Illinois Journal of Mathematics
Volume 48, Number 2, Summer 2004, Pages 491–503
S 0019-2082
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE
L-FUNCTIONS
XIAN-JIN LI
Abstract. In 1997, thePauthor proved that the Riemann hypothesis
holds if and only if λn = [1−(1−1/ρ)n ] > 0 for all positive integers n,
where the sum is over all complex zeros of the Riemann zeta function. In
1999, E. Bombieri and J. Lagarias generalized this result and obtained
a remarkable general theorem about the location of zeros. They also
gave an arithmetic interpretation for the numbers λn . In this note, the
author extends Bombieri and Lagarias’ arithmetic formula to Dirichlet
L-functions and to L-series of elliptic curves over rational numbers.
1. Introduction
Let K be a finite field with q elements, and let E be an elliptic curve over
K. In the 1930s, H. Hasse proved the inequality
√
|#E(K) − q − 1| ≤ 2 q,
where #E(K) is the number of K-rational points on E; see [12].
Let a = 1 + q − #E(K) and
LE (s) = 1 − az + qz 2 ,
where z = q −s . By Hasse’s inequality we have
LE (s) = (1 − αz)(1 − βz)
with |α| = |β| =
√
q. Hence
−
∞
X
d
log LE (s) =
λE (n + 1)z n ,
dz
n=0
where λE (n) = αn + β n . It is clear that
√
|λE (n)| ≤ 2 q n
Received April 24, 2003; received in final form October 17, 2003.
2000 Mathematics Subject Classification. Primary 11M26, 11M36.
Research supported by National Security Agency.
c
2004
University of Illinois
491
492
XIAN-JIN LI
for n = 1, 2, . . .. This estimate implies that all zeros of LE (s) lie on the line
<s = 1/2.
Let
ξ(s) = s(s − 1)π −s/2 Γ(s/2)ζ(s),
where ζ(s) is the Riemann zeta function, and let λζ (n), n = 1, 2, . . . , be a
sequence of numbers defined by
X
∞
d
1
ln ξ
=
λζ (n + 1)z n .
dz
1−z
n=0
In 1997, the author obtained the following criterion for the Riemann hypothesis.
Theorem 1 ([9]). All complex zeros of ζ(s) lie on the line <s = 1/2 if
and only if λζ (n) > 0 for n = 1, 2, . . . .
In 1952, A. Weil [13] proved a famous criterion for the validity of the
Riemann hypotheses for number fields. The following is Bombieri’s refinement
of Weil’s criterion.
Bombieri’s refinement ([2]). All complex zeros of ζ(s) lie on the line
<s = 1/2 if and only if
X
fb(ρ)fb̄(1 − ρ) ≥ 0
ρ
for every complex-valued f ∈ C0∞ (0, ∞) which is not identically 0, where the
Mellin transform of f is given by
Z ∞
b
f (s) =
f (x)xs−1 dx.
0
Let f, g ∈
C0∞ (0, ∞).
The multiplicative convolution of f and g is given by
Z ∞
dy
(f ∗ g)(x) =
f (x/y)g(y) .
y
0
If fe(x) = x−1 f (x−1 ), the Mellin transform of f ∗ fē is fb(s)fb̄(1 − s). Let gn (x)
be the inverse Mellin transform of 1 − (1 − 1/s)n for n = 1, 2, . . . . E. Bombieri
and J. Lagarias observed in [3] that
n n 1
1
1− 1−
+ 1− 1−
s
1−s
n n 1
1
= 1− 1−
1− 1−
,
s
1−s
and that
gn (x) + gen (x) = (gn ∗ gen )(x).
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS
493
Hence, the positivity in the author’s criterion has the same meaning as that
in Weil’s criterion.
In 1999, Bombieri and Lagarias obtained the following remarkable theorem.
Theorem 2 (Bombieri-Lagarias [3]). Let R be a set of complex numbers
ρ whose elements have positive integral multiplicities assigned to them, such
that 1 6∈ R and
X 1 + |<ρ|
< ∞.
(1 + |ρ|)2
ρ
Then the following conditions are equivalent:
(1) <ρ ≤ 1/2
every ρ in
R;
for
−n P
1
(2)
≥ 0 for n = 1, 2, . . . .
ρ< 1− 1− ρ
An arithmetic interpretation for the numbers λζ (n) was given in [3].
Theorem 3 (Bombieri-Lagarias [3]). We have
( N
)
n X
X Λ(k)
n (−1)j
j−1
j
λζ (n) =
lim j
(ln k)
− (ln N )
j
j! N →∞
k
j=1
k=1
n X
n
n
+ 1 − (ln 4π + γ) +
(−1)j (1 − 2−j )ζ(j)
2
j
j=2
for n = 1, 2, . . . , where γ = 0.5772 . . . is Euler’s constant and where Λ(k) =
ln p when k is a power of a prime p and Λ(k) = 0 otherwise.
Let χ be a primitive Dirichlet character of modulus r > 1, and L(s, χ) the
Dirichlet L-function of character χ. If
s+a
− 12 (s+a)
ξ(s, χ) = (π/r)
Γ
L(s, χ),
2
where
(
0, if χ(−1) = 1,
a=
1, if χ(−1) = −1,
then ξ(s, χ) is an entire function of order one and satisfies the functional
equation
ξ(s, χ) = χ ξ(1 − s, χ̄),
where χ is a constant of absolute value one. By Theorem 2 of [1] we have
Y
ξ(s, χ) = ξ(0, χ) (1 − s/ρ),
ρ
where the product is over all the zeros of ξ(s, χ) in the order given by |=ρ| < T
for T → ∞.
494
XIAN-JIN LI
For n = 1, 2, . . . let
λχ (n) =
X
[1 − (1 − 1/ρ)n ] ,
ρ
where the sum on ρ runs over all zeros of ξ(s, χ) in the order given by |=ρ| < T
for T → ∞. First, we give an arithmetic interpretation for the numbers λχ (n).
Theorem 4. Let χ be a primitive Dirichlet character of modulus r > 1.
Then we have
n ∞
X
n (−1)j−1 X Λ(k)
n r
χ̄(k)(ln k)j−1 +
ln − γ + τχ (n),
λχ (n) = −
j (j − 1)!
k
2
π
j=1
k=1
where
τχ (n) =
(Pn
n
1
j
j (−1) 1 − 2j
Pj=2
n
n
j −j
j=2 j (−1) 2 ζ(j),
ζ(j) −
n
2
P∞
1
l=1 l(2l−1) ,
if χ(−1) = 1,
if χ(−1) = −1.
Let E be an elliptic curve over Q with conductor N . For each prime p, we
denote by Ẽp the reduction of E at p. Let

p + 1 − #Ẽp (Fp ), if E has good reduction at p,



1,
if E has split multiplicative reduction at p,
ap =

−1,
if E has non-split multiplicative reduction at p,



0,
if E has additive reduction at p.
We define the L-series associated to E by the Euler product
Y
Y
LE (s) =
(1 − ap p−s + p1−2s )−1
(1 − ap p−s )−1
p|N
p-N
for <s > 3/2; see [12].
Let k and N be positive integers, and let χ be a multiplicative character of
modulus N with χ(1) = 1 and χ(−1) = (−1)k . Let Γ be the Hecke congruence
subgroup Γ0 (N ) of level N . We denote by S0 (Γ, k, χ) the space of all cusp
forms of weight k and character χ for Γ. That is, f belongs to S0 (Γ, k, χ) if
and only if f is holomorphic in the upper half-plane H, satisfies
az + b
f
= χ(d)(cz + d)k f (z)
cz + d
for all ac db ∈ Γ, satisfies the usual regularity conditions at the cusps of Γ,
and vanishes at each cusp of Γ.
The Hecke operators Tn are defined by
X az + b 1 X
(Tn f )(z) =
f
χ(a)ak
n
d
ad=n
0≤b<d
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS
495
for any function f on H. A function f in S0 (Γ, k, χ) is called a Hecke eigenform
if
Tn f = λ(n)f
for all positive integers n with (n, N ) = 1. The Fricke involution W is defined
by
(W f )(z) = N −k/2 z −k F (−1/N z),
and the complex conjugation operator K is defined by
(Kf )(z) = f¯(−z̄).
Set W̄ = KW . Then f is a newform if it is an eigenfunction of W̄ and of all
the Hecke operators Tn .
Let f be a newform in S0 (Γ, k, χ) normalized so that its Fourier coefficient
is 1. Then it has the Fourier expansion
∞
X
f (z) =
λ(n)e2πinz
n=1
with the Fourier coefficients equal to the eigenvalues of Hecke operators. Since
f is an eigenfunction of the involution W̄ , we can assume that
W̄ f = ηf
for a constant η. Let
∞
X
λ(n)
Lf (s) =
ns
n=1
for <s > (k + 1)/2. This L-series has Euler product
Y
Lf (s) =
(1 − λ(p)p−s + χ(p)pk−1−2s )−1
p
and satisfies the functional identity
√ !s
N
Γ(s)Lf (s) = ik η̄
2π
√
N
2π
!k−s
Γ(k − s)L̄f (k − s̄).
When χ is primitive, we have η = τ (χ̄)λ(N )N −k/2 with τ (χ) being the Gauss
sum for χ. For the theory of Hecke L-functions see [8].
We denote by S2 (N ) the space of cusp forms of weight 2 with the principal
character for Γ0 (N ).
Shimura-Taniyama Conjecture ([4], [14]).
S2 (N ) such that Lf (s) = LE (s).
There is a newform f ∈
The Shimura-Taniyama conjecture has now been proved ([4], [14]).
If
1
1
ξE (s) = cE N s/2 (2π)−s Γ
+ s LE
+s ,
2
2
496
XIAN-JIN LI
where cE is a constant chosen so that ξE (1) = 1, then ξE (s) is an entire
function and satisfies
ξE (s) = wξE (1 − s),
r
where w = (−1) with r being the vanishing order of ξE (s) at s = 1/2.
Let
n X
1
λE (n) =
1− 1−
ρ
ρ
for n = 1, 2, . . . , where the sum is over all the zeros of ξE (s) in the order
given by |=ρ| < T for T → ∞. Since ξE (s) is an entire function of order one,
the conditions of Bombieri-Lagarias’ theorem are satisfied, and hence all the
zeros of ξE (s) lie on the line <s = 1/2 if and only if
λE (n) > 0
for n = 1, 2, . . . .
For each prime number p, we let αp and βp be the roots of T 2 − ap T + p.
Let b(pk ) = akp if p|N and b(pk ) = αpk + βpk if (p, N ) = 1. Next, we give an
arithmetic interpretation for the numbers λE (n).
We have
!
√
n ∞
X
N
n (−1)j−1 X Λ(k)
λE (n) = n ln
−γ −
b(k)(ln k)j−1
3/2
2π
j
(j
−
1)!
k
j=1
k=1
!
∞
n
∞
X n
X
2 X
3
1
+n − +
+
(−1)j
.
3
l(2l + 3)
j
(l
+
1/2)j
j=2
Theorem 5.
l=1
l=1
The author wishes to thank Brian Conrey for his help during the preparation of the manuscript, and William Duke for his valuable suggestions. He
also wants to thank the referee for carefully reading the manuscript and for
his/her valuable suggestions.
2. Proof of Theorem 4
Weil’s explicit formula for L(s, χ) ([1], [13]).
defined on R such that
Let F (x) be a function
2F (x) = F (x + 0) + F (x − 0)
for all x ∈ R, such that F (x) exp((b + 1/2)|x|) is of bounded variation on R
for a constant b > 0, and such that
F (x) + F (−x) = 2F (0) + O(|x|` )
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS
497
as x → 0 for a constant ` > 0. Then
∞
r
X
X
Λ(n)
√ (χ(n)F (ln n) + χ̄(n)F (− ln n))
Φ(ρ) = F (0) ln − γ −
π
n
ρ
n=1
Z ∞
dx
+
F (x)e(3/2−a)|x| − F (0)
,
1 − e2|x|
−∞
where the sum on ρ runs over all zeros of ξ(s, χ) in the order given by |=ρ| < T
for T → ∞, and
Z ∞
(2.1)
Φ(s) =
F (x)e(s−1/2)x dx.
−∞
A multiset is a set whose elements have positive integral multiplicities assigned to them [3].
Lemma 2.1 ([3, (2,4)]).
Formally, if
Y
z
f (z) =
1−
ρ
ρ
and
λn =
X
[1 − (1 − 1/ρ)n ] ,
ρ
then we have
d
ln f
dz
1
1−z
=
∞
X
λ−n−1 z n .
n=0
Lemma 2.2 ([3, Corollary 1]). Let R be a multiset of complex numbers
such that
(1) 0, 1 6∈ R;
(2) if ρ ∈ R, then 1 − ρ and ρ̄ are in R and have the same multiplicity as
ρ;
P
2
(3)
ρ (1 + |<ρ|)/(1 + |ρ|) < ∞.
Then <ρ = 1/2 for all ρ ∈ R if, and only if,
X
λn =
[1 − (1 − 1/ρ)n ] ≥ 0
ρ∈R
for n = 1, 2, 3, . . ..
Lemma 2.3 ([3, Lemma 2]). For n = 1, 2, . . ., let

Pn
n xj−1
x/2

e
j=1 j (j−1)! , if −∞ < x < 0,
Fn (x) = n/2,
if x = 0,


0,
if 0 < x.
498
XIAN-JIN LI
Then
n
1
,
Φn (s) = 1 − 1 −
s
where Φn is related to Fn as in (2.1).
Proof of Theorem 4. For a sufficiently
an integer let


Fn (x),
Fn,X (x) = 12 Fn (− ln X),


0,
large positive number X that is not
if − ln X < x < ∞,
if x = − ln X,
if −∞ < x < − ln X.
Then Fn,X (x) satisfies all conditions of Weil’s explicit formula for L(s, χ). Let
Φn,X (s) =
Z
∞
Fn,X (x)e(s−1/2)x dx.
−∞
By using the Weil explicit formula, we obtain that
r
X
Φn,X (ρ) = Fn,X (0) ln − γ
π
ρ
∞
X
Λ(k)
√ (χ(k)Fn,X (ln k) + χ̄(k)Fn,X (− ln k))
k
k=1
Z ∞
dx
+
Fn,X (x)e(3/2−a)|x| − Fn,X (0)
,
1
−
e2|x|
−∞
−
where the sum on ρ runs over all zeros of ξ(s, χ) in the order given by |=ρ| < T
for T → ∞. It follows that
(2.2)
X
Φn,X (ρ)
lim
X→∞
ρ
n ∞
X
n r
n (−1)j−1 X Λ(k)
=
ln − γ −
χ̄(k)(ln k)j−1
2
π
j
(j
−
1)!
k
j=1
k=1
(Pn
P∞
n
1
n
1
j
j=2 j (−1) 1 − 2j ζ(j) − 2
l=1 l(2l−1) , if χ(−1) = 1,
+ Pn
n
j −j
if χ(−1) = −1.
j=2 j (−1) 2 ζ(j),
Note that the infinite series in the second term on the right side of (2.2)
converges by the prime number theorem for arithmetic progressions; see §1920 of [5].
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS
499
We have
(2.3)
n X
n
j−1
X
(ln X)j−k−1
s−k−1
j
(j
−
k
−
1)!
j=1
k=0
n −s X
n (− ln X)j−1
(ln X)n−2 −<s
X
+O
X
.
=
s j=1 j
(j − 1)!
|s|2
Φn (s) − Φn,X (s) = X
−s
(−1)j−1
Let ρ be any zero of ξ(s, χ) which is not the Siegel zero when χ is a real
nonprincipal character. By §14 of [5] we have
c
c
≤ <ρ ≤ 1 −
ln r(|ρ| + 2)
ln r(|ρ| + 2)
for a positive constant c. An argument similar to that made in the proof of
(3.9) of [3] shows that
X X −<ρ
(2.4)
|ρ|2
ρ
0
e−c
√
ln X
+
X −β
β2
for a positive constant c0 , where the second term on the right side of the
inequality exists only when β is the Siegel zero of L(s, χ).
Let
X
ψ0 (x, χ) =
χ(n)Λ(n)
n≤x
when x is not a prime power. By §19–20 of [5] we have
−
X X ρ̄
ρ̄
ρ
= ψ0 (X, χ̄) +
L0 (0, χ̄) 1 X + 1
− ln
L(0, χ̄)
2 X −1
when χ(−1) = −1, and
−
X X ρ̄
ρ
ρ̄
= ψ0 (X, χ̄) + b(χ̄) + ln
p
X2 − 1
when χ(−1) = 1, where b(χ) is the constant term in the expansion of L0 /L
near s = 0,
L0 (s, χ)
1
= + b(χ) + · · · ,
L(s, χ)
s
and
ψ0 (X, χ) = −
√
0
Xβ
+ O Xe−c ln X
β
500
XIAN-JIN LI
for a positive constant c0 . Since
X X −(1−ρ̄)
X X −ρ
=
ρ
1 − ρ̄
ρ
ρ
!
X X −(1−<ρ)
1 X X ρ̄
=−
+O
X ρ ρ̄
|ρ|2
ρ
√
0
X β−1
1 X X ρ̄
+ O e−c ln X +
,
=−
X ρ ρ̄
β2
we have
(2.5)
X X −ρ
ρ
ρ
0
X β−1 + e−c
√
ln X
for a positive constant c0 , where the term X β−1 exists only when β is the
Siegel zero of L(s, χ). It follows from (2.3), (2.4) and (2.5) that
X
X
lim
Φn,X (ρ) =
Φn (ρ).
X→∞
ρ
ρ
This completes the proof of the theorem.
3. Proof of Theorem 5
Explicit formula for LE (s) ([10]). Let F (x) be a function defined on
R such that
2F (x) = F (x + 0) + F (x − 0)
for all x ∈ R, such that F (x) exp(( + 1/2)|x|) is integrable and of bounded
variation on R for a constant > 0, and such that (F (x) − F (0))/x is of
bounded variation on R. Then
√
∞
X
N X Λ(n)
−
b(n) [F (ln n) + F (− ln n)]
Φ(ρ) = 2F (0) ln
2π
n
ρ
n=1
Z ∞
e−x
F (x) + F (−x)
−
−
2F
(0)
dx,
ex − 1
x
0
where the sum on ρ runs over all zeros of ξE (s) in the order given by |=ρ| < T
for T → ∞, and
Z
∞
Φ(s) =
F (x)e(s−1/2)x dx.
−∞
Lemma 3.1 ([6], [7], [11]). Let f be a newform of weight 2 for Γ0 (N ).
Then there an absolute effective constant c > 0 such that Lf (s) has no zeros
in the region
c
s = σ + it : σ ≥ 1 −
.
ln(N + 1 + |t|)
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS
501
Proof of Theorem 5. Since ξE (s) is an entire function of order one satisfying ξE (1) = 1 and ξE (s) = wξE (1 − s), we have
ξE (s) = w
Y
(1 − s/ρ),
ρ
where the product is over all the zeros of ξE (s) in the order given by |=ρ| < T
for T → ∞. If ϕE (z) = ξE (1/(1 − z)), then
∞
ϕ0E (z) X
=
λE (n + 1)z n .
ϕE (z) n=0
For a sufficiently large positive number X that is not an integer let


if − ln X < x < ∞,
Fn (x),
Fn,X (x) = 12 Fn (− ln X), if x = − ln X,


0,
if −∞ < x < − ln X,
where Fn (x) is given as in Lemma 2.3. Then Fn,X (x) satisfies all conditions
of the explicit formula for LE (s). Let
Φn,X (s) =
Z
∞
Fn,X (x)e(s−1/2)x dx.
−∞
By using the explicit formula, we obtain that
√
X
∞
N X Λ(k)
−
b(k) [Fn,X (ln k) + Fn,X (− ln k)]
2π
k
k=1
Z ∞
Fn,X (x) + Fn,X (−x)
e−x
−
−
2F
(0)
dx,
n,X
ex − 1
x
0
Φn,X (ρ) = 2Fn,X (0) ln
ρ
where the sum on ρ runs over all zeros of ξE (s) in the order given by |=ρ| < T
for T → ∞. It follows that
X
lim
Φn,X (ρ)
X→∞
ρ
√
n ∞
X
n (−1)j−1 X Λ(k)
b(k)(ln k)j−1
3/2
j
(j
−
1)!
k
j=1
k=1
!
∞
n
∞
X n
X
2 X
3
1
+n − +
+
(−1)j
.
3
l(2l + 3)
j
(l
+
1/2)j
j=2
N
= n ln
−γ
2π
!
−
l=1
l=1
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XIAN-JIN LI
We have
(3.1)
Φn (s) − Φn,X (s) = X −s
n X
n
j−1
X
(ln X)j−k−1
s−k−1
j
(j
−
k
−
1)!
j=1
k=0
n −s X
n (− ln X)j−1
(ln X)n−2 −<s
X
+O
X
.
=
s j=1 j
(j − 1)!
|s|2
(−1)j−1
Let ρ be any zero of ξE (s). By Lemma 3.1 and the Shimura-Taniyama conjecture we have
c
c
≤ <ρ ≤ 1 −
ln(N + 1 + |ρ|)
ln(N + 1 + |ρ|)
for a positive constant c. An argument similar to that made in the proof of
(3.9) of [3] shows that
(3.2)
X X −<ρ
|ρ|2
ρ
0
e−c
√
ln X
for a positive constant c0 .
Since
X X −ρ
X X −(1−ρ)
=
ρ
1−ρ
ρ
ρ
X X −(1−<ρ)
1 X Xρ
=−
+O
X ρ ρ
|ρ|2
ρ
0√
1 X Xρ
=−
+ O e−c ln X ,
X ρ ρ
!
and since
(ln X)j−1 X X ρ
=0
X→∞
X
ρ
ρ
lim
for j = 1, 2, . . . , n by Theorem 4.2 and Theorem 5.2 of [11], we have
(3.3)
lim (ln X)j−1
X→∞
X X −ρ
ρ
ρ
=0
for j = 1, 2, . . . , n. It follows from (3.1), (3.2) and (3.3) that
X
X
lim
Φn,X (ρ) =
Φn (ρ).
X→∞
ρ
This completes the proof of the theorem.
ρ
EXPLICIT FORMULAS FOR DIRICHLET AND HECKE L-FUNCTIONS
503
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Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
E-mail address: [email protected]