ETH Zürich
Wintersemester 09/10
Integral Moments of elliptic curve L-functions over Q
Author: Meriton Ibraimi
Advisor: Dr. Paul-Olivier Dehaye
1
Contents
1 Introduction
3
2 The L-function of a cusp form
6
3 Properties of L(s, E(Q))
7
3.1
Definition of elliptic curves over Q . . . . . . . . . . . . . . . .
7
3.2
Definition of L(s, E(Q)) . . . . . . . . . . . . . . . . . . . . .
8
3.3
The renormalized L-function . . . . . . . . . . . . . . . . . . . 12
4 Approximate Functional Equations for the Riemann zeta function and elliptic curve L-functions over Q
14
5 Using the recipe to conjecture the moments of the product
of shifted elliptic curve L-functions over Q
18
6 Conjecturing the moments for primitive L-functions of the
Selberg class
22
7
6.1
Definition of the Selberg class . . . . . . . . . . . . . . . . . . 22
6.2
The conjectures produced by the recipe when applied to primitive L-functions of the Selberg class . . . . . . . . . . . . . . 24
The arithmetic factor in the conjecture
7.1
27
Series for log(Ak (z1 , . . . , z2k )) in the case of elliptic curve Lfunctions over Q . . . . . . . . . . . . . . . . . . . . . . . . . 37
2
1
Introduction
The integral moments of the Riemann zeta function
1
T
ZT
| ζ(1/2 + it) |2k dt,
(1.1)
0
for k ∈ C are of great interest in number theory, since they give information about the distribution of the zeros on the critical line. The asymptotic
behavior, as T tends to infinity, of this integral was determined for k = 1
in 1918 by G. Hardy and L. Littlewood and in 1926, A. Ingham showed the
asymptotic behavior for k = 2. For a long time it was not clear, even conjecturally, what the asymptotic of higher moments should look like. In 1998,
J. B. Conrey and A. Ghosh [1] could guess the asymptotic for k = 3 and
J. B. Conrey and S. M. Gonek [2] guessed the asymptotic for k = 4. A breakthrough towards understanding the general form of the conjecture came in
2000 when J. P. Keating and N. C. Snaith [3], using Random Matrix Theory,
conjectured the moments of the zeta function for k ∈ C. Their conjecture is
1
lim
2
T →∞ T (log T )k
ZT
| ζ(1/2 + it) |2k dt = f (k)a(k),
(1.2)
0
where k ∈ C,
f (k) =
G(1 + k)2
G(1 + 2k)
(1.3)
and G denotes the Barnes G-function defined, for z ∈ C, by
z/2 −(z(z+1)+γz 2 )/2
G(1 + z) = (2π)
e
∞ h
Y
n=1
1+
z n −z+z2 /2n i
e
,
n
where γ is the Euler-Mascheroni constant, and
)
(
2
∞ Y
X
Γ(k + m)
k2
−m
a(k) =
(1 − 1/p)
p
.
Γ(m)Γ(k)
m=0
p prime
(1.4)
(1.5)
The factors f (k) is called the geometric factor and it is the factor predicted
by Random Matrix Theory, while the factor a(k) is called the arithmetic
3
factor due to its prediction by using the arithmetic properties of the zeta
function. For k ∈ N0 these factors reduce to
a(k) =
Y
−1 (k−1)2
(1 − p )
j=0
p prime
and
f (k) =
2
k−1 X
k−1
k−1
Y
j!
,
(k + j)!
j=0
j
p−j
(1.6)
(1.7)
where a(k) and b(k) are 1 for k = 0. Thus, for example, f (1) = 1, f (2) =
1/12, f (3) = 42/9!, f (4) = 24024/16!. Note that sometimes in the literature
the geometric factor f (k) is multiplied by k!2 and the product a(k)f (k) is
divided by k!2 in order that f (k). In 2005, in the paper of Conrey et al [4],
a recipe was given for conjecturing the moments not only for the Riemann
zeta function, but also for L-functions in the Selberg class, and for arbitrary
k ∈ N. The Selberg class of L-functions is the axiomatic approach to the
theory of L-functions, i.e. it is a set of Dirichlet series satisfying four axioms,
such that most of the known L-functions are (conjecture to be) included
in this set. We will define the Selberg class in Section 7. Note that all
the previously mentioned conjectures give also very good agreement when
evaluated numerically [4].
In our paper, we only consider L-functions of elliptic curves over Q, which
we always denote by E, and recapitulate the above mentioned recipe for
conjecturing their moments. One advantage we have is that in the meantime
the Sato-Tate conjecture was proved by C. Breuil, B. Conrad, F. Diamond,
R. Taylor [5], whereas Conrey et al assume Sato-Tate only conjecturally for
example in [4, Theorem 2.4.1]. We state this fact and others in Section 2
and 3. The L-function of the elliptic curve E satisfies a functional equation
of the following form
L(s) = εXL (s)L(1 − s),
(1.8)
where ε and XL are factors depending on the elliptic curve E, for details see
Corollary 3.6. We define the Z-function of E by
Z(s) = X(s)−1/2 L(s, E)
4
(1.9)
which has a functional equation of a simpler form than the L-function itself
ZL (s) = ZL (1 − s),
(1.10)
and therefore simplifies calculations. For these reasons the recipe of Conrey
et al gives the moments for the Z-function, instead of the moments for the
L-function, but since | XL ( 21 + it) |= 1, we also obtain the moments of the
L-function. The recipe, now limited to k ∈ N, is that
ZT
| Z(1/2 + it) |2k g(t)dt ∼
0
ZT
Pk (log(
t
))g(t)dt
2π
(1.11)
0
for some polynomial Pk (x) in x of degree k 2 and g(t) is a suitable weight
function. The authors of [4] do not define the property ”suitable” but they
give as an example, that g(x) can be taken to equal f (x/T ) for any function
f which is real, nonnegative, bounded, and integrable on the positive real
line. This is going to be explained more in Section 5. The polynomial Pk (x)
involves a factor of the form
Ak (z) =
k Y
k YY
p
i=1 j=1
1−
1
k
Z1 Y
p1+zi −zk+j
j=1
0
Lp
e(θ) p
1
+zj
2
Lp
e(−θ) 1
p 2 −zk+j
dθ, (1.12)
converging absolutely in some half plane, where the product is taken over
all primes and Lp is the local factor of the Euler product expansion of the
renormalized L-function of the elliptic curve E. This factor evaluated at
z1 = · · · = z2k = 0 determines the arithmetic factor ak .
In another paper of Conrey et al [6] , the lower order terms in the polynomial
Pk (x) are determined for the special case of the zeta function. In Section 6,
we try therefore to determine the lower order terms of Pk (x) for the special
case of a L-function of an elliptic curve over Q, by using the same ideas and
methods as in [6]. We were not able to give a final answer to this question
due to very complicated expressions appearing in the computation of the
multivariate Taylor series of log(Ak (z1 , ..., z2k )), but one can see that the
coefficients of Pk (x) are computable with the methods used in [6], even if
they can only be expressed in a very complicated way.
I want to thank Dr. Paul-Olivier Dehaye for his many corrections and great
support while writing this thesis. It was an invaluable experience to work
with him.
5
2
The L-function of a cusp form
We always denote by E an elliptic curve over Q and a definition of E is given
at the beginning of Section 3. In order to be able to use the recipe for the
L-function of E, we have to show that it has meromorphic continuation to
the whole plane and it satisfies a functional equation, similarly to the one
of the zeta function. This will be stated in Section 3 with the use of the
Modularity Theorem [7, Theorem 4.3], which states that every L-function
L(s, E) of an elliptic curve is the L-function L(s, f ) of a specific modular
form. Therefore we state here the main results,which will be needed later, in
the context of modular form.
P
2πinz
Definition 2.1. Let f ∈ S2 (Γ0 (N )) be a cusp form, and let f (z) = ∞
n=1 dn e
be its Fourier expansion. The L -function of f is the Dirichlet series
L(s, f ) =
∞
X
dn
n=1
ns
.
(2.1)
Theorem 2.2. (Hecke-Petersson) Let f ∈ S2 (Γ0 (N )) renormalized so d1 =
1. Then L(s, f ) converges for <(s) > 23 and has there the Euler product
representation
L(s, f ) =
Y
p|N
Y
1
1
.
−s
−s
1 − dp p
1 − dp p + pp−2s
(2.2)
p-N
Proof. See [8, Theorem 9.21].
Theorem 2.3. (Hecke) Let f ∈ S2 (Γ0 (N )). Then L(s, f ) is initially defined
for <(s) > 23 and extends to be entire in s. Moreover, the function
Λ(s, f ) = N s/2 (2π)−s Γ(s)L(s, f )
(2.3)
satisfies the functional equation
Λ(s, f ) = εΛ(2 − s, f ),
where ε = ±1 depends on f .
Proof. See [8, Theorem 9.8].
6
(2.4)
3
Properties of L(s, E(Q))
3.1
Definition of elliptic curves over Q
The set E of solutions (X, Y, Z) in the projective space
P2 (Q) = (Q3 \ (0, 0, 0))/Q× ,
where Q× operates on Q3 by left multiplication, of the equation
E:
Y 2 Z = X 3 + aXZ 2 + bZ 3 , a, b ∈ Q, ∆ := 4a3 + 27b2 6= 0
(3.1)
is called an elliptic curve over Q. The invariant ∆ is called the discriminant.
We denote by [x, y, z] an equivalence class in P2 (Q). The map from P2 (Q)
to Q2 , given by
[x, y, z] = [x/z : y/z : 1] 7→ (x/z, y/z),
(3.2)
for z 6= 0 is an embedding of P2 (Q) into Q2 and gives a bijection between
the set P2 (Q) \ {[x : y : 0] | x, y ∈ Q} and Q2 . Elements in P2 (Q) of the form
[x : y : 0] satisfy x3 = 0 by (3.4), thus x = 0. Thus the point [0 : 1 : 0] is the
only point in E having z-component equal to 0 and it is called the point at
infinity. Therefore, E is also given by the solutions of the equation
E:
Y 2 = X 3 + aX + b, a, b ∈ Q, ∆ := 4a3 + 27b2 6= 0
(3.3)
in Q2 with an additional point at infinity. We can make in (3.4) the change
of variables X 7→ X/c2 and Y 7→ Y /c3 , and choose c so that the new a and
b are integers and ∆ is minimal. Therefore, we can reduce the coefficients a
and b modulo a prime p to obtain a new equation
E:
Y 2 Z = X 3 + aXZ 2 + bZ 3 , a, b ∈ Fp ,
(3.4)
which is called the reduction of E at p. There are three cases to consider:
(a) Good reduction: If p 6= 2 and p - ∆ then E is an elliptic curve over
Fp .
(b) Cuspidal, or additive, reduction: For p 6= 2, 3 this occurs exactly when
p | ∆ and p | 2ab.
(c) split multiplicative reduction: For p 6= 2, 3 this occurs exactly when p | ∆
7
and p - −2ab and −2ab is a square in Fp .
(d) non split multiplicative reduction: For p 6= 2, 3 this occurs exactly when
p | ∆ and p - −2ab and −2ab is not a square in Fp .
The conductor Q of E is defined as
Y
Q=
pnp ,
(3.5)
p
where

 0
1
np =

≥2
if E has good reduction at p
if E has multiplicative reduction at p
if E has additive reduction at p
For a precise definition of the conductor one can consider for example [9,
Proposition 16.2]. Note that the discriminant divides the conductor.
L-functions are usually defined by a Dirichlet series and then it is shown,
if the Dirichlet coefficients are multiplicative, that it can be expressed as
a Euler product. Here we go the other way around and first define the Lfunction of an elliptic curve as a Euler product. By multiplying out the Euler
product we obtain a Dirichlet series and take these Dirichlet coefficients to
define a modular form, i.e. it has to be shown that by doing so we define
a modular form. For L-functions of certain modular forms it is proved that
they have analytic continuation, while for elliptic curves one did not know
if they have analytic continuation until the Modularity Theorem [7] proved
that each L-functions of an elliptic curve over Q is also the L-functions of a
modular form.
3.2
Definition of L(s, E(Q))
In the following we denote by #E(Fp ) the number of solutions of (3.3) in
P2 (Fp ), when the coefficients of this equation are reduced modulo p.
Definition 3.1. Let E be an elliptic curve over Q. We define the L-function
of E by
Y
Y
1
1
·
(3.6)
L(s, E) :=
−s
−s
1 − ap p
1 − ap p + pp−2s
p-∆
p|∆
8
where ∆ is the discriminant of E and
ap := p + 1 − #E(Fp )
(3.7)
for p - ∆, and for p | ∆

if E has split multiplicative reduction
 +1
−1
if E has non-split multiplicative reduction
ap :=

0
if E has additive reduction.
Remark 3.2. By a theorem of Hasse [8, Theorem 8.17], we have
1
| ap |≤ 2p 2 ,
(3.8)
so we can regard Hasse’s theorem as saying #E(Fp ) is approximately p + 1
√
with error term ap with | ap |≤ 2 p. Using (3.8) one can show that L(s, E)
converges for <(s) > 32 .
Lemma 3.3. For n ∈ N let cn ∈ C be the coefficients obtained by expanding
out the product (3.6), i.e. such that
L(s, E) =
∞
X
cn
n=1
ns
.
(3.9)
for <(s) > 23 . Then,
c p = ap
(3.10)
for any prime p, and

 cmn = cm cn if gcd(m, n) = 1
(∗) cpl+1 = cp cpl − pcpl−1 for l ≥ 1 and p - ∆

cp l
= cp l for l ≥ 1 and p | ∆.
Proof. For a prime p not dividing ∆ let
Lp (s) =
∞
X
cpn p−ns = 1 + cp p−s + cp2 p−2s + · · · .
n=0
Now the coefficients of p−s , respectively of p−(n+1)s for n ≥ 1, in the product
(1 − ap p−s + pp−2s )Lp (s)
9
is cp − ap , respectively cpn+1 − cp c(pn ) + pcpn−1 , hence
Lp (s) =
1
(1 − ap p−s + pp−2s )
if and only if we have cp = ap and cpn+1 = cp cpn − pcpn−1 . Similarly,
Lp (s) :=
∞
X
cpn p−ns =
n=0
1
1 − ap p−s
(3.11)
if and only if the third equality of (∗) holds.
Q
If nQ ∈ N has Q
prime factorization n =
pri , then the coefficient of p−ns
in Lp (s) is cpri , which equals cn if and only if the first equality in (∗)
holds.
Theorem 3.4. For n ∈ N let cn be defined as in Lemma 3.3. Define a
function f : H −→ C by
f (s) :=
∞
X
cn e2πins .
(3.12)
n=1
Then f is an eigenform in S2 (Γ0 (Q)) and
3
L(s, f ) = L(s, E), for <(s) > .
2
(3.13)
Proof. This is a consequence of the modularity theorem, proved by Breuil,
Conrad, Diamond and Taylor [7].
Remark 3.5. The L-function of an eigenform f is uniquely determined by
f , but the L-function of an elliptic curve E is uniquely determined only by
a class of L-functions, i.e. there might exist two different elliptic curves E
and E 0 such that L(s, E) = L(s, E 0 ).
Corollary 3.6. We have analytic continuation of L(s, E) to the whole plane
and
Λ(s, E) = γL (s)L(s, E),
(3.14)
where γL = (Q)s/2 (2π)−s Γ(s), satisfies the functional equation
Λ(s, E) = εΛ(2 − s, E),
where ε = ±1, depending on E.
10
(3.15)
Proof. This follows immediately from (3.13) and Theorem 2.3.
Theorem 3.7. (Deligne) For n ∈ N we have
1
cn = O(n 2 +ε )
(3.16)
for any ε > 0, which was the Ramanujan-Petersson conjecture. Further,
writing for p - ∆
1 − cp T + pT 2 = (1 − α1,p T )(1 − α2,p T )
(3.17)
α1,p + α2,p = c(p), α1,p · α2,p = p,
(3.18)
with
1
we have that α1,p and α2,p are complex conjugated with | α1,p |=| α2,p |= p 2 .
Proof. Deligne [10] proved this for L-functions of cusp forms, but by the
modularity theorem, this applies also to the L-function of an elliptic curve
over Q.
The next theorem was previously called the Sato-Tate Conjecture and
was proved in more generality than we state it here:
Theorem 3.8. (Taylor, Clozel, Harris, Shepherd-Barron) Let E be an elliptic
curve over Q. By Theorem 3.7, for any prime p there exists θp ∈ [−π, π] such
that
√ iθp
p(e + e−iθp )
1 + p − #E(Fp )
cp
=
=
= cos θp .
(3.19)
√
√
√
2 p
2 p
2 p
Then for any continuous function f (t) on the interval [−1, +1] we have that
the limit
1 X
lim
f (cos θp )
(3.20)
C→∞ π(C)
p≤C
exists, where π(·) is the prime counting function, and is equal to
2
π
Z+1
√
f (t) 1 − t2 dt.
(3.21)
−1
Proof. See Theorem 4.3 of the paper of Harris, Taylor and Shepherd-Barron
[5].
11
This theorem is formulated for elliptic curves over a totally real field with
multiplicative reduction at some prime. These conditions are satisfied for
elliptic curves over Q.
The following lemma will be used to prove Theorem 6.5. Note that contrary
to the proof of Theorem 6.5 in [4], this lemma is a direct consequence of the
Sato-Tate conjecture, while Conrey et al only assume this lemma to hold
conjecturally.
Corollary 3.9. We have
1 X
| bp |2 = 1,
N →∞ π(N )
p≤N
lim
(3.22)
where π(·) is the prime counting function.
Proof. Since bp =
f (t) = 4t2 .
3.3
cp
√
p
= 2 cos θp this follows from Theorem 3.8 by taking
The renormalized L-function
For any prime p define
α2,p
α1,p
γ1,p := √ , γ2,p := √ ,
p
p
(3.23)
and the normalized L-function
1
L(s) := L(s + , E),
2
(3.24)
where αi,p is defined as in Theorem 3.7. Normalizations like above are done
to be able to speak of L-functions in the Selberg class, which is a set of Lfunctions satisfying different axioms (see Section 7). Since L-functions appear
in different contexts, such as L-functions of elliptic curves or modular forms,
but have many properties in common, it is very reasonable to postulate the
existence and to axiomatize a set of L-functions containing all the known
L-functions. For more information on the Selberg see Section 7.
12
Theorem 3.10. For <(s) > 1 we have
L(s) =
1
Y
p
(1 − γp,1
p−s )(1
− γp,2 p−s )
(3.25)
with
γ p,1 = γp,2
(3.26)
| γp,i |= 0 or 1
(3.27)
and
for i = 1, 2.
Proof. This is an immediate consequence of Theorem 3.7.
Remark 3.11. Note that L(s) converges for <(s) > 1. Expanding out the
product (3.25) into a Dirichlet series, we obtain
L(s) =
∞
X
bn
n=1
(3.28)
ns
with bp = √cpp = 2 cos θp , where θp is defined as in Theorem 3.8, for any prime
p. Following the same steps as in Lemma 3.3, we obtain

 bmn = bm bn if gcd(m, n) = 1
(∗) bpl+1 = bp bpl − bpl−1 for l ≥ 1 and p - ∆

bp l
= bp l for l ≥ 1 and p | ∆.
Using Theorem 3.7, and the fact that bn =
ε > 0.
cn
√
,
n
we obtain bn = O(nε ) for any
Corollary 3.12. Let
1
,
(3.29)
ξL (s) := Λ s +
2
where Λ is defined as in Corollary 3.6. Then we have the following functional
equation
ξL (s) = ε · ξL (1 − s),
(3.30)
where ε = ±1 is the same ε as in Corollary 3.6.
13
Proof. Using Corollary 3.6 we obtain
1
1 1
ξL (s) = Λ s+
= εΛ 2− s+
= εΛ (1−s)+
= εξL (1−s). (3.31)
2
2
2
According to Corollary 3.12 we define
s+1/2
1
1
−(s+1/2)
2
= (Q)
γL (s) := γL s +
(2π)
Γ s+
2
2
s+1/2
1
1
1
3
1
= (Q) 2
Γ s+
,
Γ s+
2π s+1 2
2
2
4
where for the last equality we have used the duplication formula
1
Γ(2z) = 22z−1 π −1/2 Γ(z)Γ z +
2
for z := 21 s + 12 .
4
(3.32)
(3.33)
Approximate Functional Equations for the
Riemann zeta function and elliptic curve
L-functions over Q
Since for applying the recipe of Conrey et al we have to use the so called
approximate functional equation for L-functions in the Selberg class, we will
reformulate here a theorem obtained by Hardy and Littlewood [11, Theorem
A]. They used this approximate functional equation to determine the second
moment of the Riemann zeta function on the critical line. The zeta function
is represented for <(s) > 1 by
∞
X
1
ζ(s) =
ns
n=1
and for <(s) < 0 it is represented by the functional equation
ζ(s) = X (s)ζ(1 − s).
The approximate functional equation serves as a compromise of these two
representations to represent the zeta function in the strip 0 ≤ <(s) ≤ 1, as
the following Theorem of Hardy and Littlewood [11, Theorem A] shows:
14
Theorem 4.1. Let K, H ∈ R be positive constants. For all x, y, σ, t ∈ R
such that
x > K, y > K,
(4.1)
and
− H ≤ σ ≤ H, | t |= 2πxy
(4.2)
we have
| ζ(s) −
X 1
X 1 |≤ C · R(x, y, σ, t),
+
X
(s)
s
1−s
n
n
n<y
n<x
where
(4.3)
1
R(x, y, σ, t) = (x−σ + y σ−1 | t | 2 −σ ), s = σ + it,
C = C(H, K) only depends on K and H and
1
X (s) = 2(2π)s−1 sin( sπ)Γ(1 − s)
2
is the factor coming from the functional equation of ζ(s). If we take, in
addition to (4.1) and (4.2), x ≥ y then
R=
x1−σ
,
| t |1/2 +x[y]
(4.4)
where [y] denotes the absolute value of the difference between y and the integer
nearest to y. If we take, in addition to (4.1) and (4.2), y ≥ x then
R=
| t |1/2−σ y σ
.
| t |1/2 +y[x]
(4.5)
Corollary 4.2. Setting <(s) = 21 and x = y = (| t | /2π)1/2 in Theorem 4.1
we obtain
1
1
X
X
1
1
ζ
+ it =
+
X
+
it
+ O(| t |−1/4 ). (4.6)
1
1
+it
−it
2
2
2
2
1 n
1 n
|t| 2
2π
|t| 2
2π
n< √
n< √
Approximate functional equations for ζ(s) were also studied by Siegel [12],
and for a more general class of zeta functions by K. Chandreskharan and R.
15
Narasimhan [13]. A geometric approach to understand the functional equation of the Riemann zeta function is given in [14]. An approximate functional
equation for elliptic curve L-functions over Q is given in [15] and we rephrase
it in the next theorem. We will formulate this theorem in terms of the nonrenormalized L-function. Clearly, an approximate functional equation could
also be formulated for the renormalized L-function given in (3.9).
The function
Z∞
Γ(s, z) = e−t ts−1 dt
(4.7)
z
for z ∈ R, <(s) > 0 is called the incomplete Gamma function of second kind
and it can be shown that it has analytic continuation in the variable z to the
right half plane <(z) > 0. Recall the functional equation of L(s, E)
(Q)s/2 (2π)−s Γ(s)L(s, E) = ε(Q)(2−s)/2 (2π)−(2−s) Γ(2 − s)L(2 − s, E), (4.8)
which was given in Corollary 3.6.
Theorem 4.3. Let s√= σ + it be a complex number such that 21 ≤ σ ≤ 32 and
M ∈ N with M > t Q/4. Set r = ei(π/2−δ(t)) , where δ(t) is some function
satisfying 0 < δ(t) ≤ π/2 for all t ∈ R. Then
1 X cn 2πnr Γ s, √
L(s, E) =
Γ(s) n≤M ns
Q
εQ1−s (2π)2(s−1) X cn 2πn √
−
Γ
2
−
s,
Γ(s)
n2−s
Qr
n≤M
+
(2π)s
R,
Γ(s)
(4.9)
where R describes the error term which is given by
√
δ(t)(t−4 √MQ ) (1−σ)/2
| R |=e−πt/2 e
Q
M δ(t)−1 ×
log M + σ + 1 (σ − 1)(log M + 2) × 1+
+
.
2tδ(t)
4(tδ(t))2
(4.10)
Note that for <(s) bounded and <(z) > c for some c > 0 we have
| Γ(s, z) || z s |,
16
(4.11)
uniformly in s and z, by [15, (6)]. Consequently we obtain for s = 1/2 + it
π
| Γ(1/2 + it, Ar) | A1/2 e−( 2 −δ(t))t ,
(4.12)
for any A ∈ R>0 , as | t |→ ∞. Using the asymptotic
√
π
1
| Γ(σ + it) |∼ 2π | t |σ− 2 e− 2 |t|−σ
(4.13)
we obtain for σ = 1/2
π
1
Ce 2 |t| ,
| Γ(1/2 + it) |
(4.14)
for some constant C independent of t, as | t |→ ∞. Thus
| Γ(1/2 + it, Ar) |
CA1/2 eδ(t)t ,
| Γ(1/2 + it) |
as | t |→ ∞. So if we take for example δ(t) = 1t , we see that
remains bounded as | t |→ ∞.
(4.15)
|Γ(1/2+it,Ar)|
|Γ(1/2+it)|
Remark 4.4. There are two main differences between the approximate
functional equation of L(s, E) and the one of the zeta function. The first is,
that in the approximate functional equation for L(s, E) we obtain incomplete
Γ-factors in the summations. And the second is, that the upper bound of
summation, which is denoted by M in the case of the approximate functional
δ(t)(t−4 √MQ )
equation of L(s, E), does not depend on t. Thus, due to the factor e
in the error term R, we can let M → ∞ in (4.9) to obtain the following
corollary:
Corollary 4.5. Let s = σ + it be a complex number such that 21 ≤ σ ≤ 32 .
Set r = ei(π/2−δ(t)) , where δ(t) is some function satisfying 0 < δ(t) ≤ π/2 for
all t ∈ R. Then
∞
1 X cn 2πnr L(s, E) =
Γ s, √
Γ(s) n=1 ns
Q
−
∞
εQ1−s (2π)2(s−1) X cn 2πnr √
Γ
2
−
s,
.
2−s
Γ(s)
n
Q
n=1
17
(4.16)
The approximate functional equation, which is considered in the recipe
of Conrey et al, is of the following form
L(s) =
X bm
X bn
+
X
(s)
+ remainder
L
ms
n1−s
m<x
n<y
(4.17)
where xy might depend on parameters of the functional equation of L(s) and
on s and the X -factor is given by
1
1
1
3
Γ
(1
−
s)
+
(1
−
s)
+
1/2 1−2s Γ
2
2
2
4
Q
γL (1 − s)
=
.
XL (s) =
·
γL (s)
π
Γ 1s + 1 Γ 1s + 3
2
2
2
4
(4.18)
As we will see in the next section, Conrey et al will replace this L-function
by the right side of (4.17) setting x = y = ∞ and omitting the remainder,
i.e. by
∞
∞
X
X
bn
bm
+
X
(s)
.
(4.19)
L
s
1−s
m
n
n=1
m=1
The only possible area of convergence of (4.19) is the critical strip, since the
left sum converges only for <(s) > 1 and the right one converges only for
<(s) < 0. The recipe will omit some summands such that we obtain a sum,
converging on some half plane which contains the critical line.
5
Using the recipe to conjecture the moments
of the product of shifted elliptic curve Lfunctions over Q
We are going to evaluate the moments of the Z-function
1
1
ZL (s) := ε− 2 XL (s)− 2 L(s).
(5.1)
To do so, we will use the recipe given in [4]. By Stirling’s formula we have
πt
1
Γ(σ + it) = e− 2 tσ− 2
t it
e
1 iπ
1
i 1
σ σ2 e 2 (σ− 2 ) 1 −
− +
+ O 2 , (5.2)
t 12 2
2
t
18
thus
t 1−2s
1
XL (s) = N
e
1 + O( t )
2
t
= e(1−2s) log N e(1−2s) log 2 e2it+iπ/2 1 + O( 1t )
−i2t(log N +log 2t −1)
(1−2σ) log N +(1−2σ) log 2t
1
·e
=e
1 + O( t )
1−2s
2it+iπ/2
(5.3)
(5.4)
(5.5)
√
as t → ∞, where N = 2πQ . We see that as t goes to ∞, XL (s) is oscillating
due to the second exponential factor in (5.5).
The idea of the recipe is to replace in the product of shifted Z-functions each
Z-factor by its approximate functional equation obtaining then, by multiplying out, an infinite sum. In this infinite sum we omit then many terms,
so that the final infinite sum obtained this way will be the one contributing
to the asymptotic of the moments of the shifted Z-functions. It is not well
understood, even for Conrey et al, why we are allowed to omit all these terms
and still obtain the correct conjectural asymptotic, where correct means when
evaluated numerically, the recipe seems to give the right asymptotic. Now
we are ready to use the recipe:
(1) For k ∈ N we start with the product of 2k shifted Z-functions :
Z(s, α1 , ..., α2k ) = Z(s + α1 ) · · · Z(s + α2k )
(5.6)
where (α1 , ..., α2k ) ∈ C2k . The recipe will determine first the moment of this
product and then, by letting αi → 0 for i = 1, ..., 2k, we obtain the 2k-th
moment of ZL .
(2) By definition of Z,
2k
Y
j=1
Z(s + αj ) =
2k
Y
ε
− 12
− 21
XL (s + αj )
2k
Y
L(s + αj ).
(5.7)
j=1
j=1
Replace in each Z-function the factor L(s + αj ) by
∞
∞
X
X
bn
bm
+
X
(s)
,
L
s+αj
1−s−αj
n
m
n=1
m=1
(5.8)
where these two terms come from the approximate functional equation. Note
that (5.8) does not converge anywhere on C, since the left sum is divergent for
19
σ < 1 − <(αj ) and the right sum is divergent for σ > −<(αj ). Since finally,
we want to determine the moment of ZL on the critical line, the recipe will
produce, by omitting terms in the product of sums like (5.8), a convegent
sum whose half plane of convergence contains the critical line. For this see
the following two steps.
Multiply out the resulting expression to obtain 22k terms. Note that each
term is of the form
− 1
XL ((β1 + it) · · · XL (βJ + it) 2
×
(5.9)
XL (γ1 + it) · · · XL (γK + it)
X bn
X bn X bm
X bm
1
1
J
K
×
·
·
·
·
·
·
(5.10)
1−γ1 −it
1−γK −it
β1 +it
βJ +it
m
m
n
n
1
1
K
J
n1
nJ
m1
mK
with J + K = 2k and βi , γj ∈ 12 + αl ; 1 ≤ l ≤ 2k for 1 ≤ i ≤ J and
1 ≤ j ≤ K. To take one specific example, suppose the first ` factors were
chosen from the first piece of the approximate functional equation, and the
last 2k − ` factors were taken from the second piece:
1
1
ε−k XL ( 21 + α1 + it)− 2 · · · XL ( 12 + α` + it)− 2
n1
2k−`
×ε
X
×
XL ( 21
1
2
· · · XL ( 12
bn1
X
1
+α1 +it
2
n1
···
X
n`
b
n`
1
+α` +it
2
n`
1
2
+ α`+1 + it)
+ α2k + it)
X bn
bn`+1
2k
·
·
·
1
1
−α
−it
−α
`+1
2k −it
2
2
n`+1 n`+1
n2k n2k
− 12
1
1
+
α
+
it)
·
·
·
X
(
+
α
+
it)
X
(
1
L
`
L
2
2
= εk−`
XL ( 12 + α`+1 + it) · · · XL ( 21 + α2k + it)
−it
X
X
bn1 · · · bn` bn`+1 · · · bnk
n1 · · · n`
×
···
(5.11)
1
1
1
1
+α1
+α
−α`+1
−α2k
n`+1 · · · n2k
2
2
2
· · · n`2 ` n`+1
· · · n2k
n1
n2k n1
(3) Now we have to determine those terms, which have no rapidly oscillating X -factors in front since, as we will see, only these terms will have
contribution to the asymptotic of the product of shifted L-functions Z. Note
20
that by (5.5) we have
XL (β1 + it) . . . XL (βJ + it)
XL (γ1 + it) . . . XL (γK + it)
= e(
P
βi −
P
− 21
γj )(log Q+log (t/2))
×
×e−it(log Q+log t/2−1)2(J−K) (1 + O(1/t))
(5.12)
which is rapidly oscillating unless J = K, i.e. J = K = k since J + K =
2k. Thus, the recipe tell us to keep those terms which have k factors X (βj +it)
in the denominator
and k factors X (γj + it) in the nominator. This gives
2k
a total of k terms in the final answer. For our specific example this says
k = l and we obtain
− 21
XL ( 21 + α1 + it) . . . XL ( 21 + αk + it)
XL ( 12 + αk+1 + it) . . . XL ( 21 + α2k + it)
(α1 +···+αk −αk+1 −···−α2k )
Qt
(1 + O( 1t )).
(5.13)
=
2
(4) Now the recipe tells us to keep the diagonal from the remaining which
in the specific example considered in (5.11), is the terms where n1 · · · n` =
n`+1 · · · n2k . Note that we are again omitting terms like we already did in
step (3). Therefore we define
X
R(s, α) =
n1 ···nk =nk+1 ···n2k
bn1 · · · bnk bnk+1 · · · bn2k
1
ns+α
1
s−α
2k
k
· · · ns+α
nk+1 k+1 · · · ns−α
k
2k
.
(5.14)
Note that the function R(s; α1 , . . . , αk , αk+1 , . . . , α2k ) is symmetric in α1 , . . . , αk
and in αk+1 , . . . , α2k . So we can rearrange them so that the first k are in increasing order, as the
last k are. Thus, the final result will be a sum of
2k
terms indexed by k permutations in S2k such that σ(1) < . . . < σ(k) and
σ(k + 1) < . . . < σ(2k). We denote the set of such permutations by Ξ. So
for σ ∈ Ξ and z = x + iy let
W (z, α, σ) =
Qy
2
(ασ(1) +···+ασ(k) −ασ(k+1) −···−ασ(2k) )
R(x; ασ(1) , . . . , ασ(2k) ),
(5.15)
21
where the factor in front of R arises the same way as in (5.13).
(5) Finally, let
X
M (z; α) =
W (z, α, σ),
(5.16)
σ∈Ξ
which is the sum of all terms having no rapidly oscillating X -factor, and we
arrive at the conjecture
Conjecture 5.1. We have
Z∞
Z( 21 +it, α1 , . . . , α2k )g(t) dt =
−∞
Z∞
1
M ( 12 +it, α1 , . . . , α2k )(1+O(t− 2 +ε ))g(t) dt,
−∞
(5.17)
for all ε > 0, where g is a suitable weight function.
Note that we do not know where the function M converges and if it has
analytic continuation to an area containing the critical line. Theorem 6.5,
given by Conrey et al in [4], will determine a half plane of convergence for
M , containing the critical line.
6
Conjecturing the moments for primitive Lfunctions of the Selberg class
In this section we will give the moment conjecture for primitive L-functions
of the Selberg class, satisfying (6.13) and (6.14) and Conjecture 6.4. As we
will see, the renormalized elliptic curve L-function over Q is also such a Lfunction, therefore all the conjectures and results formulated in this section
also hold for renormalized elliptic curve L-function over Q.
6.1
Definition of the Selberg class
Definition 6.1. The Selberg class is the set of all Dirichlet series
L(s) =
∞
X
an
n=0
22
ns
(6.1)
satisfying the following axioms:
(1) The Dirichlet series converges absolutely for σ > 1.
(2)Analytic continuation: There exists an integer m ≥ 0 such that
(s − 1)m L(s) is entire.
(3) Functional equation: L satisfies the following functional equation
Φ(s) = δ Φ̄(1 − s)
where
Φ(s) = Q
s
r
Y
Γ(λj s + µj )L(s)
(6.2)
(6.3)
j=1
and r ≥ 0, Q > 0, λj > 0, <µj > 0, | δ |= 1 depend on L.
(4) Ramanujan hypothesis: For every ε > 0 we have an nε .
(5) Euler product: For σ > 1 we have
Y
Lp (1/ps )
L(s) =
(6.4)
p
and
s
log Lp (1/p ) =
∞
X
bp n
n=0
pn
(6.5)
with bn nθ/2 for some θ < 1/2.
P
The number 2 rj=1 λj is called the degree of the L-function. Note that
r can also be 0 and in this case we take the L-function to be the constant 1.
This is the only constant function in the Selberg class.
Definition 6.2. A L-function in the Selberg class is called primitive if it is
not the product of non-trivial L-functions in the Selberg class.
Remark 6.3. One immediately realizes that renormalized elliptic curve Lfunctions over Q are primitive L-functions of the Selberg class of degree 2,
since λ1 = λ2 = 21 by Corollary 3.12 and (3.32).
23
6.2
The conjectures produced by the recipe when applied to primitive L-functions of the Selberg class
In [4], the recipe is used to conjecture the moments of primitive L-functions
of the Selberg class which satisfy (6.13) and (6.14) and Conjecture 6.4. We
define
X
an1 · · · ank ank+1 · · · an2k
,
(6.6)
R(s, α) =
s+α1
2k
k s−αk+1
nk+1 · · · ns−α
· · · ns+α
2k
k
n1 ···nk =nk+1 ···n2k n1
which corresponds to the function R defined in 5.14 for the case of an elliptic
curve L-function over Q.
Conjecture 6.4. Let
L(s) =
∞
X
an
n=0
ns
be a primitive L-function in the Selberg class, satisfying (6.13) and (6.14).
Then
1 X
| ap |2 = 1,
(6.7)
lim
N →∞ π(N )
p≤N
where π(·) is the prime counting function.
Theorem 6.5. Suppose |αj | < δ for j = 1, . . . , 2k and
1 X
| ap |2 = 1,
N →∞ π(N )
p≤N
lim
(6.8)
where π(·) is the prime counting function. Then R(s; α1 , . . . , α2k ) converges
absolutely for σ > 21 + δ and has meromorphic continuation to σ > 14 + δ.
Furthermore,
R(s; α1 , . . . , α2k ) =
k
Y
ζ(2s + αi − αk+j ) Ak (s; α1 , . . . , α2k )
(6.9)
i,j=1
where
Ak (s; α1 , . . . , α2k ) =
k
Y Y
p
(1 − p−2s−αi +αk+j Bp (s; α1 , . . . , α2k )
i,j=1
24
(6.10)
with
Bp (s; α1 , . . . , α2k ) =
Z1 Y
k
0
Lp
j=1
e(θ)
ps+αj
Lp
e(−θ)
ps−αk+j
dθ.
(6.11)
Proof. See [4].
Remark 6.6. Note that elliptic curve L-functions over Q satisfy the condition that | ap |2 is 1 on average, by Corollary 3.9. Since the function M
in Conjecture 5.1 involves finitely many expressions of the form (6.9), we
see by Theorem 6.5 that M has meromorphic continuation to a half plane
containing the critical line, and it is shown in [4, Section 2.5] that M is in
fact a polynomial. Therefore, we will rephrase Conjecture 5.1 in Conjecture
6.7 in terms of this polynomial. The value Ak ( 12 ; 0, . . . , 0) is the arithmetical
factor for primitive L-functions, satisfying (6.13) and (6.14) and Conjecture
6.4, as we will see later. We see that in the case of a renormalized elliptic curve L-function over Q, this value is well defined by Theorem 6.5 and
Corollary 3.9. For primitive L-functions, satisfying (6.13) and (6.14), the
value Ak ( 21 ; 0, . . . , 0) is only defined, if Conjecture 6.4 holds (it is believed
that it holds also for primitive L-functions). Assuming that Conjecture 6.4
also holds for primitive L-functions in the Selbeg class , satisfying (6.13) and
(6.14), we will determine in Theorem 7.4 the value of Ak ( 12 ; 0, . . . , 0), i.e. of
Ak (0, . . . , 0).
Now let
Y
∆(z1 , ..., zm ) =
(zi − zj )
(6.12)
1≤i<j≤m
denote the Vandermonde determinant. Let L(s) be a primitive L-function
in the Selberg class and assume that L(s) satisfies the functional equation
r
Y
r
Y
1
1
1−s
Q
Γ( s + µj )L(s) = δQ
Γ( (1 − s) + µj )L(1 − s),
2
2
j=1
j=1
s
(6.13)
i.e. with λj = 21 . Further assume that it satisfies
L(s) =
Y
p
25
Lp (x),
(6.14)
where
r
Y
Lp (x) =
(1 − γj,p x)−1
(6.15)
j=1
with | γi,p |= 0 or 1 for some r ≥ 1 and all primes p. The following conjecture
is given in [4]:
Conjecture 6.7. Let L(s) be a primitive L-function in the Selberg class,
satisfying (6.13) and (6.14) and Conjecture 6.4. Then
Z∞
Z( 21 +it, α1 , . . . , α2k )g(t) dt
Z∞
Pk (r log
=
Q2/r t 2
1
, α)(1+O(t− 2 +ε ))g(t)dt
−∞
−∞
(6.16)
where Z is defined as in (5.6) and Pk is the polynomial (in x) of degree k 2
given by the 2k-fold residue
(−1)k 1
Pk (x, α) =
k!2 (2πi)2k
I
I
···
G(z1 , . . . , z2k )∆2 (z1 , . . . , z2k )
×
2k Y
2k
Y
(zj − αi )
j=1 i=1
×
x Pk
e 2 j=1 zj −zk+j
dz1 . . . dz2k ,
(6.17)
with the path of integration being small circles around the poles αi , with
G(z1 , . . . , z2k ) = Ak (z1 , . . . , z2k )
k Y
k
Y
ζ(1 + zi − zk+j ),
(6.18)
i=1 j=1
and Ak is the Euler product
Ak (z1 , . . . , z2k ) =
k Y
k YY
p
i=1 j=1
1−
1
k
Z1 Y
p1+zi −zk+j
0
j=1
Lp
e(θ) 1
p 2 +zj
Lp
e(−θ) 1
p 2 −zk+j
dθ.
(6.19)
Remark 6.8. We will show in Theorem 7.2 that Pk (x) = Pk (x, 0, . . . , 0) is
really a polynomial in x of degree k 2 . Note that the factor Ak in Theorem
26
6.5 and the factor Ak in Conjecture 6.7 are not the same, we only have the
equality
1
1
1
1
Ak (s; α1 , . . . , α2k ) = Ak (s− +α1 , . . . , s− +αk , −s+αk+1 , . . . , −s+α2k ),
2
2
2
2
(6.20)
and therefore we obtain
1
Ak ( ; 0, . . . , 0) = Ak (0, . . . , 0).
2
7
(6.21)
The arithmetic factor in the conjecture
In this section we will show that Pk , defined in (6.17), is a polynomial of
degree k 2 , by following the proof of Conrey et al given in [6]. Conrey et
al prove this in the case of the zeta function, but as we will see the same
methods also apply for primitive L-function in the Selberg class, satisfying
(6.13) and (6.14) and Conjecture 6.4. The next lemma is formulated and
proved in Section 2.1 of [6] and we are going to give a proof only of Theorem
7.2, which is similar to the one in [6], but giving more explanations. Recall
the definition of the Vandermonde in (6.12).
Lemma 7.1. Let f : Cn → C be a function. Then
∆(∂/∂x1 , ..., ∂/∂x2k )∆(∂/∂x1 , ..., ∂/∂xk )∆(∂/∂xk+1 , ..., ∂/∂x2k )
2k
Y
f (xi )
i=1
(7.1)
evaluated at x1 = · · · = xk = 1, xk+1 = · · · = x2k = −1 equals
f (1)
f (1) (1) . . . f (2k−1) (1) f (1) (1)
f (2) (1) . . .
f (2k) (1) ..
..
..
..
.
.
.
.
(k−1)
(k)
(3k−2)
f
(1)
f
(1)
.
.
.
f
(1)
k!2 (1)
(2k−1)
f (−1) . . . f
(−1) f (−1)
(1)
f (−1) f (2) (−1) . . . f (2k) (−1) ..
..
..
..
.
.
.
.
f (k−1) (−1) f (k) (−1) . . . f (3k−2) (−1) 2k×2k.
27
Theorem 7.2. Let Pk (x) = Pk (x, 0, ..., 0) be given as in (6.17). Then Pk (x)
is a polynomial of degree k 2 and has leading coefficient
c0 (k) = ak
k−1
Y
j!
,
(k
+
j)!
j=0
where
ak := Ak (0, ...0),
(7.2)
Ak (z1 . . . , z2k ) is given in (6.19).
Proof. We have
I
I
(−1)k 1
· · · Ak (z1 , ..., z2k )×
Pk (x) =
k!2 (2πi)2k
Qk Qk
2
P
j=1 ζ(1 + zi − zj+k )∆ (z1 , . . . , z2k ) x kj=1 zj −zk+j
i=1
2
×
dz1 . . . dz2k
e
2k
Y
zi2k
i=1
k
(−1)
1
=
2
k! (2πi)2k
I
···
I Ak (0, ..., 0) + {higher order terms in z1 , . . . , z2k } ×
(7.3)
× (1 + {higher order terms in z1 , . . . , z2k })∆(z1 , . . . , z2k )∆(z1 , . . . , zk )
(7.4)
x Pk
× ∆(zk+1 , . . . , z2k )
e2
j=1 zj −zk+j
2k
Y
dz1 . . . dz2k
(7.5)
zi2k
i=1
where in (7.3) we use the multivariate Taylor expansion of Ak (z
z2k ) and
Q1k, ...,Q
in (7.4) the Taylor expansion of ζ and the fact that the product i=1 kj=1 ζ(1+
zi − zj+k ) has k 2 poles of the form
ζ(1 + zi − zj+k ) =
1
+ γ0 + γ1 (zi − zj+k ) + higher order terms (7.6)
zi − zj+k
and these poles cancel k 2 factors of the Vandermonde ∆(z1 , ..., z2k )2 such that
the factor
28
∆(z1 , ..., z2k )2
(7.7)
∆(z1 , ..., z2k )∆(z1 , ..., zk )∆(zk+1 , ..., z2k )
(7.8)
reduces to
by using the fact
∆(z1 , ..., z2k ) = ∆(z1 , ..., zk )∆(zk+1 , ..., z2k )
Y
(zi − zj+k ).
(7.9)
1≤i,j≤k
Now consider the following residue, which follows form (7.4) by omitting
terms of higher order:
I
I
(−1)k ak 1
· · · ∆(z1 , . . . , z2k )∆(z1 , . . . , zk )∆(zk+1 , . . . , z2k )×
k!2 (2πi)2k
x Pk
×
e2
j=1 zj −zk+j
2k
Y
dz1 . . . dz2k ,
(7.10)
zi2k
i=1
where ak := Ak (0, ...0). The term ∆(zk+1 , . . . , z2k )2 has 2
factors, and since k 2 are canceled from it,
2k
2
= 2k(2k − 1)
∆(z1 , . . . , z2k )∆(z1 , . . . , zk )∆(zk+1 , . . . , z2k )
(7.11)
has degree 2k(2k − 1) − k 2 . Define a function q(z1 , ..., z2k ) by
∆(z1 , . . . , z2k )∆(z1 , . . . , zk )∆(zk+1 , . . . , z2k )
1
=
, (7.12)
Q2k
Q2k 2k
( i=1 zi )q(z1 , ..., z2k )
i=1 zi
i.e.
Q2k
q(z1 , ..., z2k ) =
2k−1
i=1 zi
∆(z1 , . . . , z2k )∆(z1 , . . . , zk )∆(zk+1 , . . . , z2k )
.
(7.13)
Since the numerator of (7.13) has 2k(2k − 1) − k 2 = 3k 2 − 2k factors, we see
that each summand in
1
q(z1 , ..., z2k )
29
(7.14)
must be of the form
Q2k
zili
Q2ki=1 2k−1 ,
i=1 zi
with
P2k
i=1 li
(7.15)
= 2k(2k − 1) − k 2 = 3k 2 − 2k, i.e. of the form
1
Q2k
(7.16)
ri
i=1 zi
with
P2k
i=1 ri
= k 2 . So in the integrand
∆(z1 , . . . , z2k )∆(z1 , . . . , zk )∆(zk+1 , . . . , z2k ) x Pkj=1 zj −zk+j
e2
Q2k 2k
z
i=1 i
(7.17)
only terms of degree k 2 in the Taylor expansion of the exp contributes to
the residue. We also see that by considering in (7.10) terms of higher order
instead of only the constant terms, then doing steps similar to the ones above,
2
we obtain coefficients of lower powers than xk . For example, for calculating
c1 we have to consider in (7.3) the next higher order term, which is
Ak,1 (0, ..., 0)
2k
X
zi ,
(7.18)
i=1
where Ak,1 (0, ..., 0) denotes the coefficient of the first non-constant term in
the multivariate Taylor expansion of Ak (z1 , ..., z2k ). In (7.4) we have also to
consider the next higher order term, which is
γ0
k
X
zi − zi+k .
(7.19)
i=1
So the integrand we have to consider, is now of the form
∆(z1 , . . . , z2k )∆(z1 , . . . , zk )∆(zk+1 , . . . , z2k )
×
(7.20)
Q2k 2k
i=1 zi
2k
k
x Pk
X
X
× Ak,1 (0, ..., 0)
zi + Ak (0, ..., 0)γ0
zi − zi+k e 2 j=1 zj −zk+j . (7.21)
i=1
i=1
30
Now since the summands of the factor in (7.20) are of the form as stated
in (7.16), we see that we have only to consider terms of order k 2 − 1 in the
Taylor expansion of the exponential, since on order is coming from the left
factor in (7.21).
The last steps also show that Pk (x) is of degree k 2 in x. Consequently, we
have shown that the leading coefficient is given by
I
I
1
ak
k2
· · · ∆(z1 , . . . , z2k )∆(z1 , . . . , zk )×
(7.22)
c0 (k)x = 2
k! (2πi)2k
x Pk
× ∆(zk+1 , . . . , z2k )
j=1 zj −zk+j
e2
2k
Y
dz1 . . . dz2k ,
(7.23)
zi2k
i=1
where the factor (−1)k vanishes because the k 2 poles of the ζ product have
sign opposite from the Vandermonde factors that they cancel. Changing
variables ui = xzi /2 and then relabeling ui with zi gives
I
I
2
1
x k ak
k2
· · · ∆(z1 , . . . , z2k )∆(z1 , . . . , zk )×
c0 (k)x = k2 2
2 k! (2πi)2k
Pk
× ∆(zk+1 , . . . , z2k )
e
j=1 zj −zk+j
2k
Y
dz1 . . . dz2k .
(7.24)
zi2k
i=1
To evaluate (7.24) we have to use a trick by introducing new variables xi and
considering the residue
Pk
I
I
1
e i=1 xi zi
dz1 . . . dz2k .
(7.25)
· · · p(z1 , ..., z2k ) 2k
(2πi)2k
Y
zi2k
i=1
for a polynomial p(z) in z1 , ..., z2k . Pulling out the polynomial p in (7.25) we
get
I
I P k xi z i
1
e =1
p(∂/∂x1 , ..., ∂/∂x2k )
·
·
·
dz1 . . . dz2k ,
(7.26)
2k
(2πi)2k
Y
zi2k
i=1
31
and taking the residue gives
p(∂/∂x1 , ..., ∂/∂x2k )
2k
Y
i=1
xi2k−1
.
(2k − 1)!
(7.27)
Therefore (7.24) equals
2
2k
Y
x k ak
x2k−1
i
p(∂/∂x
,
...,
∂/∂x
)
1
2k
2
(2k
− 1)!
2k k!2
i=1
(7.28)
evaluated at x1 = 1, . . . , xk = 1, . . . , xk+1 = −1, . . . , x2k = −1, with
p(∂/∂x1 , ..., ∂/∂x2k ) =∆(∂/∂x1 , ..., ∂/∂x2k )∆(∂/∂x1 , ..., ∂/∂xk )×
× ∆(∂/∂xk+1 , ..., ∂/∂x2k ).
(7.29)
Using Lemma 7.1 with
f (x) =
x2k−1
(2k − 1)!
we see that
p(∂/∂x1 , ..., ∂/∂x2k )
2k
Y
i=1
equals
Γ(2k)−1
Γ(2k − 1)−1
Γ(2k − 1)−1
Γ(2k − 2)−1
..
..
.
.
−1
Γ(k + 1)
Γ(k)−1
k!2 −1
−Γ(2k)
Γ(2k − 1)−1
−Γ(2k − 2)−1
(Γ(2k − 1)−1
..
..
.
.
(−1)k Γ(k + 1)−1 (−1)k+1 Γ(k)−1
−1
(7.30)
x2k−1
i
(2k − 1)!
...
...
..
.
Γ(1)−1
Γ(0)−1
..
.
...
...
...
..
.
Γ(−k + 2)−1
Γ(1)−1
−Γ(0)−1
..
.
. . . (−1)3k−1 Γ(−k + 2)−1
(7.31)
2k×2k,
(7.32)
= 0 if m ∈ {0, −1, −2, −3, . . .} and the i, j entry
where we take Γ(m)
above is given by
(
Γ(2k − i − j + 2)−1 ,
(−1)i+j−k−1 Γ(3k − i − j + 2)−1
32
if 1 ≤ i ≤ k;
if k + 1 ≤ i ≤ 2k.
(7.33)
Lemma 3.1 together with Theorem 3.2 of [6] show how the determinant in
(7.32) can be evaluated, so that to obtain
c0 (k) = ak
k−1
Y
l=0
l!
(k + l)!
(7.34)
Remark 7.3. Note that Theorem 7.2 gives the factor f (k) explicitly, while
the factor a(k) = Ak (0, . . . , 0) is not given explicitly. The next theorem will
allow us to compute explicitly the arithmetic factor ak .
Theorem 7.4. Let L(s) be a primitive L-function in the Selberg class, satisfying (6.13) and (6.14) and Conjecture 6.4. Then the arithmetic factor of
L is given by
∞
Y
X
−1 k2
(1 − p )
| cn |2 p−n/2 ,
ak = Ak (0, . . . , 0) =
p
(7.35)
n=0
where
jr + k − 1 jr−1 − jr + k − 1
cj =
× ···
jr
jr−1 − jr
0≤jr ≤jr−1 ≤...≤j2 ≤j
j − j2 + k − 1 jr jr−1 −jr
j1 −j2
×
γr,p γr−1,p · · · γ1,p
(7.36)
j − j2
X
for j ∈ N0 .
Proof. The proof is similar to the proof of Lemma 7.10. Recall that
∞ X
n+k−1
1/2 −k
(1 − γl,p e(θ)t ) =
(γl,p e(θ)t1/2 )n ,
(7.37)
n
n=0
therefore
r
Y
Lp (t) =
(1 − γl,p e(θ)t1/2 )−k
k
l=1
r X
∞
Y
nl + k − 1
=
(γl,p e(θ)t1/2 )nl
nl
l=1 n =0
l
33
(7.38)
(7.39)
n1
∞ X
X
=
n1 =0 n2 =0
nr−1
···
X nr + k − 1nr−1 − nr + k − 1
nr =0
nr−1 − nr
nr
× ···
n1 − n2 + k − 1 nr nr−1 −nr
n1 −n2
e(n1 θ)tn1 /2
×
γr,p γr−1,p · · · γ1,p
n1 − n2
∞ X
X
nr + k − 1 nr−1 − nr + k − 1
=
× ···
n
n
−
n
r
r−1
r
n1 =0 0≤nr ≤nr−1 ≤...≤n1
n1 − n2 + k − 1 nr nr−1 −nr
n1 −n2
e(n1 θ)tn1 /2
×
γr,p γr−1,p · · · γ1,p
n1 − n2
∞
X
=:
c̃n ,
(7.40)
(7.41)
(7.42)
(7.43)
(7.44)
n=0
where
nr + k − 1 nr−1 − nr + k − 1
c̃n =
× ···
n
n
−
n
r
r−1
r
0≤nr ≤nr−1 ≤...≤n
n − n2 + k − 1 nr nr−1 −nr
n1 −n2
e(nθ)tn/2 .
×
γr,p γr−1,p · · · γ1,p
n − n2
X
Similarly we obtain
k
L(t) =
∞
X
d˜m ,
(7.45)
(7.46)
(7.47)
m=0
where
mr + k − 1 mr−1 − mr + k − 1
× ···
mr
mr−1 − mr
0≤mr ≤mr−1 ≤...≤m
m − m2 + k − 1 mr mr−1 −mr
1 −m2
×
γ r,p γ r−1,p
· · · γm
e(−mθ)tm/2 .
1,p
m − m2
d˜m =
X
34
(7.48)
(7.49)
Set cj := (e(jθ)tj/2 )−1 c̃j and dj := (e(jθ)tj/2 )−1 d˜j for j ∈ N0 . Note that
dj = cj . Then we obtain
Z1
1/2 k
Lp (e(θ)t
1/2 k
) Lp (e(θ)t
) dθ =
0
Z1 X
∞
0
=
n=0
Z1 X
∞ X
n
0
=
c̃n ·
∞
X
d˜m
(7.50)
c̃m d˜n−m dθ
(7.51)
m=0
n=0 m=0
Z1 X
∞ X
n
0
cm dn−m e((2m − n)θ) tn/2 dθ
n=0 m=0
(7.52)
=
∞
X
cn dn tn/2
(7.53)
cn cn tn/2 .
(7.54)
n=0
=
∞
X
n=0
By definition of Ak , we have
1
Z
Y
1
1
−1 k2
(1 − p )
Ak (0, . . . , 0) =
Lp (e(θ)( )1/2 )k Lp (e(−θ)( )1/2 )k dθ (7.55)
p
p
p
0
∞
Y
X
−1 k2
(1 − p )
| cn |2 tn/2 ,
=
p
(7.56)
n=0
as asserted in the theorem.
Corollary 7.5. The arithmetic factor of an elliptic curve L-function over Q
is given by
2
∞
Y
X
j+k−1
−1 k2
2
ak := Ak (0, ...0) = (1 − p )
×
γ2,p
j
p
j=0
2 2 −j/2
× 2 F1 (1 − k − j, −j; 1 − k − j; −γ1,p
)p
,
35
(7.57)
Ak (z1 . . . , z2k ) is given in (6.19) and 2 F1 (a, b; c; t) is the Gauss hypergeometric
function
∞
Γ(c) X Γ(a + n)Γ(b + n) tn
.
(7.58)
F
(a,
b;
c;
t)
=
2 1
Γ(a)Γ(b) n=0
Γ(c + n)
n!
Proof. This follows from Theorem 7.4, or more directly from Lemma 7.10.
Note that we have used the fact that γ1,p = γ 2,p and | γ1,p |=| γ1,p |= 1 for
all primes p - ∆, where ∆ is the discriminant of E, and therefore we have
γ1,p
2
= γ1,p
.
γ2,p
Theorem 7.2, Theorem 7.4 and Conjecture 6.7 predict the following conjecture:
Conjecture 7.6. Let k ∈ N and L(s) be a primitive L-functions in the
Selberg class, satisfying (6.13), (6.14) and Conjecture 6.4. Then
1
lim
2
T →∞ T (log T )k
Z1
1
|L
+ it |2k dt = a(k)f (k),
2
(7.59)
0
where
a(k) = Ak (0, . . . , 0) =
∞
Y
X
2
(1 − p−1 )k
| cn |2 p−n/2
p
(7.60)
n=0
with cn defined as in (7.36) and
f (k) =
k−1
Y
l=0
l!
.
(k + l)!
(7.61)
We see that the function f remains the same in the moment conjecture of
primitive L-functions as in the moment conjecture for the zeta function. The
only factor which changes is the arithmetical factor a(k). From the moment
conjecture of the zeta function, we know how to extend f (k) to be defined
also for k ∈ C. So the goal is to extend the factor a(k) for k ∈ C. One
possible way to do this is to replace in cn each binomial coefficient by its
expression as a Γ-factor. Therefore we claim the following conjecture:
36
Conjecture 7.7. Let k ∈ C and L(s) be a primitive L-functions in the
Selberg class, satisfying (6.13), (6.14) and Conjecture 6.4. Then
1
lim
2
T →∞ T (log T )k
Z1
1
+ it |2k dt = a(k)f (k),
|L
2
(7.62)
0
where
∞
X
Y
−1 k2
| cn |2 p−n/2
a(k) =
(1 − p )
(7.63)
n=0
p
with
cj =
X
0≤jr ≤jr−1 ≤...≤j2 ≤j
×
Γ(jr + k − 1) Γ(jr−1 − jr + k − 1)
× ···
Γ(jr )Γ(k − 1) Γ(jr−1 − jr )Γ(k − 1)
Γ(j2 − j3 + k − 1) Γ(j − j2 + k − 1) jr jr−1 −jr
j1 −j2
γ γ
· · · γ1,p
Γ(j2 − j3 )Γ(k − 1) Γ(j − j2 )Γ(k − 1) r,p r−1,p
and
f (k) =
G(1 + k)2
,
G(1 + 2k)
(7.64)
(7.65)
where G is the Barnes function defined in (1.4).
7.1
Series for log(Ak (z1 , . . . , z2k )) in the case of elliptic
curve L-functions over Q
As we explained in the previous section, to determine the coefficient cr (k) of
the polynomial Pk (x) we need to compute the Taylor series of Ak (z1 , . . . , z2k )
up to order r. In the following we will determine the multivariate Taylor
series of log(Ak (z1 , . . . , z2k )) in the case of the elliptic curve L-functions over
Q. This is done in [6] for the case of the ζ function and we will use the same
methods for obtaining the Taylor series of log(Ak (z1 , . . . , z2k )) in the case
of the renormalized elliptic curve L-function L(s). Note that by using the
method for multiplying series, given in Section 2.1.3 of [6], one could recover
the series for
exp(log(Ak (z1 , . . . , z2k )),
but we are not going to do this here.
We use the same notation as in [6] and therefore, we we recapitulate it
37
here again: Since Ak (z1 , . . . , z2k ) is symmetric in z1 , . . . , zk and separately in
zk+1 , . . . , z2k , we distinguish two sets of variables, and let
Bk (α1 , α2 , . . . , αk ; β1 , β2 , . . . , βk )
(7.66)
denote the coefficient of
β
z αk z 1
z βk
z1α1
· · · k k+1 · · · 2k .
α1 !
αk! β1 !
βk !
(7.67)
in the multivariate Taylor expansion of log(Ak (z1 , . . . , z2k )). Therefore we
use the convention, when writing Bk (α; β) of only listing the non-zero αi ’s
and βi ’s, and writing them in decreasing order. Also, because
Ak (z1 , . . . , z2k ) = Ak (−zk+1 , . . . , −z2k , −z1 , . . . , −zk+1 )
(7.68)
we have
Bk (α; β) = (−1)|α|+|β| Bk (β; α)
(7.69)
Pk
where |α| = 1 αi . We list here the first few terms of the Taylor expansion,
using the above conventions and (7.69):
log(Ak (z1 , . . . , z2k )) =
k
X
X
zi zj + zi+k zj+k
zi − zi+k + Bk (1, 1; )
log(ak ) + Bk (1; )
1
+Bk (1; 1)
X
1≤i<j≤k
zi zj+k + Bk (2; )
+Bk (1, 1, 1; )
i
1
1≤i,j≤k
X
2k
X
z2
2
zi zj zl − zi+k zj+k zl+k + · · · .
1≤i<j<l≤k
(7.70)
The term log(ak ) comes from the value of the function at the origin log(Ak (0, . . . , 0)) =
log(ak ). Note that because of the symmetry the coefficients Bk can be taken
out of the summations in (7.70).
Next, let l = l(α) denote the number of non-zero αi ’s, and m = m(β)
denote the number of non-zero βi ’s. Since we are interested in extracting
cr (k), we only need to consider the power series expansion of Ak up to degree
r, i.e. l + m ≤ r. Since we are assuming in our evaluation of Bk (α; β) that
38
the αi ’s and βi ’s are in decreasing order, we focus on the first l zi ’s (m zi+k ’s
respectively) and we have, together with (6.19),
l
m
XY
∂ αi Y ∂ βi
Bk (α; β) =
log
(f
(1/p;
z))
(7.71)
k,p
αi
β
i
∂z
∂z
i
z=0
i+k
p i=1
i=1
where
fk,p (t; z) =
Y
Z1 Y
k −1 −1
1
1
1+zi −zk+j
)
1 − γ1 e(θ)t 2 +zj
1 − γ2 e(θ)t 2 +zj
(1 − t
×
1≤i,j≤k
0
j=1
−1 −1
1
1
−zk+j
−zk+j
2
2
1 − γ2 e(−θ)t
dθ.
× 1 − γ1 e(−θ)t
(7.72)
Note that in contrast to the case of the ζ function, the form of the function
fk,p really depends on a prime p. Since we are assuming αl+1 = . . . = αk = 0,
βm+1 = . . . = βk = 0, with l + m ≤ r, we may as well immediately set
zr+1 = . . . = zk = 0, zk+r+1 = . . . = z2k = 0. We see that calculating the
Taylor series of log(fk,p ) involves calculating the Taylor series of
X
log 1 − t1+zi −zk+j ,
1≤i,j≤k
which is done in [6, (2.29)] and therefore we give only the terms up to degree
2:
k
X
kt log(t) X
1+zi −zk+j
2
zi − zi+k
log 1 − t
= k log(1 − t) −
1−t
1
1≤i,j≤k
2k
t log(t)2 X
kt log(t)2 X zi2
+
zi zj+k −
+ ...
(1 − t)2 1≤i,j≤k
(1 − t)2 1 2
(7.73)
Next, applying
k Y
k
Y
∂ ci ∂ d j
∂zi ∂zk+j
i=1 j=1
(7.74)
to
log
k Z1 Y
0
1
1 − γ1 e(θ)t 2 +zj
−1 1
1 − γ2 e(θ)t 2 +zj
−1
×
j=1
× 1 − γ1 e(−θ)t
1
−zk+j
2
−1 1 − γ2 e(−θ)t
39
1
−zk+j
2
−1
dθ
(7.75)
we end up, by the chain rule, with a rational expression involving partial
derivatives of the form
1 k
−1 −1
Y ∂ ci Y ∂ di Z Y
1
1
+zj
+zj
2
2
1
−
γ
e(θ)t
1
−
γ
e(θ)t
×
1
2
di
∂zici i=1 ∂zi+k
i=1
j=1
0
−1 −1 1
1
−z
−z
k+j
k+j
1 − γ2 e(−θ)t 2
dθ
× 1 − γ1 e(−θ)t 2
.
z=0
(7.76)
By the Leibniz rule we have
−1 −1 1
1
∂ c +z
+z
1 − γ1 e(θ)t 2
1 − γ2 e(θ)t 2
∂z c
z=0
c l −1 c−l X
−1
1
1
c
∂
∂
+z
+z
=
1 − γ1 e(θ)t 2
1 − γ1 e(θ)t 2
l
c−l
l
∂z
∂z
z=0
l=0
(7.77)
Now,
∞
∂l −1 X
1
1
+z
l
1/2+z −(l+1)
2
1 − γi e(θ)t
= log(t) (1−γi e(θ)t
)
(γi e(θ)t 2 +z )m ml
l
∂z
m=0
(7.78)
Definition 7.8. For n ∈ N and m ∈ N0 the Eulerian numbers are defined
as
m
X
l n+1
(−1)
E(n, m) =
(m + 1 − l)n ,
(7.79)
l
l=0
where the sum is taken to be 1 if m = 0.
It is well known that the Eulerian numbers satisfy the following recurrence
relation:
E(n, m) = (n + 1 − m)E(n, m − 1) + (m + 1)E(n, m),
and this is going to be used in the proof of the next lemma.
40
(7.80)
Lemma 7.9. Let x ∈ C with | x |< 1. For the Eulerian numbers we have
the identity
Pn−1
∞
X
E(n, m)xm+1
n l
,
(7.81)
l x = m=0
n+1
(1
−
x)
l=1
where the sum in the numerator is taken to be 1 if n = 0.
Proof. We prove this by induction on n. For n = 0, (7.81) reduces to
∞
X
xl =
l=0
1
1−x
(7.82)
which obviously holds by the geometric series. So assume that (7.81) is true
for some n ∈ N. We want to show that it is also true for n + 1:
The r.h.s. of (7.81) is
∞
X
l=1
l
∞
d X n l x =x·
l x
dx l=1
P
m+1 d n−1
m=0 E(n, m)x
=x·
,
dx
(1 − x)n+1
n+1 l
(7.83)
(7.84)
where for the last equality we have used the induction hypothesis. Applying
the derivative in (7.84) we obtain
Pn−1
Pn−1
m
n+1
E(n, m)xm+1 · (n + 1)(1 − x)n
+ m=0
m=0 E(n, m)(m + 1)x · (1 − x)
x·
(1 − x)2(n+1)
(7.85)
Pn−1
P
n−1
m+2
E(n, m)(m + 1)xm+1
(n + 1)
m=0 E(n, m)x
= m=0
+
(7.86)
n+1
n+2
(1 − x)
(1 − x)
41
=
n−1
n−1
X
X
1
m+1
m+2
E(n,
m)(m
+
1)x
(1
−
x)
+
E(n,
m)x
(n
+
1)
(1 − x)n+2 m=0
m=0
(7.87)
n−1
n−1
X
X
1
m+1
=
E(n, m)(m + 1)x
−
E(n, m)(m + 1)xm+2 (7.88)
(1 − x)n+2 m=0
m=0
+
n−1
X
E(n, m)xm+2 (n + 1)
(7.89)
m=0
n−1
n−1
X
X
1
m+1
m+2
E(n,
m)(m
+
1)x
+
(n
+
1
−
(m
+
1))E(n,
m)x
=
(1 − x)n+2 m=0
m=0
(7.90)
n
n−1
X
X
1
m+1
m+1
(n
+
1
−
m)E(n,
m
−
1)x
=
E(n,
m)(m
+
1)x
+
(1 − x)n+2 m=0
m=1
(7.91)
n−1
X
1
(E(n, m)(m + 1) + (n + 1 − m)E(n, m − 1)) xm+1
=
1
·x +
|{z}
|
{z
}
(1 − x)n+2
E(n,0)=E(n+1,0)
m=1
=E(n+1,m)
(7.92)
+ E(n, n − 1) xn+1
|
{z
}
(7.93)
=1=E(n+1,n)
n
X
1
E(n + 1, m)xm+1
·
=
n+2
(1 − x)
m=0
(7.94)
The sum in (7.78) is of the form
∞
X
ml xm ,
m=0
42
(7.95)
1
with x = γi e(θ)t 2 +z . Using Lemma 7.9 we see that (7.77) equals
c X
c
l=0
l
1
2
l
−(l+1)
log(t) (1 − γ1 e(θ)t )
l−1
X
E(l, m)γ1m+1 e((m + 1)θ)t(m+1)/2 ×
m=0
(7.96)
1
× log(t)c−l (1 − γ2 e(θ)t 2 )−(c−l+1)
c−l−1
X
E(c − l, m)γ2m+1 e((m + 1)θ)t(m+1)/2
m=0
(7.97)
= log(t)
c
c X
c
l
l=0
×
l−1
X
1
1
(1 − γ1 e(θ)t 2 )−(l+1) (1 − γ2 e(θ)t 2 )−(c−l+1) ×
E(c − l, m)γ1m+1 e((m + 1)θ)t(m+1)/2 ×
m=0
×
c−l−1
X
E(c − l, m)γ2m+1 e((m + 1)θ)t(m+1)/2
(7.98)
m=0
= log(t)
×
c
c X
c
l=0
l−1
c−l−1
X X
l
1
1
(1 − γ1 e(θ)t 2 )−(l+1) (1 − γ2 e(θ)t 2 )−(c−l+1) ×
(7.99)
E(l, m)E(c − l, r)γ1m+1 γ2r+1 e((m + r + 2)θ)t(m+r+2)/2 (7.100)
m=0 r=0
Likewise,
−1 −1 1
1
∂d 1 − γ1 e(−θ)t 2 −z
1 − γ2 e(−θ)t 2 −z
∂z d
z=0
(7.101)
d
= (− log(t))
d X
d
l=0
l
1
1
(1 − γ1 e(−θ)t 2 )−(l+1) (1 − γ2 e(−θ)t 2 )−(d−l+1) ×
(7.102)
×
l−1 d−l−1
X
X
E(l, m)E(d − l, r)γ1m+1 γ2r+1 e(−(m + r + 2)θ)t(m+r+2)/2 .
m=0 r=0
(7.103)
43
Finally, we have shown that
−1 −1
1
1
∂ c ∂ d −z
−z
2
2
1
−
γ
e(−θ)t
1
−
γ
e(−θ)t
×
1
2
∂z c ∂z d
−1 −1 1
1
−z
−z
2
2
1 − γ2 e(−θ)t
× 1 − γ1 e(−θ)t
(7.104)
z=0
c
d
= log(t) (− log(t))
c X
d X
c d
s=0 l=0
1
s
l
×
1
1
× (1 − γ1 e(θ)t 2 )−(s+1) (1 − γ2 e(θ)t 2 )−(c−s+1) (1 − γ1 e(−θ)t 2 )−(l+1) ×
1
× (1 − γ2 e(−θ)t 2 )−(d−l+1)
s−1 c−s−1
l−1 d−l−1
X
X X
X
i=0
×
γ1i+m+2 γ2j+r+2 e((i
E(s, i)E(c − s, j)E(l, m)E(d − l, r)×
j=0 m=0 r=0
+ j + 2)θ)e(−(m + r + 2)θ)t(m+r+2)/2 ,
and this generalizes to
44
(7.105)
k −1 −1
Y ∂ ci Y ∂ di Y
1
1
+zj
+zj
2
2
1
−
γ
e(θ)t
×
1
−
γ
e(θ)t
2
1
di
∂zici i=1 ∂zi+k
i=1
j=1
−1 −1 1
1
−z
−z
1 − γ2 e(−θ)t 2 k+j
× 1 − γ1 e(−θ)t 2 j+k
z1 =z2 =···=z2k =0
= log(t)
P
ci
P
(− log(t))
di
c1
X
···
P
−(k+ li )
× (1 − γ1 e(θ)t )
1
d1
X
dk
X
c1
ck
d1
dk
···
···
×
l
l
m
m
1
k
1
k
=0
···
mk
P
−(k+ ci −li )
lk =0 m1 =0
l1 =0
1
2
ck
X
1
2
(1 − γ2 e(θ)t )
1
P
×
P
−(k+ di −mi )
× (1 − γ1 e(−θ)t 2 )−(k+ mi ) (1 − γ2 e(−θ)t 2 )
×
c
−l
−1
l
−1
c
−l
−1
l
−1
kX
k
k
1X
1
1
X
X
···
E(l1 , r1 ) · · · E(lk , rk )×
···
×
r1 =0
rk =0
r̃1 =0
r̃k =0
P
× E(c1 − l1 , r̃k ) · · · E(ck − lk , r̃k )γ1
×
1 −1
mX
s1 =0
···
m
k −1 d1 −m
X1 −1
X
sk =0
···
dk −m
Xk −1
ri +k
P
γ2
r̃i +k
X
P
( ri +r̃i +k)/2
e((
ri + r̃i + k)θ)t
×
E(m1 , s1 ) · · · E(mk , sk )×
s̃k =0
s̃1 =0
P
× E(d1 − m1 , s̃1 ) · · · E(dk − mk , s̃k )γ1
X
P
× e(−(
si + s̃i + k)θ)t( si +s̃i +k)/2
si +k
P
γ2
s̃i +k
×
(7.106)
In (7.106) we see that we need to evaluate integrals of the form
Z1
1/2 −(k+
1 − γ1 e(θ)t
P
li )
P
1/2 −(k+ ci −li )
1 − γ2 e(θ)t
×
0
× 1 − γ1 e(−θ)t
1/2 −(k+
P
mi )
1 − γ2 e(−θ)t
1/2 −(k+
P
di −mi )
e(Cθ) dθ
(7.107)
for all C ∈ Z with
X
X
X
−
(mi − 1) + (di − mi − 1) ≤ C =
ri + r̃i + k − (
si + s̃i + k)
X
≤
(li − 1) + (ci − li − 1) ,
(7.108)
45
i.e. for all C ∈ Z with 2k −
following lemma
P
di ≤ C ≤
P
ci − 2k. Therefore we need the
Lemma 7.10. Let A1 , A2 , B3 , B4 ∈ N0 , C ∈ Z and γ1 , γ2 ∈ C, γ2 6= 0. Then
Z1
(1 − γ1 e(θ)t1/2 )−A1 (1 − γ2 e(θ)t1/2 )−A2 ×
0
× (1 − γ1 e(−θ)t1/2 )−B1 (1 − γ2 e(−θ)t1/2 )−B2 e(Cθ)dθ
|C|/2
=t
F (A1 , A2 , B1 , B2 , C, γ1 , γ2 ; t),
(7.109)
(7.110)
where
P∞
cj dj+C tj/2 if C ≥ 0
F (A1 , A2 , B1 , B2 , C, γ1 , γ2 ; t) = Pj=0
∞
j/2
if C < 0.
j=0 cj+|C| dj t
with
cj =
γ2j
γ1
j + A2 − 1
)
2 F1 (1 − A1 − j, −j, 1 − A2 − j, −
j
γ2
(7.111)
dj =
γ2j
γ1
j + B2 − 1
)
2 F1 (1 − B1 − j, −j, 1 − B2 − j, −
j
γ2
(7.112)
and
for j ∈ N0 .
Remark 7.11. Note that later we will fix a prime p and set γ1 = γ1,p and
γ2 = γ2,p , where γi,p is defined in (3.23). If one of the γi,p ’s is 0 then we take
it to be γ1,p and we set γ2,p = 1. Therefore we can assume in Lemma 7.10
that γ2 ∈ C \ 0.
Proof. By the binomial series we have
(1 − γ1 e(θ)t1/2 )−A1 (1 − γ2 e(θ)t1/2 )−A2
∞ ∞ X
X
m + A2 − 1
n + A1 − 1
1/2 n
(γ2 e(θ)t1/2 )m
=
(γ1 e(θ)t )
m
n
m=0
n=0
(7.113)
∞
X
=
cl e(lθ)tl/2 ,
(7.114)
l=0
46
where, by the Cauchy product of series,
l X
m + A1 − 1 l − m + A2 − 1 m l−m
cl =
γ1 γ2 ,
k
l−m
m=0
(7.115)
for l ∈ N0 . Similarly, we obtain
(1 − γ1 e(−θ)t1/2 )−B1 (1 − γ2 e(−θ)t1/2 )−B2 =
∞
X
dl e(−lθ)tl/2 ,
(7.116)
l=0
where
l X
m + B1 − 1 l − m + B2 − 1 m l−m
dl =
γ1 γ2 ,
m
l
−
m
m=0
(7.117)
for l ∈ N0 . Thus, (7.109) equals
Z1 X
∞
0
n=0
Z1 X
∞
∞
X
m/2
e(Cθ)dθ =
gn · e(Cθ)dθ
·
dm e(−mθ)t
c e(nθ)t
|
{z
}
|n {z }
n/2
m=0
:=c̃n
0
:=d˜n
n=0
(7.118)
where
gn =
n
X
n
X
˜
c̃l dn−l =
cl dn−l e((2l − n)θ) tn/2 ,
l=0
(7.119)
l=0
by the Cauchy product of infinite series. The integral will pull out the coefficient of e(−Cθ), of the series
∞ X
n
∞
X
X
cl dn−l e((2l − n)θ) tn/2 .
(7.120)
gn =
n=0
This coefficient is
∞
X
X
n=0
l=0
X
cl dn−l tn/2 =
n=0 0≤l≤n
n−2l=C
=
c (n−C) dn− (n−C) tn/2
0≤n≤∞
(n−C)/2∈N0
∞
X
2
cn dn+C t(n+C)/2
(7.121)
2
(7.122)
n=0
C/2
=t
∞
X
n=0
47
cn dn+C tn/2
(7.123)
P
P
P
Applying
this lemma
k+
ci − li , A3 =
P
P for A1 = k + Pli , A2 =
k + mi , A4 = k + di − mi and C =
ri + r̃i − (si + s̃i ) wee see that
(7.106) is equal to
log(t)
P
ci
P
(− log(t))
di
c1
X
···
lk =0 m1
l1 =0
×
1 −1
lX
r1 =0
···
lX
−l1 −1
k −1 c1X
rk =0
r̃1 =0
···
dk X
c1
ck
d1
dk
···
···
×
···
l1
lk
m1
mk
m =0
=0
ck X
d1
X
ckX
−lk −1 m
1 −1
X
r̃k =0
k
···
m
k −1 d1 −m
X
X1 −1
sk =0
s1 =0
s̃1 =0
···
dk −m
Xk −1
×
s̃k =0
× E(l1 , r1 ) · · · E(lk , rk )E(c1 − l1 , r̃k ) · · · E(ck − lk , r̃k )×
× E(m1 , s1 ) · · · E(mk , sk )E(d1 − m1 , s̃1 ) · · · E(dk − mk , s̃k )×
X
X
X
X
X
×F k+
li , k +
ci − li , k +
mi , k +
di − mi ,
ri + r̃i − (si + s̃i ),
P
P
ri +r̃i −(si +s̃i )+k)/2
γ1 , γ2 ; t |t( ri +r̃i +si +s̃i +k)/2+(
(7.124)
{z
} .
P
t
ri +r̃i +k
Using (7.124) one can write out the Taylor expansion of (7.76), but due to
the complex structure of this expression it is very difficult to write down even
the first terms in the multivariate Taylor expansion of log(Ak (z1 , . . . , z2k ).
So it would be valuable to invest more time in giving at least the first terms
explicitly and also to follow the same steps as in [6] to obtain a formula
for the coefficients cr (k) of the polynomial Pk (x). Another point which is
worth to mention, is the question if the methods, given in [6] to compute the
coefficients of Pk (x), also apply to a L-function in the Selberg class instead of
only elliptic curve L-functions. One can see that using these methods involves
applying a general Leibniz rule, as we did in (7.77), to a product of more
than two functions. Doing this will in the end give much more complicated
expressions than we have in (7.124). Therefore it would be better to think
of a method giving results which have more closed expressions.
48
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