)
In order to bound the number of real quadratic fields Q(√d) with discriminant
d x for which the class number h(d) is extremely large (that is, h(d) satisfies (1.11)),
≤
we shall first bound the number of small solutions to the Pell equations m2 dn2 = 4.
− ±
We prove the following lemma.
Lemma 3.1. Let x be large, and let d x be a positive integer. For a real number
≤
θ (1/2,3/2) denote by S (d) the set of positive solutions (m,n) to the Pell equations
θ
∈
(3.1) m2 dn2 = 4,
− ±
such that m dθ. Then, we have
≤
S (d) x1/2 +xθ 1/2 (logx)2.
θ −
| | ≪
d x
X≤ (cid:0) (cid:1)
EXTREME VALUES OF CLASS NUMBERS OF REAL QUADRATIC FIELDS 9
Proof. LetP (d)bethesetofsolutions(m,n)tothepositivePellequationm2 dn2 = 4,
θ
−
such that m dθ. Similarly, let N (d) be the set of solutions (m,n) to the negative
θ
≤
Pell equation m2 dn2 = 4, such that m dθ. We shall only bound P (d)
− − ≤ d x| θ |
≤
since the treatment for N (d) is similar. Let (m,n) P (d). Then, note that
d x| θ | ∈ θ P
≤
dn2 m2 d2θ, and hence n dθ 1/2. Furthermore, for a fixed n xθ 1/2, if
≤ ≤ P ≤ − ≤ −
(m,n) P (d) for some d x then m n√x+2 and m2 4 (mod n2). Therefore,
θ
∈ ≤ ≤ ≡
we deduce that
P (d) m n√x+2, such that m2 4 (mod n2) .
θ
| | ≤ |{ ≤ ≡ }|
Xd≤x n≤Xxθ−1/2
Let ℓ(q) be the number of solutions m (mod q) of the congruence m2 4 (mod q).
≡
Then, ℓ(q) is a multiplicative function, and
2 if p > 2,
ℓ(pk)
≤ 4 if p = 2.
(
Hence, we derive
m n√x+2, such that m2 4 (mod n2)
|{ ≤ ≡ }|
n≤Xxθ−1/2
√x ℓ(n2)
ℓ(n2) +1 x1/2 +xθ 1/2 .
−
≪ n ≪ n
n≤Xxθ−1/2 (cid:18) (cid:19) (cid:0) (cid:1)n≤Xxθ−1/2
The lemma follows upon noting that
ℓ(n2) 1 −2
1 (logx)2.
n ≪ − p ≪
n≤Xxθ−1/2 2<Yp<x(cid:18) (cid:19)
(cid:3)
The second ingredient in the proof of part (b) of Theorem 1.2 is to show that
L(1,χ ) (2eγ +o(1))loglogd for all but at most x1/2+o(1) fundamental discriminants
d
≤
0 < d < x. To this end, we shall use Heath-Brown’s quadratic large sieve to show that
Proposition 2.3 can be improved if we restrict our attention to quadratic characters.
More precisely, we prove that L(1,χ ) can be approximated by an Euler product over
d
the primes p (logx)A, for all but at most x1/A+o(1) fundamental discriminants 0 <
≤
d < x.
Proposition 3.2. Let A > 1 be fixed. Then for all but at most x1/A+o(1) fundamental
discriminants 0 < d < x we have
1
χd(p) − 1
L(1,χ ) = 1 1+O .
d
− p loglogx
p (logx)A(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
≤Y
10 YOUNESSLAMZOURI
Proof. First, it follows from Proposition 2.3 that
1
χd(p) − 1
L(1,χ ) = 1 1+O ,
d
− p loglogx
p (logx)2A(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
≤ Y
for all except at most x1/A+o(1) fundamental discriminants 0 < d < x. To prove the
result, we are going to show that
χ (p) 1
d
(3.2) = O ,
p loglogx
(logx)A<p<(logx)2A (cid:18) (cid:19)
X
for all but at most x1/A+o(1) fundamental discriminants 0 < d < x. To this end, we will
exploit Heath-Brown’s quadratic large sieve (see Corollary 2 of [6]) which asserts that
2
♭
(3.3) a(n)χ (n) (xN)ǫ(x+N) a(n )a(n ) .
d ǫ 1 2
(cid:12) (cid:12) ≪ | |
0<Xd<x(cid:12)(cid:12)nX≤N (cid:12)(cid:12) nn11X,nn22=≤(cid:3)N
(cid:12) (cid:12)
(cid:12) (cid:12)
♭
where the is taken over fundamental discriminants, and the a(n) are arbitrary
complex numbers.
X
For 0 j J := [Aloglogx/log2], we define z = 2j(logx)A, and put z =
j J+1
≤ ≤
(logx)2A. Also, we let k = [(logx)/(Aloglogx)]+1 so that zk x for all 0 j J+1.
j ≥ ≤ ≤
Now, similarly to (2.8) we have
k
χ (p) b (m;z ,z )χ (m)
d k j j+1 d
= ,
 p  m
zj<Xp<zj+1 zjk<Xm<zjk+1
 
where the coefficient b (m;z ,z ) is defined in (2.9). Then, by (3.3) we obtain
k j j+1
(3.4)
2k 2
♭ χ (p) ♭ b (n;z ,z )χ (n)
d k j j+1 d
=
 p   n 
0<Xd<x zj<Xp<zj+1 0<Xd<x zjk<Xn<zjk+1
   
b (m;z ,z )b (n;z ,z )
(z )k(1+ǫ) k j j+1 k j j+1 .
ǫ j+1
≪ mn
zjk<nX,m<zjk+1
mn=(cid:3)
Let m and n be positive integers such that Ω(m) = Ω(n) = k and put d = (m,n).
Also, put n = dn and m = dm where (m ,n ) = 1. Since mn = (cid:3) then both m and
1 1 1 1 1
n are squares. Let n = ℓ2 and m = ℓ2, and put s = Ω(ℓ ). Since Ω(n) = Ω(m) = k,
1 1 1 1 2 1