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Pair correlation of zeros of quadratic
L-functions near the real axis
Keiju Sono (宗野惠樹)
Abstract
In this article, we investigate the non-trivial zeros of quadratic Lfunctions near the real axis. Assuming the Generalized Riemann Hypothesis, we give an asymptotic formula for the weighted pair correlation
function of quadratic L-functions. From this formula, we prove that there
exists a number of “ close low lying zeros ”.
1
Introduction
About forty years ago, H. L. Montgomery [9] published his famous paper titled
“ The pair correlation of zeros of the zeta function ”. Under the assumption of
the Riemann Hypothesis (RH), he investigated the function
(
F (α, T ) =
T
log T
2π
)−1
∑
′
′
T iα(γ−γ ) w(γ − γ ),
0<γ,γ ′ ≤T
′
where w(u) = 4/(4 + u2 ) and γ, γ run over the set of the imaginary parts
of the non-trivial zeros of the Riemann zeta-function ζ(s) in 0 < Im(s) ≤ T .
(Note that the number of non-trivial zeros of ζ(s) in this domain is asymptotic
to (1/2π)T log T .) He obtained some asymptotic formula for F (α, T ) (0 < α ≤
1 − ϵ), and using this formula, he obtained several results on the distances of
the non-trivial zeros. For example, under the assumption of the RH, he proved
that at least 67% of the non-trivial zeros are simple , and that
lim inf
n→∞
(γn+1 − γn ) log γn
≤λ<1
2π
holds for speciﬁc λ, where γn denotes the imaginary part of the n-th non-trivial
zero of ζ(s) in the upper half plane.
Later, Montgomery’s idea was extended to many types of L-functions or
¨ uk [10] investigated the non-trivial zeros of
other situations. For example, Ozl¨
the Dirichlet L-functions near the real axis. Assuming the Generalized Riemann
Hypothesis (GRH), he proved that at least 86% of such zeros are simple in some
sense. One of the other interesting generalizations is the work of Hejhal [5].
1
From his explicit formula of the Riemann zeta-function, he constructed certain
asymptotic formula for the function involving the pairs of three distinct zeros of
ζ(s). Further, the result of Hejhal was generalized by Rudnick and Sarnak [14],
and the n-level correlation of the zeros of principal L-functions was obtained.
In particular, their results agree with the prediction for the Gaussian unitary
ensemble of random matrix theory. Today there are several papers considering
such kind of problem (n-level density). For example, see [1], [2], [3], [6], [7], [8],
[12], [13].
Our aim in this paper is to investigate the pair correlation of the zeros of the
¨ uk and Snyder
quadratic L-functions near the real axis. As a prior research, Ozl¨
[11] investigated such zeros. Under the assumption of GRH, they studied the
asymptotic behavior of the function
(
GK (α, D) =
1
K
2
( ) )−1 ∑
πd2 ∑
1
D
e− D2
K(ρ)Diαγ
2
d̸=0
ρ∈Zd
as D → ∞ for |α| < 2, where ρ = 1/2 + iγ runs over the set of all non-trivial
zeros of L(s, χd ), the quadratic L-function associated to the Kronecker symbol
χd = (d/·), and K(s) is some weight function. From their asymptotic formula,
they proved that assuming the GRH, not more than 6.25% of all integers d
have the property that L(s, χd ) vanishes at the central point s = 1/2. Subsequently Soundararajan [16] unconditionally proved that L(1/2, χd ) ̸= 0 for at
least 87.5% of all fundamental discriminants d.
In this paper, assuming the GRH (including RH), we investigate the function
FK (α, D) deﬁned as follows. Let K(s) be analytic in −1 < Re(s) < 2 and satisfy
K(1/2 − it) = K(1/2 + it) for any t ∈ R. Moreover, we assume that its Mellin
inverse transform
∫ c+i∞
1
a(x) :=
K(s)x−s ds
(1.1)
2πi c−i∞
converges absolutely for any −1 < c < 2, x > 0, and that a(x) is real, nonnegative, belongs to C 1 class, and has a support in [A, B] for some 0 < A <
B < ∞. Then, K(s) is given by the Mellin transform of a(x):
∫ ∞
dx
K(s) =
a(x)xs .
(1.2)
x
0
For d ∈ Z, let χd = (d/·) be the Kronecker symbol and L(s, χd ) be the Lfunction associated to χd . We denote the set of non-trivial zeros of L(s, χd ) by
Zd . For x > 0, D > 0, we put
fK (x, D) =
∑
d
πd2
e− D2
∑
ρ1 ,ρ2 ∈Zd
2
K(ρ1 )K(ρ2 )xρ1 +ρ2 ,
and for α ∈ R, we deﬁne the correlation function FK (α, D) by
[
]
1
FK (α, D) =
fK (x, D)
xD log D
x=Dα
∑ − πd2 ∑
1
e D2
K(ρ1 )K(ρ2 )Diα(γ1 −γ2 ) ,
=
D log D
(1.3)
ρ1 ,ρ2 ∈Zd
d
where ρj = 1/2 + iγj for j = 1, 2. Then, the main theorem is stated as follows:
Theorem 1.1. Assuming the Generalized Riemann Hypothesis, for any small
δ > 0, we have
FK (α, D) = L(1)α + a(D−α )2 D−α log D + a(D−α ) · O(αD− 2 log D)
α
+ a(D−α )2 · O(D−α ) + O(min{1, αD−α log2 D})
α
−1
+ O(min{D (log D)
2
, α D
−α
(1.4)
3
log D}) + o(1)
uniformly for 0 < α < 1 − δ as D → ∞, where
∫ ∞
L(1) =
a(x)2 dx.
0
The implied constants depend only on K(s) and δ > 0.
In the next section, we introduce the outline of the proof. The author recommends the reader to see the preprint [15] to check the detailed computations.
Several results on the average gaps of the non-trivial zeros can be obtained.
Among others, in Section 3, we prove that there are quite a few pairs of zeros (1/2 + iγ1 , 1/2 + iγ2 ) of L(s, χd ) (d ∈ Z\{0}) near the real axis satisfying
0 < |γ1 − γ2 | ≤ (2πλ)/ log D, if λ is large to a certain extent.
2
The proof of Theorem 1.1 (outline)
¨ uk’s explicit formula
We start from Ozl¨
∑
ρ∈Zd
( )
∞
(n)
∑
d
Λ(n)
a
x
n
n=1
( )
( )
1
|d|
+a
log
+ O(min{x, log |d| log x})
x
π
K(ρ)xρ = K(1)E(χd )x −
(x ≥ 1),
introduced in [11]. Here, E(χ) = 1 if χ is a principal character, and otherwise
E(χ) = 0. The error term is interpreted as O(1) if x = 1. Since the main terms
of the right hand side are real, we have
3
∑
K(ρ1 )K(ρ2 )xρ1 +ρ2
ρ1 ,ρ2 ∈Zd
{
( )
∞
(n)
∑
d
= K(1)E(χd )x −
a
Λ(n)
x
n
n=1
( )
( )
}2
1
|d|
+a
log
+ O(min{x, log |d| log x})
x
π
( )( )
∞
(n) (m)
∑
d
d
2
2 2
= K(1) E(χd ) x +
a
a
Λ(n)Λ(m)
x
x
n
m
n,m=1
( )2
( )
( )
∞
(n)
∑
|d|
d
1
log2
− 2K(1)E(χd )x
a
Λ(n)
+a
x
π
x
n
n=1
( )
( )
( )
( )
( )∑
∞
(n)
1
|d|
1
|d|
d
+ 2K(1)E(χd )xa
log
− 2a
log
Λ(n)
a
x
π
x
π n=1
x
n
(
{
( ) ( )
( )})
∞
(n)
∑
d
1
|d|
+ O max K(1)E(χd )x,
Λ(n)
,a
log
a
x
n
x
π
n=1
× O (min{x, log |d| log x})
(
)
+ O min{x2 , log2 |d| log2 x} .
πd2
By multiplying both sides by e− D2 and taking the sum over d, we have
fK (x, D) =
6
∑
Mi +
i=1
where
M1 = K(1)2 x2
∑
4
∑
Oi ,
i=1
πd2
E(χd )2 e− D2 ,
d
(n) (m)
∑ − πd2 ( d ) ( d )
M2 =
a
Λ(n)Λ(m)
e D2
,
a
x
x
n
m
n,m=1
∞
∑
d
( )2 ∑
( )
2
1
|d|
− πd
2
2
M3 = a
e D log
,
x
π
d
( )
∞
(n)
∑
∑ − πd2
d
2
D
M4 = −2K(1)x
a
Λ(n)
e
E(χd )
,
x
n
n=1
d
( )∑
( )
2
1
|d|
− πd
2
M5 = 2K(1)xa
,
e D E(χd ) log
x
π
d
( )∑
( )
∞
(n)
∑ − πd2 ( d )
1
|d|
2
D
M6 = −2a
Λ(n)
e
log
,
a
x n=1
x
n
π
d
4
(2.1)
O1 = O (min {O11 , O12 }) ,
O2 = O (min {O21 , O22 }) ,
O3 = O (min {O31 , O32 }) ,
O4 = (min {O41 , O42 }) ,
with
O11 = K(1)x2
∑
πd2
e− D2 E(χd ),
O12 = K(1)x log x
d
∑
πd2
e− D2 E(χd ) log |d|,
d
∞
(n)
∑ − πd2 ( d )
∑
Λ(n)
e D2
,
a
x
n
n=1
O21 = x
d
( )
d
log |d|,
x
n
n=1
d
( )∑
( )
2
|d|
1
− πd
2
O31 = xa
e D log
,
x
π
d
( )
( )
∑ − πd2
|d|
1
log x
e D2 log |d| log
,
=a
x
π
O22 = log x
O32
∞
∑
a
(n)
Λ(n)
∑
πd2
e− D2
d
O41 = x2
∑
2
− πd
D2
e
,
O42 = log2 x
d
∑
πd2
e− D2 log2 |d|.
d
First, by a standard technique, we ﬁnd that the error terms O1 , O2 , O3 , O4 are
evaluated by
O1 ≪ min{x2 D1/2 , xD1/2 log x log D},
O2 ≪ min{x2 D, xD log x log D, x3/2 log2 x log D},
O3 ≪ min{xD log D, D log x log2 D},
O4 ≪ min{x2 D, D log2 x log2 D}.
Next, by using partial summation and prime number theorem, the main terms
except for M2 are obtained as follows:
1
M1 = IK(1)2 x2 D1/2 − K(1)2 x2 + O(x2 D−1/2 ),
2
( )
1
M3 = a
{D log2 D + O(D log D)},
x
M4 = −2IK(1)2 D1/2 x2 + O(D1/2 x3/2 log2 x) + O(min{x2 , x3 D−1/2 }),
( )
1
M5 ≪ xa
D1/2 log D,
x
( )
1
M6 ≪ a
{x3/2 log x log D + Dx1/2 log x log D},
x
5
where I = 4−1 π −1/4 Γ(1/4). These are obtained by some computations similar
¨ uk and Snyder [11], hence we omit the detail. Finally,
to those in the paper of Ozl¨
we compute
∞
∑
∑ ( pk ) ( q l )
∑ − πd2 ( d ) ( d )
M2 =
a
a
(log p)(log q)
e D2
,
x
x
pk
ql
k,l=1 p,q∈P
d
where P denotes the set of all prime numbers. It will be convenient to keep in
mind that only k, l satisfying k, l ≪ log x contribute to the sum above, since
a(x) has a support in such range. First, we evaluate the contribution of the part
p = 2 to M2 . The contribution of the part p = q = 2 is
( k) ( l)∑
∞
∑
πd2
2
2
a
e− D2 ≪ D log2 x.
≪
a
(2.2)
x
x
d
k,l=1
The contribution of the part p = 2, q ≥ 3, l ≥ 2 is
∑ ∑ ∑ ( 2k ) ( q l )
∑ − πd2
≪
a
(log q)
e D2
a
x
x
l≥2 q∈P
k
1
2
d
(2.3)
≪ Dx log x.
2
Since (·/2k q) is a non-principal character whose conductor is at most 2q, by
combining partial summation and P´olya-Vinogradov inequality for the sum involving quadratic characters, we ﬁnd that
∑ − πd2 ( d ) ( d )
1
e D2
≪ q 2 log q
2k
q
d
for primes q ≥ 3. Therefore, the contribution of the part p = 2, q ≥ 3, l = 1 is
∑ ∑ ( 2k ) ( q )
∑ − πd2 ( d ) ( d )
a
(log 2)(log q)
e D2
a
x
x
2k
q
k q∈P≥3
d
∑ (q)
1
(2.4)
(log q) · q 2 log q
≪ (log x)
a
x
q∈P
3
2
≪ x log x,
2
where P≥3 denotes the set of all prime numbers greater than 2. By (2.2), (2.3)
and (2.4), the contribution of the part p = 2 to M2 is at most O(Dx1/2 log2 x +
x3/2 log2 x). The contribution of the part q = 2 is the same. Hence
( k) ( l)
∞
∑
∑
∑ − πd2 ( d ) ( d )
p
q
a
M2 =
a
(log p)(log q)
e D2
x
x
pk
ql
k,l=1 p,q∈P≥3
1
2
d
2
3
2
2
+ O(Dx log x + x log x)
∞
∑
1
3
(k,l)
=:
M2
+ O(Dx 2 log2 x + x 2 log2 x),
k,l=1
6
(2.5)
say. Moreover, by prime number theorem and partial summation, we have
∑ ( pk ) ( q l )
∑ − πd2
(k,l)
M2
≪
a
a
(log p)(log q)
e D2
x
x
p,q
d
≪ klDx
1
1
k+ l
uniformly for each k, l. Hence the contribution of the part k ≥ 3, l ≥ 2 or
k ≥ 2, l ≥ 3 is at most O(Dx5/6 log4 x). Therefore,
∑ (1,l)
5
3
(1,1)
(2,2)
M2 = M2
+ M2
+2
M2
+ O(Dx 6 log4 x + x 2 log2 x).
(2.6)
l≥2
(2,2)
By the computation above, M2
is evaluated by
(2,2)
M2
≪ Dx.
(2.7)
(1,l)
Next, we compute M2
for l ≥ 1.
A) First, we consider the case that l is odd. We decompose


(1,l)
M2
 ∑

=

( p ) ( ql )
∑ 
∑ − πd2 ( d ) ( d )

a
(log p)(log q)
e D2
a
x
x
p
q

+
p,q∈P≥3
p=q
(1,l)
p,q∈P≥3
p̸=q
d
(1,l)
=: M2,1 + M2,2 .
(2.8)
Easily, we ﬁnd that
(1,1)
M2,1
= L(1)Dx log x + O(Dx + x log x),
(1,l)
M2,1 ≪ lDx1/l log x (l ≥ 2).
(1,l)
Next, we compute M2,2 .
a) If x = o(D1/2 ), by prime number theorem and P´olya-Vinogradov inequality,
we obtain
(1,l)
M2,2 ≪ lx3/2(1+1/l) log2 x.
b) If D1/2−δ ≪ x ≪ D1−δ (δ > 0), by the translation formula of twisted theta
function, we ﬁnd that
( )
2 2
∑ ∑ ( p ) ( ql )
1 ∑ m
− πm D
(1,l)
M2,2 = D
a
a
(log p)(log q) √
e p2 q2 .
x
x
pq m pq
p≥3 q≥3
q̸=p
We decompose this by
(1,l)
(1,l)
M2,2 = Ms(1,l) − Mp(1,l) − Mq(1,l) + Mpq
+ E,
7
(2.9)
where
Ms(1,l) = D
2 D2
∑ ∑ ( p ) ( ql )
1 ∑ − πm
a
a
(log p)(log q) √
e p2 q 2 ,
x
x
pq
p≥3 q≥3
q̸=p
Mp(1,l)
Mq(1,l)
2 D2
∑ ∑ ( p ) ( ql )
1 ∑ − πm
=D
a
a
e p2 q 2 ,
(log p)(log q) √
x
x
pq
E=D
p≥3 q≥3
q̸=p
m=
p|m
p≥3 q≥3
q̸=p
m=
q|m
p≥3 q≥3
q̸=p
m=
pq|m
2 D2
∑ ∑ ( p ) ( ql )
1 ∑ − πm
a
=D
e p2 q 2 ,
a
(log p)(log q) √
x
x
pq
(1,l)
Mpq
=D
and
m=
2 D2
∑ ∑ ( p ) ( ql )
1 ∑ − πm
e p2 q 2 ,
a
a
(log p)(log q) √
x
x
pq
( )
2 2
∑ ∑ ( p ) ( ql )
1 ∑ m
− πm D
a
a
(log p)(log q) √
e p2 q2 .
x
x
pq
pq
m̸=
p≥3 q≥3
q̸=p
Here, ¤ denotes the square of natural numbers. Using partial summation and
prime number theorem, we obtain
Ms(1,l) = ID1/2 l−1 K(1)K(1/l)x1+1/l +O(l−1 D1/2 x1+1/2l log2 x)+O(Dx1/2+1/2l ),
Mp(1,l) ≪ lD1/2 x1/l + Dx1/2+1/2l ,
Mq(1,l) ≪ l−1 D1/2 x log x + Dx1/2+1/2l ,
(1,l)
≪ Dx1/2+1/2l ,
Mpq
and after slightly complicated computations (we use the assumption of the GRH
to evaluate the sum involving quadratic symbols), the error term E is evaluated
by
E ≪ Dx1/l + x1+1/l log4 x.
Hence we obtain
M2,2 = ID 2 l−1 K(1)K
(1,l)
1
( )
1
1
1
1
1
1
x1+ l + O(l−1 D 2 x1+ 2l log2 x) + O(Dx 2 + 2l )
l
1
+ O(x1+ l log4 x)
(2.10)
for odd l and D1/2−δ ≪ x ≪ D1−δ . Therefore we have
( )
1
1
1
1
1
1
(1,l)
−1
2
M2
x1+ l + O(l−1 D 2 x1+ 2l log2 x) + O(lDx l log x)
= ID l K(1)K
l
1
1
1
+ O(Dx 2 + 2l ) + O(x1+ l log4 x)
(2.11)
8
for odd l and D1/2−δ ≪ x ≪ D1−δ . If l = 1, D1/2−δ ≪ x ≪ D1−δ , we have
(1,1)
M2
1
1
3
= L(1)Dx log x+IK(1)2 D 2 x2 +O(Dx+D 2 x 2 log2 x+x2 log4 x). (2.12)
On the other hand, for odd l ≥ 3 and x = o(D1/2 ), we have
(1,l)
M2
1
3
1
≪ lDx l log x + lx 2 (1+ l ) log2 x,
(2.13)
and for l = 1, x = o(D1/2 ), we have
(1,1)
M2
= L(1)Dx log x + O(Dx + x3 log2 x).
B) Next, we consider the case that l is even. In this case, we have
∑ ( p ) ( ql )
∑ ( d ) − πd2
(1,l)
M2
=
a
a
(log p)(log q)
e D2
x
x
p
p,q
d
∑ ( d ) − πd2
∑ ( p ) ( ql )
−
a
a
(log p)(log q)
e D2 .
x
x
p
p,q
(2.14)
(2.15)
q|d
∑ (d)
Since
d
p
πd2
e− D2 ≪
√
p log p,
the ﬁrst term of the right hand side of (2.15) is
∑ ( p ) ( ql )
∑ ( d ) − πd2
a
a
(log p)(log q)
e D2
x
x
p
p,q
d
∑ (p)√
∑ ( ql )
≪
a
p log2 p
a
log q
x
x
p
q
(2.16)
1
3
≪ x 2 log x · lx l
3
1
≪ lx 2 + l log x.
The second term of the right hand side of (2.15) is
∑ ( p ) ( ql )
∑ ( d ) − πd2
a
a
(log p)(log q)
e D2
x
x
p
p,q
q|d
∑ (p)
∑ ( q ) ( ql )
∑ ( d ) − πq2 d2
=
a
(log p)
a
(log q)
e D2
x
p
x
p
p
q
d
¯
¯
¯ ∑ ( d ) πq2 d2 ¯
∑ (p)
∑ ( ql )
¯
− D2 ¯
a
a
≪
(log p)
(log q) ¯
e
¯.
¯
¯
x
x
p
p
q
(2.17)
d
Now, since q satisﬁes q ≤ (Bx)1/l ≪ D1−δ ≪ D, we have D/q ≫ 1. Therefore,
by using P´
olya-Vinogradov inequality, we have
∑ ( d ) − πq2 d2
√
e D2 ≪ p log p.
(2.18)
p
d
9
Therefore,
∑ ( p ) ( ql )
∑ ( d ) − πd2
a
a
(log p)(log q)
e D2
x
x
p
p,q
q|d
∑ (p)√
∑ ( ql )
≪
a
p log2 p
a
log q
x
x
p
q
3
(2.19)
1
≪ lx 2 + l log x.
By combining (2.16) and (2.19), we obtain
(1,l)
M2
3
1
≪ lx 2 + l log x
(2.20)
(1,l)
for even l. Now, we have computed or evaluated M2
for all l. If x = o(D1/2 ),
by (2.13) and (2.20), we have
∑ (1,l)
1
M2
≪ Dx 3 log x + x2 log2 x.
(2.21)
l≥2
By inserting (2.7), (2.14), (2.21) into (2.6), we obtain
M2 = L(1)Dx log x + O(Dx + x3 log2 x).
If D1/2−δ ≪ x ≪ D1−δ , by (2.11),
∑
1
4
2
4
(1,l)
≪ D 2 x 3 + Dx 3 + x 3 log4 x.
M2
(2.22)
l≥3, odd
(Notice that K(1/l) ≪ l.) On the other hand, by (2.20),
∑
(1,l)
≪ x2 log x.
M2
(2.23)
l≥2, even
By inserting (2.7), (2.12), (2.22), (2.23) into (2.6), we obtain
1
M2 = L(1)Dx log x + IK(1)2 D 2 x2
1
3
+ O(Dx + D 2 x 2 log2 x + x2 log4 x)
for D1/2−δ ≪ x ≪ D1−δ .
Now, we have computed or evaluated all the terms appearing in (2.1), hence
we obtain the asymptotic formula for fK (x, D). By dividing this by xD log D
and putting x = Dα (α > 0), we obtain the asymptotic formula in Theorem 1.1.
3
The pairs of close zeros near the real axis
The asymptotic formula (1.4) of Theorem 1.1 is useful to investigate the distribution of low lying zeros of quadratic L-functions. As a corollary, we prove that
10
assuming the GRH and simple zero conjecture for each relevant L-functions,
there exists a number of“ close zeros ”near the real axis. First, we mention to
the L-functions associated to Kronecker symbols. If d ̸≡ 3 (mod 4), χd = (d/·)
becomes a Dirichlet character modulo 4|d| or |d|. In this case, we denote the
conductor of χd by d∗ . If d ≡ 3 (mod 4), the L-function associated to χd is
expressed by
∏
1
1
(2)
( )
,
L(s, χd ) =
−s
1 − d 2 p≥3 1 − η4 (p) dp p−s
where η4 is the non-principal character of modulo 4. In this case, we denote the
conductor of η4 (·)(·/d) by d∗ . We deﬁne the constants A∗+ , A∗− by
∑ − πd2
1
e D2 log d∗ .
D→∞
D
log
D
D→∞
d
d
(3.1)
Notice that since log d∗ ≤ log d + O(1), we have A∗+ ≤ 1.
A∗− = lim inf
∑ − πd2
1
e D2 log d∗ ,
D log D
A∗+ = lim sup
Corollary 3.1. Assume the Generalized Riemann Hypothesis and that all nontrivial zeros of L(s, χd ) are simple. Then, for 0 < λ < 1, we have
∑
∑ − πd2
1
e D2
K(ρ1 )K(ρ2 )
D log D
ρ1 ,ρ2 ∈Zd
d
2πλ
(3.2)
0<|γ1 −γ2 |≤ log
D
≥
2
2
cos 2πλ sin 2πλ
∗
λ − λ2 −
+
− B+
+ o(1)
3
9
6π 2
12π 3 λ
∗
as D → ∞, where B+
= A∗+ /3 and
(
K(s) =
es− 2 − e−s+ 2
2s − 1
1
1
)2
.
In particular, if λ > λ0 = 0.6073, we have
∑
∑ − πd2
e D2
K(ρ1 )K(ρ2 ) ≫ D log D
d
(3.3)
ρ1 ,ρ2 ∈Zd
2πλ
0<|γ1 −γ2 |≤ log
D
as D → ∞.
Proof. We use Selberg’s minorant function
)2
(
1
sin πu
.
h(u) =
πu
1 − u2
This function is bounded and satisﬁes h(u) ≤ 1, h(u) < 0 if |u| > 1. The Fourier
transform of h(u) is given by
{
2π|α|
1 − |α| + sin 2π
(|α| ≤ 1)
ˆ
h(α)
=
0
(|α| > 1)
11
(for example, see [4]). For 0 < λ < 1, we give lower and upper bounds for the
integral
∫ ∞
ˆ
FK (α, D) · λh(λα)dα.
−∞
First, since the integrant is non-negative and 1/λ > 1, by (1.4), we have
∫ ∞
ˆ
FK (α, D) · λh(λα)dα
−∞
∫
≥
1
−1
ˆ
FK (α, D) · λh(λα)dα
{
}
sin 2πλ|α|
(3.4)
= λL(1)
|α| 1 − |λα| +
dα
2π
−1
{
}
∫ 1
sin 2πλ|α|
+ λ log D
a(D−|α| )2 D−|α| 1 − |λα| +
dα + o(1)
2π
−1
2
2
cos 2πλ sin 2πλ
+
+ o(1).
= λ − λ2 −
3
9
6π 2
12π 3 λ
∫
1
In the computation above, we used
∫
L(1) =
∫
∞
a(v)2 dv =
0
∞
a(D−α )2 D−α dα =
−∞
1
log D
∫
1
,
3
1
a(v)2 dv =
0
1
.
6 log D
On the other hand,
∫ ∞
ˆ
FK (α, D) · λh(λα)dα
−∞
∑ − πd2
1
=
e D2
D log D
d
(
∑
K(ρ1 )K(ρ2 )h
ρ1 ,ρ2 ∈Zd
(γ1 − γ2 ) log D
2πλ
)
(3.5)
.
Now, since h((γ1 − γ2 ) log D/(2πλ)) is negative if |γ1 − γ2 | > (2πλ)/ log D, by
(3.5), we have
∫ ∞
ˆ
FK (α, D) · λh(λα)dα
−∞
≤
∑ − πd2
1
e D2
D log D
d
=
(
∑
K(ρ1 )K(ρ2 )h
ρ1 ,ρ2 ∈Zd
2πλ
|γ1 −γ2 |≤ log
D
(γ1 − γ2 ) log D
2πλ
)
∑ − πd2 ∑
1
e D2
mρ K(ρ)2
D log D
d
+
1
D log D
∑
d
ρ∈Zd
πd2
e− D2
(
∑
K(ρ1 )K(ρ2 )h
ρ1 ,ρ2 ∈Zd
2πλ
0<|γ1 −γ2 |≤ log
D
12
(γ1 − γ2 ) log D
2πλ
)
∗
≤ B+
+
∑ − πd2
1
e D2
D log D
d
∑
K(ρ1 )K(ρ2 ) + o(1),
ρ1 ,ρ2 ∈Zd
2πλ
0<|γ1 −γ2 |≤ log
D
where mρ is the multiplicity of the zero of L(s, χd ) at s = ρ, and by our
assumption, this equals 1. By combining (3.4) and above, we obtain (3.2).
∗
Since B+
= A∗+ /3 ≤ 1/3, the right hand side of (3.2) becomes positive if
λ > λ0 = 0.6073. Therefore, we obtain (3.3).
4
Acknowledgement
The author of this article would like to express his gratitude to Professors Taku
Ishii and Hiro-aki Narita for giving him the opportunity to talk about this
research at the RIMS symposium in 2014.
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Tokyo Denki University
Muzaigakuendai, Inzai,
Chiba, Japan
E-mail address:[email protected]
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