The distribution of values of logarithmic derivatives of real
L-functions
by
Mariam Mourtada
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
c 2013 by Mariam Mourtada
Copyright Abstract
The distribution of values of logarithmic derivatives of real L-functions
Mariam Mourtada
Doctor of Philosophy
Graduate Department of Mathematics
University of Toronto
2013
We prove in this thesis three main results, involving the distribution of values of
L0 /L(σ, χD ), D being a fundamental discriminant, and χD the real character attached to
it. We prove two Omega theorems for L0 /L(1, χD ), compute the moments of L0 /L(1, χD ),
and construct, under GRH, for each σ > 1/2, a density function Qσ such that
#{D fundamental discriminants, such that |D| ≤ Y, and α ≤ L0 /L(σ, χD ) ≤ β}
6
∼ √ Y
π 2 2π
Z
β
Qσ (x)dx.
α
ii
Dedication
To the Man who stood by my side during the hard times
To my little angels for the love they brought into our home
To my Mom and Dad for their endless sacrifice
To my parents in law for their love and support
I wouldn’t imagine myself graduating today without my husband Hassan’s encouragement
and patience during the 7 years we have been together, through all the storms, through
all the love, our bond is always growing stronger. No matter what happens, I know that
he is there, ready to wait, love and care.
I wouldn’t imagine myself graduating today without my little ones Hadi and Nour smiling
at me and giving me full strength to continue forward.
I wouldn’t imagine myself graduating today without my parents education since my toddler
years and the indescribable love of learning they implemented in my heart. Dad, I will
always remember you sitting with me, and helping me read carefully and understand more
and more stories. Mom I will never forget your unbelievably big heart, and selflessness.
I would like to thank my parents in law Khalil and Zeinab for their kindness and prayers,
for their support and big heart.
iii
Acknowledgements
I would like, first of all, to thank my great supervisor, Prof. V.Kumar Murty, for
everything. Basically, this thesis is due to his great supervision, in terms of pushing me
to always try, without mentioning his valuable knowledge and ability to manage all of
Ganita members’ work. You will never see Prof. Murty, the Chair, stressed, or upset. His
smile never leaves his face, and he looks calm all the time. I wonder how this could be
with all the responsibilities he is tackling. But I have no wonder how this was pushing
me to work more and more, regarding him all the way as an example.
I will forever cherish every single moment I spent at the Math Department at the University
of Toronto, with the great staff in charge, and my colleagues at Ganita.
I would love to express all my respects and thanks to Ida who is the living heart of our
department. Everyone, without exception, needs her, and she is always ready to help,
with such a great kindness!
I also would like to thank Pamela Britain for her continuous help, Sarah, Jemaima,
Patrina, Diana and Donna for being so nice!
Finally, I would like to express my sincere thanks to Prof. John Friedlander as well as
my PhD supervisory committee members, Prof. Steven Kudla, and Prof. James Arthur,
and my external examiner Prof. Kanaan Soundararajan for their valuable comments and
suggestions.
iv
Contents
1 Introduction
1.1
1.2
History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
Work on L0 /L . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Construction of Qσ . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.2.1
1.3
1
Summary of Chapters
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Operations on Dirichlet series
2.1
2.2
7
Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.1
Operations on Power series . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2
Tests of convergence . . . . . . . . . . . . . . . . . . . . . . . . .
8
Dirichlet series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Operations on Dirichlet series . . . . . . . . . . . . . . . . . . . .
9
Infinite Product and Euler Product . . . . . . . . . . . . . . . . . . . . .
11
2.3.1
Infinite Product
. . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3.2
Tests of convergence for infinite products . . . . . . . . . . . . . .
12
2.3.3
Euler Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.2.1
2.3
3 Known results on the distribution of L0 /L
3.1
6
History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
14
14
3.2
L-functions and L0 /L . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.3
Zero-free region for L(s, χ) . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.4
Recent progress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Connection to the distribution of L0 /L(1, χ) . . . . . . . . . . . .
23
Ihara’s study of the distribution of values of L0 /L . . . . . . . . . . . . .
24
3.4.1
3.5
4 Omega Theorems for L0 /L(1, χD )
26
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4.2
Unconditional bounds
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.2.1
Choice of Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
32
4.2.2
Application of the Explicit Formula . . . . . . . . . . . . . . . . .
33
4.2.3
The sum over zeros
. . . . . . . . . . . . . . . . . . . . . . . . .
36
4.2.4
Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . .
39
Conditional bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.3.1
Proof of Theorem 17 . . . . . . . . . . . . . . . . . . . . . . . . .
44
Moments of L0 /L(1, χD ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.4.1
Exceptional Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.4.2
The main term . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.4.3
The sum over zeros . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3
4.4
5 On Distribution functions
53
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.2
Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.3
Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.3.1
Properties of Fourier Transforms . . . . . . . . . . . . . . . . . .
56
5.3.2
Some properties of Mr . . . . . . . . . . . . . . . . . . . . . . . .
58
5.3.3
Convergence of infinite convolutions . . . . . . . . . . . . . . . . .
59
vi
6 Distribution of values of L0 /L(σ, χD )
62
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
6.2
Admissible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.3
Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
6.4
Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6.5
Proof of Theorem 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6.6
Toward a Density Function . . . . . . . . . . . . . . . . . . . . . . . . . .
70
6.6.1
Construction of Mσ (x) . . . . . . . . . . . . . . . . . . . . . . . .
72
6.6.2
Construction of Qσ (x) . . . . . . . . . . . . . . . . . . . . . . . .
72
6.6.3
Fourier transform of distributions and connection with Fourier
6.7
transform of functions . . . . . . . . . . . . . . . . . . . . . . . .
74
Average of L0 /L(σ, χD ), σ > 1/2 . . . . . . . . . . . . . . . . . . . . . .
79
6.7.1
The main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6.7.2
The main term . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.7.3
Estimation of the sum over zeros under GRH . . . . . . . . . . .
82
6.7.4
Estimation of the sum over zeros, unconditionally . . . . . . . . .
83
vii
Notation
we say f (x) = O(g(x)) if |f (x)| ≤ cg(x), as x → ∞ for some constant c, where
O
g(x) ≥ 0.
f (x) g(x) if f (x) = O(g(x)).
o
f (x) = o(g(x) means
f (x)
g(x)
→ 0 as x → ∞.
∞
P
1
denotes the Riemann zeta function ζ(s) =
.
ns
n=1
P
χ(n)
denotes L-functions defined by L(s, χ) = ∞
n=1 ns , s here is a complex number
ζ
L
with <s > 1, and χ a complex or real character.
denotes mainly an algebraic number field of finite degree over the field of rational
K
numbers Q.
ζK
is the Dedekind zeta function ζK (s) =
Q
P (1
− N (P −s ))−1 , P runs over prime
ideals of K, and <s > 1.
ρ
is used for non-trivial zeros of ζ, L-functions, or ζK .
( m. )
denotes the Jacobi symbol.
γ
is Euler’s constant, except when stated otherwise(denotes sometimes the imaginary
part of the zeros ρ).
γK
is the Euler-Kronecker constant.
GRH
Λ
stands for the Generalized Riemann Hypothesis.
is the von Mangoldt function.
p, q
φ
D
P0
will always denote prime numbers, except when stated otherwise.
is Euler’s totient function.
will always represent a fundamental discriminant.
P∗
or
represent a restricted sum over a set of integers.
N (σ, T, χ)
logk
is the number of zeroes ρ = β + it of L(s, χ) such that β ≥ σ and t ≤ T .
stands for log(logk−1 ), where log1 := log and k ≥ 2.
Ω, Omega
we say that f (x) is Omega of g(x) and we write f (x) = Ω(g(x)) if
f (x) g(x) for arbitrarily large values of x.
viii
δ
is the usual Dirac-delta function.
this is equal to #{Fundamental discriminants D; |D| ≤ Y } ∼ 6/π 2 Y .
N (Y )
this represents #{Prime p; p ≤ x} ∼ x/ log x by the Prime Number Theorem.
π(x)
N (χ, σ0 , T )
fY (n)
this represents #{ρ = σ + it; L(ρ, χ) = 0, σ ≥ σ0 and |t| ≤ T }.
X∗
is equal to
χD (n) = π62 + O(Y 1/2 ) and sometimes this represents
|D|≤Y
X∗
χD (n) =
0<βD≤Y
p
d
3
π2
+ O(Y
1/2
), for β = ±1.
R
|f (x)|p dx < ∞.
L (R )
represents the set of all functions such that
L∞ (Rd )
denotes the set of functions that are bounded almost everywhere (i.e bounded
up to a set of measure zero).
ix
Rd
Chapter 1
Introduction
In this chapter we present the contents of this thesis.
1.1
1.1.1
History
L-functions
A classical result of Chowla [3] states that for infinitely many fundamental discriminants
D we have
L(1, χD ) ≥ (1 + o(1))eγ log log |D|
where χD is the quadratic Dirichlet character of conductor D. Before Chowla, Littlewood
[23], showed that under GRH
1
ζ(2)
+ o(1) γ
≤ L(1, χD ) ≤ (2 + o(1))eγ log2 |D|
2
e log2 |D|
and that for infinitely many fundamental discriminants D, we have
L(1, χD ) ≥ (1 + o(1))eγ log2 |D|
and for infinitely many fundamental discriminants D, we have
L(1, χD ) ≤
(1 + o(1))ζ(2)
.
eγ log2 |D|
1
2
Chapter 1. Introduction
After Chowla, several authors refined these results. In particular, Granville and Soundararajan [6] studied the distribution of values of L(1, χD ), and provided a stronger version of
Chowla’s Omega Theorem. Conditionally, they proved that ([6], Theorem 5a) assuming
GRH there are infinitely many primes q such that
.
L(1,
) ≥ eγ (log2 q + log3 q − log(2 log 2) − )
q
and infinitely many primes q
.
ζ(2)
L(1,
) ≤ γ (log2 q + log3 q − log(2 log 2) − ).
q
e
Unconditionally, they proved ([6], Theorem 5b) that
.
L(1,
) ≥ eγ (log2 x + log3 x − log4 x − 10)
d
1
holds for x 10 fundamental discriminants d ≤ x. Here log1 x = log x, and logk+1 x =
log(logk x).
Recently, R. Vaughan [33] and H. Montgomery and Vaughan [24], have pursued a probabilistic model, which led them to formulate the following conjectures.
(i)If we let P1 (x) denote the proportion of fundamental discriminants |D| ≤ x such that
L(1, χD ) ≥ eγ log2 |D|, then
exp(−C log x/ log2 x) < P1 (x) < exp(−c log x/ log2 x)
for appropriate constants 0 < c < C < ∞. A similar estimate should hold for the
proportion of D such that L(1, χD ) ≤ ζ(2)/(eγ log2 |D|).
(ii)If we let P2 (x) denote the proportion of fundamental discriminants |D| ≤ x s.t
L(1, χD ) ≥ eγ (log2 |D| + log3 |D|), then
xθ < P2 (x) < xΘ
for some 0 < θ < Θ < 1. A similar estimate should hold for the proportion of D such
that L(1, χD ) ≤ ζ(2)/(eγ (log2 |D| + log3 |D|)).
3
Chapter 1. Introduction
(iii)For any > 0, there are only finitely many D such that
L(1, χD ) ≥ eγ (log2 |D| + (1 + ) log3 |D|)
or such that
L(1, χD ) ≤
eγ (log
ζ(2)
.
2 |D| + (1 + ) log3 |D|)
The first of these conjectures was established by Granville and Soundararajan, and the
second almost follows from GRH. A more general work on the distribution of extreme
values of L-functions at the edge of the critical strip was done recently by Y. Lamzouri in
[21], [22].
1.1.2
Work on L0 /L
Ihara [11] has initiated the study of the distribution of values of the logarithmic derivatives
of L-functions L0 /L, and introduced what is called the Euler-Kronecker constant γK of
a global field K. This constant is related to L0 /L, and in the case of a quadratic field
√
Q( D), Ihara-Murty-Shimura [17] proved that under GRH
0
L
(1, χD ) ≤ 2 log log |D| + O(1).
L
They also computed the moments of L0 /L(1, χ), where χ runs over all non-principal
multiplicative characters χ mod m. They proved
b
1 X 0
(L (1, χ)/L(1, χ))a L0 (1, χ)/L(1, χ) = Ca,b
m→∞ |Xm |
χ
lim
where Xm denotes the set of all non principal multiplicative characters χ mod m, and
the sum is over such set. Here Ca,b is a well defined constant.
1.2
Statement of results
In this thesis we prove three main results. The first result is on Omega theorems for
L0 /L(1, χD ), where D is a fundamental discriminant, and χD is the real character attached
Chapter 1. Introduction
4
to D. These theorems describe large values of L0 /L, and their influence on the behaviour
of the Euler-Kronecker constant. We prove unconditionally that
L0
(1, χD ) ≥ log log |D| + O(1)
L
holds for infinitely many fundamental discriminants D. Furthermore, we prove that for
infinitely many fundamental discriminants D, we have
L0
(1, χD ) ≤ − log log D + O(1).
L
We also prove under GRH, that for infinitely many primes q ≡ 1 mod 4, we have
L0
(1, χq ) ≥ log log q + log log log q + O(1)
L
and for infinitely many primes q ≡ 1 mod 4, we have
L0
(1, χq ) ≤ − log log q − log log log q + O(1).
L
In the second result, we prove that the moments of L0 /L are constant. In other words,
for each non-negative integer k, there is a constant Ck so that
k
X∗ L0
− (1, χD ) ∼ Ck Y.
L
0<βD≤Y
Here Y > 1, β = ±1, and the asterisk on the sum indicates that we are summing over
fundamental discriminants D.
The Omega results, stated precisely in Chapter 4, justify the following conjecture.
Conjecture. For β = ±1
L0
(1, χD ) = log2 x + log3 x + O(1)
x≤βD≤2x L
max
and
L0
(1, χD ) = − log2 x − log3 x + O(1).
x≤βD≤2x L
min
5
Chapter 1. Introduction
The max and min are being taken over fundamental discriminants D.
The third result which is on the distribution of values of L0 /L(σ, χD ) shows that, assuming
GRH, for each σ > 12 , there is a density function Qσ such that
#{D fundamental discriminants, such that |D| ≤ Y, and α ≤ L0 /L(σ, χD ) ≤ β}
6
∼ √ Y
2
π 2π
1.2.1
Z
β
Qσ (x)dx.
α
Construction of Qσ
Write
∞
exp(
X λz (n)
iz ζ 0
(s)) =
2 ζ
ns
n=1
Here ζ(s) is the Riemann zeta function.
We note that this is well defined, since in Chapter 2 of the thesis, we prove that if f (s) is
a Dirichlet series, then so is exp(f (s)). Let us set
Q
∞
X
λ2a (n2 ) p|n (1 + p1 )−1
fσ (a) :=
M
.
n2σ
n=1
Then, we prove that
1 X∗
L0
fσ (a).
lim
exp ia (σ, χD ) = M
Y →∞ N (Y )
L
|D|≤Y
Here N (Y ) := #{D fundamental discriminants, such that |D| ≤ Y }. We also prove that
fσ (a) = Oσ (exp(−Cσ |a|1/σ ))
M
for some positive constant Cσ . We then define its Fourier inverse to be the density function
we are looking for
bσ (a) = M
fσ (a).
Q
We further compute the average over fundamental discriminants D, of L0 /L(σ, χD ), under
GRH, and we show that it is a constant dependent only on σ. We also compute this
average unconditionally, for certain values of σ.
Chapter 1. Introduction
1.3
6
Summary of Chapters
Chapter 2 presents some basic but useful results on Dirichlet series that we will need later.
We cover the differences as well as similarities, and provide some lemmas on possible
operations on them.
Chapter 3 covers known results on the distribution of L0 /L(s, χ), s being a complex
number, and χ a complex character. It is noteworthy that most of the work has been
initiated mainly by Ihara in [11], Ihara-Matsumoto [16], and Ihara-Murty-Shimura [17].
We also cover new results from different authors.
Chapter 4 is based on a recent paper by M. Mourtada and V. Kumar Murty [27], it covers
all the Omega results presented above.
Chapter 5 summarizes most results from a work of B.Jessen and A.Wintner [18], on
distribution functions that will be needed in Chapter 6. This last chapter gives the actual
distribution results on L0 /L(σ, χD ).
Chapter 2
Operations on Dirichlet series
In this chapter, we present some useful facts about Dirichlet series. We present a
comparison between power series and Dirichlet series, and study some operations on
Dirichlet series. This will be in particular helpful to understand the type of obstacles we
are facing when we follow methods used to study the distribution of L-functions and try
to apply these to the case of L0 /L . We finish the chapter by reviewing some properties
of infinite products and Euler products.
2.1
Power Series
A complex power series is a series of the form
f (z) =
X
an z n .
n
Here the coefficients an are complex, and the variable z is a complex variable as well.
When the series converges for all |z| < R and diverges for all |z| > R, we call R the radius
of convergence of the series.
7
8
Chapter 2. Operations on Dirichlet series
2.1.1
Operations on Power series
We recall some elementary properties of power series: For two power series f (z) =
P
and g(z) = n bn z n and c a complex number,
1. f (z) + g(z) =
P
n (an
P
n
an z n
+ bn )z n is again a power series.
2. f (z)g(z) is a power series of the form
3. cf (z) is a power series of the form
P
P Pn
n
n(
k=1 ak bn−k )z .
n (can )z
n
.
4. If a0 6= 0 then f (z) has an inverse power series.
5. If R is the radius of convergence of f (z) then, for all |z| < R, f (z) is differentiable
and its derivative is again a power series. Similarly, f (z) is integrable and its
antiderivative is also a power series that is convergent for all |z| < R .
2.1.2
Tests of convergence
There are many ways to test the convergence of a series. However we will present here
tests that might be useful to proofs related to this thesis.
Theorem 1. (Dirichlet’s test) If an is a sequence of real numbers and bn a sequence of
complex numbers satisfying
1. an ≥ an+1
2. lim an = 0
3.
PN
n=1 bn
as n −→ ∞
≤M
then the series
for every integer N > 0
P
n
an bn is convergent. Here M is some constant.
9
Chapter 2. Operations on Dirichlet series
2.2
Dirichlet series
A Dirichlet series is a series of the form
f (s) =
X an
ns
n
; s ∈ C.
Dirichlet series enjoy most properties listed above for power series, except that the domain
of convergence is a half plane rather than a disc. Within wedge shaped regions in the half
plane of convergence, the convergence is uniform, and so can differentiate and integrate
term by term, in particular,
f 0 (s) =
X an log n
ns
n
.
Proposition 1. ([31], p.291) The region of absolute convergence of a Dirichlet series is
a half-plane.
2.2.1
Operations on Dirichlet series
Lemma 1. Let f (s) =
P∞
an
n=1 ns
be a Dirichlet series, then exp(f (s)) is a formal Dirichlet
series.
Proof. Formally, we have
∞
∞
∞
∞
X
X
1 X an k X 1 X 1
exp(f (s)) =
(
)
=
(
an . . . ank ) .
k! n=1 ns
ns k=1 k! n=n ...n 1
n=1
k=1
1
k
P
P
1
We have to show now that for each n, Tn := ∞
k=1 k! (
n=n1 ...nk an1 . . . ank ) converges.
Q
Let Mn be the number of ways one can write n = r cr , where the cr are non trivial
divisors, then for all k ≥ d(n) =: dn , the number of ways one can write n = n1 . . . nk is
k!
Mn (k−d
. Indeed, for each such writing of n as a product of non-trivial divisors, one
n )!
can deduce a writing of n as a product n1 . . . nk , for k large enough, where the empty
places are filled up with 1. This shows that our series
Tn Mn Nn
∞
X
k=dn
(a1 )k
(k − dn )!
+1
10
Chapter 2. Operations on Dirichlet series
Here Nn := max{
Q
r
ac r ; n =
Q
r cr
the cr being non trivial divisors}.
Clearly such series converges. Hence the lemma is proved.
Lemma 2. Let f (s) =
P∞
an
n=1 ns
be a Dirichlet series convergent in some half plane. Then
the following are equivalent:
1. f (s) has a formal Dirichlet inverse.
2. a1 6= 0.
3. log(f (s)) has a formal Dirichlet series.
Proof. (1 ⇒ 2) Let g(s) =
P∞
bn
n=1 ns
be such that f (s)g(s) = 1. Then by comparison of
coefficients one can show that a1 b1 = 1, then a1 6= 0.
(2 ⇒ 1) Suppose a1 6= 0, then one can construct the inverse as follows: b1 = a11 , bn =
P
− a11
bd . Thus g(s) = (f (s))−1 exists as a Dirichlet series, at least formally.
d|n a n
d
d<n
P
bn
0
(1 ⇒ 3) Suppose f (s) has a Dirichlet inverse g(s) = ∞
n=1 ns , then f /f (s) has a Dirichlet
P
cn
series, this shows that d(log(f (s)))/ds has a Dirichlet series f 0 /f (s) = ∞
n=1 ns , this
implies that
log(f (s)) =
X −cn 1
+ constant
s
log
n
n
n≥2
where the constant can be considered as a term of the series, so we conclude that log(f (s))
itself has a Dirichlet series expansion.
(3 ⇒ 1) Suppose now that log(f (s)) has a Dirichlet series, then log(1/f (s)) = − log(f (s))
has a Dirichlet series, this implies according to Lemma 1, that exp(log(1/f (s))) has a
Dirichlet series, in other terms 1/f (s) has a Dirichlet series. This finishes the proof of the
Lemma.
Remark 1. We know from the above lemmas, that −ζ 0 (s) =
∞
P
n=1
log n
ns
does not have a
Dirichlet inverse because its constant term is 0, thus log(−ζ 0 (s)) does not have a Dirichlet
11
Chapter 2. Operations on Dirichlet series
series expansion and neither does log(−ζ 0 (s)/ζ(s)). This will prevent us from regarding
the complex moments of −ζ 0 (s)/ζ(s) as Dirichlet series, since these are defined as being
0 −ζ (s)
, z ∈ C.
exp z log
ζ(s)
2.3
2.3.1
Infinite Product and Euler Product
Infinite Product
An infinite product is an expression of the form [31]
P =
∞
Y
(1 + an )
n=1
where the an are supposed to be different from -1. This infinite product is said to be
convergent if the partial product
Pn :=
n
Y
(1 + am )
m=1
has a limit as n −→ ∞. Otherwise the product is said to be divergent.
Definition 1. The product
Q
(1 + |an |) is convergent.
Q
(1 + an ) is said to be absolutely convergent if the product
Proposition 2. If an ≥ 0 for all values of n, the product
Q
(1 + an ) and the series
P
an
converge or diverge together.
Q
P
Corollary 1. (1 + an ) converges absolutely if and only if
|an | is convergent (in other
P
terms
an is absolutely convergent).
Definition 2. The infinite product
P =
∞
Y
(1 + un (z))
n=1
Chapter 2. Operations on Dirichlet series
12
where the factors are functions of the complex variable z, is uniformly convergent if the
partial product
Pn (z) :=
n
Y
(1 + um (z))
m=1
is convergent uniformly, in a certain region of values of z, to a limit.
Q
Proposition 3. The product (1 + un (z)) is uniformly convergent in any region where
P
the series
|un (z)| is uniformly convergent.
2.3.2
Tests of convergence for infinite products
We state in this subsection some tests for convergence of infinite products, that are
relevant to the present thesis.
Proposition 4. ([30], p.141) If
P
|an | < ∞, then the infinite product
∞
Y
(1 + an )
n=1
converges. Moreover, the product converges to 0 if and only if one of its factors is 0.
2.3.3
Euler Product
An Euler product is an expansion of a Dirichlet series into an infinite product over the
prime numbers. Leonhard Euler was the first to write such a product for the Riemann
Zeta function.
Definition 3. An arithmetic function
a : N −→ C
is said to be multiplicative, if for every two coprime integers m and n, we have
a(mn) = a(m)a(n).
It is called totally multiplicative if this is true for any integers m and n.
13
Chapter 2. Operations on Dirichlet series
Definition 4. Let a be an arithmetic multiplicative function, then the Dirichlet series
∞
X
a(n)
ns
n=1
has the following product expansion
Y
P (p, s)
p
where for each prime p
P (p, s) =
∞
X
a(pk )
k=0
pks
.
Such an expansion is called the Euler product of the Dirichlet series
P
a(n)/ns .
Remark 2. If a is totally multiplicative, then
P (p, s) =
1
1 − a(p)p−s
this is true because in such case the sum of P (p, s) is a geometric series. For example,
when a(n) = 1 for all n, then we get the Euler product of the Riemann zeta function,
and when a(n) = χ(n) for some multiplicative character χ then this is the product of the
corresponding L-function L(s, χ).
Chapter 3
Known results on the distribution of
L0/L
In this chapter we cover in detail what has been done on the distribution of values of
L0 /L and the techniques used. We also present some conjectures and the progress that
has been made on them so far.
3.1
History
Historically, L- functions have been studied in detail in many aspects, ranging from their
functional equation, to zero-free regions, to special values, and so on. The study of the
distribution of values of L-functions has gained a comparable attention as well, from
the work of Selberg, to Chowla’s great Omega Theorem, to Montgomery and Vaughan,
and recently to the work of Granville-Soundararajan, as well as many other authors.
Recently, Y.Ihara [11, 10], has initiated a study of the distribution of values of logarithmic
derivatives of L-functions, L0 /L. Those differ from L-functions in being meromorphic
functions rather than holomorphic functions.
The logarithmic derivatives of L-functions, L0 /L, are important in many applications,
and are expected to be related to periods of Abelian varieties.
14
Chapter 3. Known results on the distribution of L0 /L
3.2
15
L-functions and L0/L
Dirichlet L-functions are given by
L(s, χ) =
∞
X
χ(n)
n=1
ns
,
<s > 1.
Here χ is a multiplicative character. The functional equation of L-functions has been
initially given by Hurwitz in 1882 [4], and next by de La vallée Poussin in 1896. The
latter’s method was a generalization of what Riemann did in his memoir on the zeta
function. The equation can be presented as follows. If we let
π 1
s+a
ξ(s, χ) = ( )− 2 (s+a) Γ(
)L(s, χ)
q
2
where a = 0 (resp. a = 1) if χ is even (resp. odd) then the mentioned functional equation
is
1
ia q 2
ξ(1 − s, χ) =
ξ(s, χ).
τ (χ)
where for any character χ to the modulus q, the Gaussian sum τ (χ) is defined by
τ (χ) =
q
X
χ(m)eq (m)
m=1
). This functional equation is valid for all complex numbers s.
where eq (m) := exp( 2πim
q
From the functional equation, we can easily see that L-functions have zeros coming from
the poles of the Gamma function Γ( s+a
). These zeros are called the trivial zeros. All
2
other zeros lie in the critical strip defined by
0 < <s < 1.
From the above functional equation, and given the fact that the Γ function has poles at 0,
-1, -2, -3,... we conclude that for a = 0, L-functions have the following trivial zeros
s = 0, − 2, − 4, − 6, . . .
and for a = 1, the trivial zeros of L-functions are
s = −1, − 3, − 5, . . .
Chapter 3. Known results on the distribution of L0 /L
16
All other zeros lie in the critical strip, and these are called non-trivial zeros.
In addition to their functional equation, L-functions enjoy the property of having an
Euler product that is valid for all complex s, with <s ≥ 1
L(s, χ) =
Y
χ(p)
(1 − s )−1 .
p
p
The logarithmic derivatives of L-functions do not have, however, a functional equation,
nor an Euler product, and given the fact that they have poles at the zeros of L(s, χ), this
makes it more difficult to work with them. Despite these facts, one can work on L0 /L(s, χ)
either outside the critical strip, for <s > 1 to get unconditional results, or inside the
critical strip, for 0 ≤ <s ≤ 1, with the assumption of the Generalized Riemann Hypothesis
GRH. From this we can see that expanding the zero-free region for L-functions, would
help us getting unconditional results on the distribution of values of L0 /L(s, χ), within
that zero-free region. By differentiating the logarithm of the Euler product of L(s, χ), we
get
∞
X
L0
Λ(n)χ(n)ns
− (s, χ) =
L
n=1
for <s > 1. Here
Λ(n) =



log p if n = pα , p prime


0
elsewhere.
Taking logarithm of both sides of the functional equation for L-functions, we get
1
ia q 2
log ξ(1 − s, χ) = log(
) + log ξ(s, χ).
τ (χ)
Differentiating this equality with respect to s, we get
−ξ 0 /ξ(1 − s, χ) = ξ 0 /ξ(s, χ).
On the other hand, we quote the following equation from [4], p. 83 that is useful in many
occurrences.
X 1
L0
1
q
1 Γ0 s + a
1
(s, χ) = − log −
(
) + B(χ) +
+
L
2
π 2Γ
2
s−ρ ρ
ρ
Chapter 3. Known results on the distribution of L0 /L
17
where a depends on the character χ, and has been defined above. The sum over ρ
represents the sum over all non-trivial zeros of L(s, χ), in other terms zeros within the
critical strip. As for the number B(χ), it is given by
B(χ) =
ξ0
ξ0
(0, χ) = − (1, χ)
ξ
ξ
and since
B(χ) = B(χ)
we get
<B(χ) = −
3.3
X 1
< .
ρ
Zero-free region for L(s, χ)
First of all, it is noteworthy that
L(1, χ) 6= 0
thus is natural to think of a zero-free region of L(s, χ), around s = 1. The focus will be
on the region within the critical strip, since for <s > 1, we already know that there are
only the trivial zeros of L(s, χ). Many theorems have been proved in this direction, like
the following theorem, which is due partly to Gronwall and partly to Titchmarch [4].
Theorem 2. ([4], p.93) There exists a positive absolute constant c with the following
property. If χ is a complex character modulo q, then L(s, χ), s = σ + it, has no zero in
the region defined by
σ≥



1 −
c
log q|t|
if |t| ≥ 1,


1 −
c
log q
if |t| ≤ 1.
If χ is a real non principal character, the only possible zero of L(s, χ) in this region is a
single (simple) real zero.
Chapter 3. Known results on the distribution of L0 /L
18
We present now a theorem of Landau, which ensures that if there exist values of q for
which L(s, χ), where χ is a real primitive character
β >1−
c
,
log q
mod q, has a real zero β with
then such values of q are rare.
Theorem 3. (Landau, [4]) If χ1 and χ2 are two distinct real primitive characters to
the moduli q1 , q2 respectively, and if the corresponding L-functions have real zeros β1 , β2
respectively, then
min(β1 , β2 ) < 1 −
C
log(q1 q2 )
where C is some positive absolute constant. The possibility that q1 = q2 is not excluded.
Another important theorem, which is due to Page states the following
Theorem 4. (Page Theorem, [4], p.95) If c is a suitable positive constant, then there is
at most one real primitive character χ to a modulus q ≤ z for which L(s, χ) has a real
zero β satisfying
β >1−
c
.
log z
We state now two forms of Siegel’s Theorem.
Theorem 5. (Siegel, [4], p. 126) For any > 0, there exists a positive number C1 ()
such that, if χ is a real primitive character to the modulus q, then
L(1, χ) > C1 ()q − .
Theorem 6. (Siegel, [4], p. 126) For any > 0, there exists a positive number C2 ()
such that, if χ is any real non principal character, with modulus q, then L(s, χ) 6= 0 for
s > 1 − C2 ()q − .
3.4
Recent progress
In a paper [11] dedicated to V.Drinfeld, Ihara introduced his definition of what is called
the Euler-Kronecker constant γK of a global field K which generalizes Euler’s constant γ
Chapter 3. Known results on the distribution of L0 /L
19
in many ways. Here
γ = γQ = lim (1 +
n→∞
1
1
+ · · · + − log n) = 0.57721566 . . .
2
n
Ihara considered this constant more as an invariant of the field K and defined it as follows.
For a number field K, define the Dedekind zeta function ζK (s), <s > 1, by the Dirichlet
series
ζK (s) =
X
I
1
(N I)s
where the sum ranges over integral ideals of K, and N I denotes the norm of the integral
ideal I. It has an analytic continuation for all s with simple pole at s = 1. Write the
expansion of the Dedekind zeta function near s = 1 as
ζK (s) = c−1 (s − 1)−1 + c0 + O(s − 1).
Then set
γK =
c0
.
c−1
In the same paper [11], Ihara proved the following explicit formula for γK
γK =
1X
1
1
r1
− log |d| + (γ + log 4π) + r2 (γ + log 2π) − 1
2 ρ ρ(ρ − 1) 2
2
where d = dK is the discriminant of K, and r1 , r2 are respectively the number of real, (resp.
imaginary) places of K(in other terms, the number of real, resp. complex embeddings).
P
The sum ρ being taken over non-trivial zeros of ζK (s). Ihara proved the following upper
bound for γK . Let
αK =
1
log |dK |.
2
Theorem 7. (Theorem 1, [11]) Under GRH, we have
γK <
αK + 1
2
(2 log αK + 1 − ΦK (αK
))
αK − 1
Chapter 3. Known results on the distribution of L0 /L
≤
20
αK + 1
(2 log αK + 1)
αK − 1
provided that the degree
[K : Q] > 2
or
[K : Q] = 2
|dK | > 8.
and
Here ΦK is the prime counting function given by
1
ΦK (x) =
x−1
X N (P )k ≤x
x
N (P )k − 1
log N (P )
for x > 1, and P runs over the non-archimedean primes of K. He then deduced that
under GRH
γK ≤ 2 log log
p
|d|.
He further provided the following unconditional lower bound. If we let
βK = −{
r1
(γ + log 4π) + r2 (γ + log 2π)}
2
then
Proposition 5. (Proposition 3, [11]) In the number field case,
γK > −αK − βK − 1.
This can be read as
γK > − log
p
|d|
Ihara further proved that the above lower bound can be improved under some conditions.
Theorem 8. (Theorem 3, [11]) Let K be an extension of Q of degree N > 1. Put
∗
αK
= αK /(N − 1)
Chapter 3. Known results on the distribution of L0 /L
21
p
log |d|
=
N −1
∗
and assume αK
> 1. Then under GRH
∗
αK
+1
∗
(γK + 1) > −2(N − 1)(log αK
+ 1).
∗
αK − 1
∗
From this theorem, Ihara deduced that when αK
→ ∞ and N is fixed or grows slowly
enough
p log |d|
γK > −2(N − 1) log
.
N −1
In the same paper [11], Ihara’s main tool was the explicit formula for the prime counting
function ΦK (x), for which he proved that when x is large enough, we have
lim (log x − ΦK (x)) = γK + 1.
x→∞
An important special case of Ihara’s general work, was when K is either the cyclotomic
field Q(µm ) or its maximal real subfield Q(µm )+ . He conjectured that
γQ(µm ) > 0.
This has been made more precise in conclusions Ihara presented in his paper [10], drawn
from numerical computations. Let
+
Km := Q(µm ) and Km
:= Q(µm )+ .
Conjecture.([10],Conjecture 1)
(i)γKm and γKm+ are positive.
+
(ii)There exist positive constants c1 , c2 , c+
1 , c2 , all ≤ 2, such that for any > 0,
(c1 − ) log m < γKm + 1 < (c2 + ) log m,
and
+
+ + 1 < (c
(c+
1 − ) log m < γKm
2 + ) log m,
Chapter 3. Known results on the distribution of L0 /L
22
hold for all sufficiently large m.
(iii) When m is restricted to primes, one can choose
+
c1 = 1/2, c+
1 = 1 and c2 = c2 = 3/2.
Furthermore, Ihara [10] proved that assuming GRH for Km , and for each m consider the
“weighted average” of cos(γ log m),
c(m) :=
X mρ−1/2 X
1
/
ρ(ρ − 1) ρ ρ(ρ − 1)
ρ
=
X cos(γ log m)
ρ
1/4 + γ 2
∗
where ρ = 1/2 + iγ. Now, set γK
:= γK + 1.
Proposition 6. ([10], Proposition 3) (Under GRH) Let M be any given infinite set
∗
of prime numbers, and let m run over M . Then, |γKm∗ / log m| (resp. |γK
+ / log m|) is
m
√
√ +
bounded if and only if | mc(m) − 1| (resp. | mc (m) − 1|) is so, and when these
conditions are satisfied, we have
∗
γK
/ log m =
m
∗
γK
+ / log m =
m
√
3
1
+ ( mc(m) − 1) + O(
)
2
log m
(m ∈ M )
√
3
1
+ ( mc+ (m) − 1) + O(
)
2
log m
(m ∈ M )
√
respectively. In particular, the above Conjecture 1 (iii) is equivalent to that mc(m) − 1
√
(resp. mc+ (m) − 1 ) belongs to the interval (−1 − , ) when m is sufficiently large.
Tsfasman [32] proved, assuming GRH, that
lim inf
γK
≥ −0.13024...
log dK
where the lim inf is taken over number fields K with dK → ∞.
Chapter 3. Known results on the distribution of L0 /L
3.4.1
23
Connection to the distribution of L0 /L(1, χ)
It is noteworthy that an equivalent definition of γK is
0
1
ζK
(s) +
.
γK = lim
s→1 ζK
s−1
In their paper [17], Ihara, Murty and Shimura showed that
X
γQ(µm ) = γ +
L0 /L(1, χ)
χ6=χ0
and
X
γQ(µm )+ = γ +
L0 /L(1, χ)
χ6=χ0
χ(−1)=1
where χ runs over all non-principal multiplicative characters
χ : (Z/m)× −→ C×
and χ0 being the trivial character. They also proved under GRH
|γQ(µm ) |, |γQ(µm )+ | = O((log m)2 )
and unconditionally, for any > 0
|γQ(µm ) |, |γQ(µm )+ | = O(m ).
More strongly, Badzyan [2] showed that under GRH, we have
γQ(µm ) = O(log m log log m).
Recently, Murty [26] has proved unconditionally that Ihara’s conjecture is true on average.
Theorem 9. ([26], Theorem 1.1) We have
X
|γq | π ∗ (Q) log Q
1
Q<q≤Q
2
where π ∗ (Q) denotes the number of primes in the interval ( 12 Q, Q] and the sum is over
primes q in this interval.
Fouvry [5] has refined this to an asymptotic formula.
Chapter 3. Known results on the distribution of L0 /L
24
Ihara’s study of the distribution of values of L0/L
3.5
In his paper [10], Ihara constructed and studied a function Mσ (z) on C parametrized
fσ (z). This function is closely related
by σ > 1/2, together with its Fourier transform M
to the density measure for the distribution of values on C of the logarithmic derivatives
of L-functions L(s, χ) , where s is fixed, <s = σ, and χ runs over an infinite family of
Dirichlet characters. Let P be any non-empty finite set of non-archimedean prime divisors
Q
of K, TP := P∈P C1 and let gσ,P : TP −→ C be defined by
gσ,P (tP ) =
X
gσ,P (tP )
P∈P
where
gσ,P (tP ) =
tP log N (P)
tP − N (P)σ
and tP = (tP )P∈P . Ihara [10] proved the following
Theorem 10. ([10], Theorem 1) Let σ > 0. There exists a unique function Mσ,P (z) of
z ∈ C, which is a hyperfunction (Schwartz distribution) when |P | = 1, that satisfies
Z
Z
Mσ,P (ω)Φ(ω)|dω| =
Φ(gσ,P (tP ))d∗ tP
TP
C
for any continuous function Φ(ω) on C, where |dω| = (2π)−1 dxdy (ω = x + yi), and d∗ tP
is the normalized Haar measure on TP . It is compactly supported, and satisfies
Z
Mσ,P ≥ 0,
Mσ,P (ω)|dω| = 1
C
Let Py := {P ∈ P ;
N (P) ≤ y} and D(a,b) :=
∂ a+b
.
∂z a ∂z b
Theorem 11. ([10], Theorem 2) Let σ > 1/2, P = Py , and let y → ∞. Then
1. Mσ,P (z) converges uniformly to a non-negative real valued C∞ -function Mσ (z).
2. Each D(a,b) Mσ,P (z) converges uniformly to D(a,b) Mσ (z) (starting with |P | sufficiently
large).
Chapter 3. Known results on the distribution of L0 /L
3. For any n ≥ 1, |z|n Mσ (z) belongs to L2 .
4. The function Mσ (z) is not identically zero; in fact
Z
Mσ (z)|dz| = 1.
C
It satisfies
Mσ (z) = Mσ (z) = Mσ (z).
25
Chapter 4
Omega Theorems for L0/L(1, χD )
In this chapter we present our results on Omega theorems for L0 /L. This is based on our
paper [25].
4.1
Introduction
A classical result of Chowla [3] states that for infinitely many fundamental discriminants
D we have
L(1, χD ) ≥ (1 + o(1))eγ log log |D|
where χD is the quadratic Dirichlet character of conductor D. In other words,
L(1, χD ) = Ω(log log |D|).
Before Chowla, Littlewood [23] proved, assuming GRH, that
L(1, χ) ≤ (2 + o(1))eγ log log m
holds for all sufficiently large conductors m, χ being a character of conductor m.
After Chowla, several authors refined this result. In particular, Granville and Soundararajan [6] studied the distribution of values of L(1, χD ), and provided a stronger version of
Chowla’s Omega Theorem. Conditionally, they proved the following
26
Chapter 4. Omega Theorems for L0 /L(1, χD )
27
Theorem 12. ([6], Theorem 5a)Assume GRH. For any > 0, and all large x, there are
1
x 2 primes q ≤ x such that
.
) ≥ eγ (log2 q + log3 q − log(2 log 2) − )
L(1,
q
1
and x 2 primes q such that
.
ζ(2)
L(1,
) ≤ γ (log2 q + log3 q − log(2 log 2) − ).
q
e
Unconditionally, they proved
1
Theorem 13. ([6], Theorem 5b) For large x there are at least x 10 square-free integers
d ≤ x such that
.
L(1,
) ≥ eγ (log2 x + log3 x − log4 x − 10).
d
Granville and Soundararajan [6] further proved the following important theorem about
the distribution of L(1, χD ). Let us set
Φx (τ ) :=
X ∗
1 /
1
X∗
|d|≤x
L(1,χd )>eγ τ
|d|≤x
and
Ψx (τ ) :=
X ∗
1
1 .
X∗
|d|≤x
|d|≤x
2
L(1,χd )≤ 6eπγ τ
Where
P∗
stands for the sum over fundamental discriminants.
Theorem 14. ([6], Theorem 4) Let x be large and let log2 x ≥ A ≥ e be a real number.
Uniformly in the range τ ≤ R1 (x) − log2 (x) we have
eτ −C1
1
1
Φx (τ ) = exp −
(1 + O( + ))
τ
A τ
and the same asymptotic holds for Ψx (τ ). Here we may take R1 (x) = log2 x + log4 x − 20
unconditionally, and R1 (x) = log2 x + log3 x − 20 if the GRH is true.
Chapter 4. Omega Theorems for L0 /L(1, χD )
In this chapter, we prove analogous results for the logarithmic derivatives
28
L0
(1, χD ),
L
and investigate the growth of the logarithmic derivatives of real Dirichlet L-functions.
This work was motivated by the lack of literature on the subject, despite the fact that a
lot of work has been done on the study of the value distribution theory of L-functions
(for example the work of Selberg and others).
Ihara began a systematic study of the value distribution of the logarithmic derivatives
of Dirichlet L-functions [11] and has obtained many interesting results. In particular, he
introduced the Euler-Kronecker constant γK of a number field K. It generalizes the Euler
√
constant γ and if K = Q( D), then
L0
γK = γ + (1, χD ).
L
Thus, our result restated gives
γK = Ω(log log |D|).
When K = Q, this is just Euler’s constant, and when K = Q(ζm ) a cyclotomic field,
recent result has been obtained by Murty [26], concerning the average of γK . Our result
is in fact consistent with a previous conditional result by Ihara, Murty, and Shimura in
[17]. They proved the following theorem assuming the Generalized Riemann Hypothesis.
Theorem 15. ([17], Theorem 3) Let χ be a non-principal primitive Dirichlet character
of a number field K with conductor fχ . Then
p
log |dK | + log log dχ
∗
|L (1, χ)/L(1, χ)| < 2(log log dχ + 1) − γK + O
.
log dχ
0
∗
Here, dχ = |dK |N (fχ ) and γK
= γK + 1, γK being the Euler-Kronecker invariant of K.
The above theorem implies that under GRH
0
L
(1, χD ) ≤ 2 log log |D| + O(1).
L
Set log1 x = log x , and logk+1 x = log(logk x). Then we prove unconditionally that
Chapter 4. Omega Theorems for L0 /L(1, χD )
29
Theorem 16. There are infinitely many fundamental discriminants D (both positive and
negative) such that
L0
(1, χD ) ≥ log2 |D| + O(1).
L
1
Furthermore, for x large enough, there are ≥ x 10 fundamental discriminants 0 < D ≤ x
such that
L0
(1, χD ) ≤ − log2 D + O(1).
L
Remark 3. In fact, the bound obtained in the proof of the second part of Theorem 16 is
− log2 x + O(1) which is stronger than what is stated.
The key point in proving Theorem 16 is the use of the explicit formula from [17], p.261:
L0
1 X yρ − 1
− (1, χ) = Φ(y) −
+O
L
y − 1 ρ ρ(1 − ρ)
log y
y
(4.1.1)
where y > 1, the sum is taken over all non-trivial zeros ρ of L(s, χ), and
Φ(y) =
1 X y
( − 1)Λ(m)χ(m).
y − 1 m<y m
As a motivation for where this formula comes from, we recall that
∞
X
Λ(m)χ(m)
L0
− (s, χ) =
, <s > 1.
s
L
m
m=1
One would expect that
P
Λ(m)χ(m)/ms is an approximation for −L0 /L(s, χ). We may
m<y
smoothen it by integrating
1
y−1
Z
1
y
X Λ(m)χ(m)
m<t
ms
dt.
When s = 1 this is equal to Φ(y).
The first part of Theorem 16 is proved by averaging over a certain special set of
integers. The second part of Theorem 16 together with the following stronger conditional
result are motivated by a result of Granville-Soundararajan on L(1, χD ) [6].
Chapter 4. Omega Theorems for L0 /L(1, χD )
30
1
Theorem 17. Assume GRH. For x large enough, there are x 2 primes q ≤ x such
that
L0
(1, χq ) ≥ log2 x + log3 x + O(1)
L
1
and x 2 primes q ≤ x such that
L0
(1, χq ) ≤ − log2 x − log3 x + O(1).
L
In [6], Granville and Soundararajan further provided a probabilistic model for the value
distribution of L(1, χD ) giving uniform results. We did investigate their method, and it
does not seem so far that one can construct a similar model for the value distribution of
L0 /L(1, χD ) following the same procedure. The main trouble comes from the fact that
we cannot compute the complex powers of L0 /L(1, χD ), without lots of loss of generality,
since
(
L0
L0
(1, χD ))z := exp(z log( (1, χD ))).
L
L
However for this to be defined, we need to investigate when is
L0
(1, χD ) 6= 0.
L
and this is not as direct as the fact that for all characters χ one has
L(1, χ) 6= 0.
In contrast with the omega results of Theorem 16 and Theorem 17, we will show unconditionally in Theorem 18 that the moments of L0 /L(1, χD ) are constants.
Theorem 18. For Y > 1, k ≥ 1 and β = ±1, we have
k
X ∗ L0
− (1, χD ) = Ck Y + Ok (Y
L
0<βD≤Y
where
∞
3 X Λk (n2 ) Y
1
Ck = 2
(1 + )−1 .
2
π n=1 n
p
p|n
5
+
6
)
Chapter 4. Omega Theorems for L0 /L(1, χD )
31
Here the asterisk on the sum indicates that we are summing over fundamental discriminants
0 < βD ≤ Y , and the functions Λk will be defined in Section 4.4 of this chapter.
The moments of L0 /L(1, χ), where χ is a non-principal multiplicative character to a prime
modulus m have been computed unconditionally by Ihara, Murty and Shimura in [17],
where they showed that
Theorem 19. ([17], Theorem 5) We have, unconditionally,
1 X (a,b) 0
P
(L (1, χ)/L(1, χ)) = (−1)a+b µ(a,b) + O(m−1 )
|Xm | χ∈X
m
for any > 0. In particular the limit formula
1 X (a,b) 0
P
(L (1, χ)/L(1, χ)) = (−1)a+b µ(a,b)
m→∞ |Xm |
χ∈X
lim
m
±
.
holds unconditionally. The same remain valid if Xm is replaced by Xm
+
Here Xm denotes the set of all non-principal multiplicative characters χ mod m, Xm
−
(resp. Xm
) the subset of Xm consisting of even (resp. odd) characters, P (a,b) (z) := z a z b
and
µ(a,b) =
∞
X
Λa (n)Λb (n)
n=1
n2
.
Notice that as we are working here with real Dirichlet characters, the moments are reduced
to powers of L0 /L(1, χD ). We note that our method follows closely the argument in [17]
pages 267-272.
Putting our results together, it seems evident enough that one might expect the following.
Conjecture. For β = ±1,
L0
(1, χD ) = log2 x + log3 x + O(1)
x≤βD≤2x L
max
and
L0
min
(1, χD ) = − log2 x − log3 x + O(1)
x≤βD≤2x L
where the min and max are taken over fundamental discriminants D of the appropriate
sign.
Chapter 4. Omega Theorems for L0 /L(1, χD )
4.2
32
Unconditional bounds
In this section we prove Theorem 16.
4.2.1
Choice of Parameters
Let x be a sufficiently large positive integer, k a positive integer, η a suitably small
positive real number, and let
log x
.
g= η
log log x
Denote by pi the ith odd prime with p1 = 3. Set
a = p1 p2 ...pg .
Then by the Prime Number Theorem, for any > 0, we have:
a = eθ(pg ) ≤ e(1+)g log g xη ,
where θ(y) :=
P
log p. Choose a positive integer b so that
p≤y
b
= −1
pi
for
1≤i≤g
and
b≡1
mod 8.
Notice that the existence of such b is guaranteed by the Chinese Remainder Theorem. we
may assume 1 ≤ b < 8a.
n
The Jacobi symbol ( m
) is defined to be zero if m is even or (m, n) > 1, and as a product
of Legendre symbols corresponding to the prime factors of m elsewhere:
n
m
=
n
q1
α1 n
q2
α2 αr
n
...
qr
where m = q1α1 q2α2 ...qrαr .
If 8an + b is square free, it is a fundamental discriminant (as b ≡ 1 mod 4). In this case,
set
χn (m) =
m
.
8an + b
Chapter 4. Omega Theorems for L0 /L(1, χD )
33
For m odd, we have
χn (m) =
8an + b
m
,
by the Jacobi Reciprocity Law. Now consider the Dirichlet series
∞
X
L0
χn (m)Λ(m)
− (s, χn ) =
, <s > 1
s
L
m
m=1
where L(s, χn ) is the Dirichlet L-function corresponding to the character χn .
4.2.2
Application of the Explicit Formula
We will need the following lemma:
Lemma 3. Let Φ(y) be as in (4.1.1), then
Φ(y) =
X χ(p) log p
p
p<y
+ O(1)
Proof. Write
Φ(y) =
X
1
+
X
2
where
j
log 2 X y
2
=
( j − 1)
,
2
y−1
2
8an
+
b
m=2j <y
P
P
and 1 is the sum over odd m < y. We see that 2 = O(1). Now, write
X
X
1
where in
P
A,
=
X
A
+
X
m ranges over primes p < y, and in
B
P
,
B
m ranges over prime powers. We
have
X
B
X log p
X log y log p
1 X
y
log p( j − 1) +
1.
y − 1 j≥2
p
p2
√
√ log p y
p< y
pj <y
This proves the lemma.
p< y
Chapter 4. Omega Theorems for L0 /L(1, χD )
34
Let y > pg be a real number to be chosen later, choose χ = χn and write
X χn (p) log p
p
p<y
=
X
11
+
X
12
P
p ranges over primes p ≤ pg and in 12 p ranges over primes pg < p < y.
Since p|a and pb = −1 for all p ≤ pg , the contribution of primes p ≤ pg is
where in
P
11 ,
X
11
=−
1 X y
( − 1) log p.
y − 1 p≤p p
g
In turn this is equal to ([4], p.57, and (1), p.111) :
X
1
log p X
=
−
log p
y
y−1
p
p≤pg
p≤pg
X
1
=
y(log pg + O(1)) − Ψ(pg ) +
log p
y−1
j
p ≤pg
j≥2
X
1
=
y (log pg + O(1)) − Ψ(pg ) +
log p
y−1
√
p≤ p
j≤
g
log pg
log p
1
pg
=
y(log pg + O(1)) − pg + O
y−1
log pg
= log pg + O(1).
As it follows from the prime number theorem and the fact that pg < y. Summing this
over n ∈ (x, 2x], with 8an + b square free, it becomes
X0 X
11
x<n≤2x
= −N log pg + O(N ),
(4.2.1)
where N = #{n ∈ (x, 2x], such that 8an + b is square free}. Then one can show easily
that N x i.e it is bounded from above and below by a constant multiple of x ([3],
Lemma 3 in Collected Papers). As for primes pg < p < y we have to estimate
X0 X
x<n≤2x
12
X y
X0 8an + b 1
=
( − 1) log p
.
y − 1 p <p<y p
p
x<n≤2x
g
(4.2.2)
Chapter 4. Omega Theorems for L0 /L(1, χD )
35
We have
X
X
X
X m
1
m
=
χ(b)χ(m)
µ(d)
p
φ(8a) χ mod 8a z ≤m≤z
p
2
m≡b(8a)
z1 ≤m≤z2
m sqfree
1
(4.2.3)
d |m
2
µ being the Mobius function, and φ the Euler’s totient function.
Writing m = d2 u in the sum, we find that the right hand side of (4.2.3) is
X
√
1≤d≤ z2
(d,8ap)=1
X
X
u
1
2
χ(d )µ(d)χ(b)
χ(u)
.
φ(8a) χ mod 8a
p
z1
z2
d2
≤u≤
d2
The inner sum is
p
8ap log (8ap),
by the Polya Vinogradov Theorem ([4], p.135), which assumes that χ(.)
.
p
6= 1, where
χ is a character mod 8a, and (8a, p) = 1.
Remark 4. Notice that 8a = 8
Y
pi , and p > pg , therefore χ(.) 6=
.
p
, as they have
pi ≤p
g
.
different conductors. Thus χ(.) p 6= 1.
Hence the entire quantity is
√ p
z2 8ap log (8ap),
If z1 = 8ax + b, and z2 = 16ax + b, this can be rewritten as
√
a px log (8ap).
Hence, inserting this into (4.2.2), we find that the contribution of primes pg < p < y is
Chapter 4. Omega Theorems for L0 /L(1, χD )
36
y
√
− 1 (log p)a px log (8ap)
p
g
X
√
log p √
a x(log (xy))
p
p
pg <p<y
√
a xy log (xy).
X
1
y − 1 p <p<y
(4.2.4)
(4.2.5)
(4.2.6)
Hence, putting together (4.2.1), and (4.2.6), we get
√
Φn (y) = −N log pg + O(N ) + O(a xy log (xy)).
X0
(4.2.7)
x<n≤2x
Here Φn is defined as in 4.1.1, with χ = χn .
4.2.3
The sum over zeros
Next, we consider the contribution of the sum over zeros, we need to estimate
X0 1 X y ρ − 1
log y
+O
.
y
−
1
ρ(1
−
ρ)
y
ρ
x<n≤2x
If we truncate the sum over ρ = β + iγ at |γ| ≤ T , we introduce an error of ([17], 5.4.6,
p.269)
X0 log (qT ) (log q)2 +
T
y
x<n≤2x
where q = 8an + b. As q ax, this is
x
x
log (axT ) + (log (ax))2 .
T
y
The contribution of non-exceptional zeros with β ≤
X0 1
2
1 (log (axT )) x
x<n≤2x
y6
5
6
is ([17], 5.4.4)
(log (axT ))2
1
y6
.
Chapter 4. Omega Theorems for L0 /L(1, χD )
37
The term ‘exceptional zeros’ refers to those exceptions to Page theorem ([4], p.95),
concerning the set of positive integers q ≤ z, where z ≥ 3: If c is a suitable positive
constant, there is at most one real primitive character χ to a modulus q ≤ z, for which
L(s, χ) has a real zero β satisfying
β >1−
c
.
log z
If such a character exists, it is called ‘exceptional’, and so is called the zero β.
For the zeros with β >
5
6
and γ ≤ T , we separate exceptional and non-exceptional zeros.
For all non-exceptional zeros, we use Jutila’s zero-density estimate [19], Theorem 2(see
below), which implies that for σ ≥ 56 ,
X0
3
N (σ, T, χn ) (axT ) 4 + ,
x<n≤2x
where the sum is over fundamental discriminants, and
N (σ, T, χ) := #{ρ = β + iγ a non-trivial zero of L(s, χ) such that β ≥ σ and |γ| ≤ T }.
Then
X
1
Z
β
y σ dN (σ, T, χn )
y =−
5
6
|γ|≤T
β≥ 65
is
1 Z 1
= −y N (σ, T, χn ) +
N (σ, T, χn )y σ log y dσ.
5
5
σ
6
6
Summing this over n ∈ (x, 2x] such that 8an + b is square free, and noticing that b ≡ 1
mod 4, hence 8an + b is a fundamental discriminant. We get
X0 X
y
β
5
6
y (axT )
3
+
4
+ (axT )
3
+
4
1
Z
y σ log ydσ
(4.2.8)
5
6
x<n≤2x |γ|≤T
β≥ 56
5
3
3
y 6 (axT ) 4 + + (axT ) 4 + y
3
y(axT ) 4 + .
(4.2.9)
(4.2.10)
Chapter 4. Omega Theorems for L0 /L(1, χD )
38
We use this to estimate
X0 X∗
x<n≤2x |γ|≤T
β≥ 56
yρ
,
ρ(1 − ρ)
(4.2.11)
where the asterisk indicates that we sum only over non-exceptional zeros, again we need
only to consider β ≥ 56 . Thus we have
|ρ(1 − ρ)| ≥ β(1 − β) + γ 2
and
1
c
<β <1−
2
log (axT )
([4],Page Theorem, p.95). Hence, the contribution of zeros with |γ| ≤ 2 to (4.2.11) is by
(4.2.10) :
3
y(log (axT ))(ax) 4 +
and the contribution of zeros with j < |γ| ≤ j + 1 is
3
1
y(axj) 4 + .
2
j
Summing this over j ≥ 1, this is
3
y(ax) 4 + .
Hence
3
1 X∗
yρ
(ax) 4 + log (axT ).
y−1
ρ(1 − ρ)
x<n≤2x
X0
|γ|≤T
β≥ 65
Theorem 20. (Jutila’s zero-density estimate [19], Theorem 2) Let N (α, T, χ) be the
number of zeros (ρ = σ + it) of the function L(s, χ) in the rectangle
α ≤ σ ≤ 1,
|t| ≤ T.
Chapter 4. Omega Theorems for L0 /L(1, χD )
39
Then for 1/2 ≤ α < 1, T ≥ 1, > 0 we have
X∗
N (α, T, χD ) (XT )(7−6α)/(6−4α)+ .
|D|≤X
The asterisk on the sum means that we are summing over fundamental discriminants
|D| ≤ X.
4.2.4
Proof of the Theorem
From the identities (4.1.1), (4.2.7), and the calculations of Subsections 4.2.2 and 4.2.3, we
deduce that :
X0 L0
1
y β0 − 1
N log y
− (1, χn ) +
=O
L
y − 1 β0 (1 − β0 )
y
x<n≤2x
+O (ax)
3
+
4
(log (axT ))2
√
log (axT ) − N log pg + O(N ) + O(a xy log (xy)) + x
.
1
y6
Now, N x, and log pg ∼ log(g log g). Moreover, recall that
P
a=e
p≤pg
log p
≤ e(1+)pg xη .
We choose y so that
1
√
a xy log (xy) = o(N log log x) = o(x log log x) and (log (axT ))2 = o(y 6 ).
Clearly, any y = xα with 0 < α < 1 satisfies this. With such a choice, we get
X 0 L0
y β0 − 1
− (1, χn ) +
= −N log pg + O(x)
L
(y − 1)β0 (1 − β0 )
x<n≤2x
where, β0 > 1 −
1
log (8an+b)
([4], p. 95). Notice that the contribution of the possible
Landau-Siegel zero in this identity is positive, so we can remove it as it points in the right
direction. Then there exists x < n ≤ 2x so that for D = 8an + b
−
L0
(1, χn ) ≤ − log pg + O(1) ≤ − log2 D + O(1).
L
Chapter 4. Omega Theorems for L0 /L(1, χD )
40
Remark 5. In fact, if for some n0 we have a Landau-Siegel zero, then for D = 8an0 + b,
we get
X 1 X
L0
1
β
(1, χn0 ) + log D =
+
+ c0
2 + γ2
L
2
β
β
real zeros
β,γ
where c0 > 0. This can be derived from differentiating the Hadamard product for L(s, χn0 ),
or by referring to (10) of [7]. Hence ,
L0
1
1
1
(1, χn0 ) + log D >
+
> log D.
L
2
β0 1 − β0
Thus,
1
L0
(1, χn0 ) > log D.
L
2
This is one of the few cases where having a Landau-Siegel zero actually gives us a better
result!
Thus, by letting x → ∞, we get infinitely many D > 0 such that:
L0
(1, χD ) ≥ log2 D + O(1).
L
In the above argument, if we choose n ∈ (−2x, −x) we get D < 0 satisfying a similar
bound. Thus, there are infinitely many fundamental discriminants D (both positive and
negative) so that
L0
(1, χD ) ≥ log2 |D| + O(1).
L
This proves the first part of Theorem 1. We now prove the second part, for which we will
need the following lemma.
Lemma 4. If χq =
.
q
−
is not a Landau-Siegel character and x ≥ q, then
L0
(1, χq ) =
L
X
p≤exp((log x)3 )
χq (p) log p
+ O(1).
p
Chapter 4. Omega Theorems for L0 /L(1, χD )
41
Proof. This is simply the identity (4.1.1) and Lemma 3, together with the fact that if we
only look at those q ≤ x which are not Landau-Siegel moduli, then these satisfy
c
1 X y 1− log x
1 X yρ − 1
,
y − 1 ρ ρ(1 − ρ)
y − 1 ρ ρ(1 − ρ)
(4.2.12)
for some suitable constant c > 0, so if we take y large enough (eg. y = exp ((log x)3 ) then
(4.2.12) is 1, and the lemma is proved.
Let x be large, and L = [c1 log2 x], z = c2 log x, for some suitably chosen positive
constants c1 and c2 . Then
1
Lemma 5. ([6], Lemma 10.1) There are at least x 3L pairwise coprime integers of the
1
form pq with p and q primes below x 3L such that ( pql ) = 1 for all primes l ≤ z.
Now using this result and adapting Lemma 10.2 from [6] we prove the following
Lemma 6. For x large, and L = [c1 log2 x], z = c2 log x, with c2 as above and c1 large
enough. Let d1 , ...dL be any L numbers constructed as in Lemma 5. Then there exists a
square free integer D ≤ x which is the product of at least L/3 of these di ’s such that
X
l
1
χD (p) log p
= 1 for all primes l ≤ z; and
≥ −c3
D
p
log x
3
z≤p≤exp((log x) )
for some positive constant c3 .
We note that we may from the beginning make the choices of c1 and c2 in such a way
they satisfy the requirements of Lemmas 5 and 6.
Proof. Let D be a product of distinct di and let ν(D) denote the number of di ’s involved
in this product. Notice that
L
X
z≤p≤exp((log x)3 )
log p Y
(1 +
p i=1
p
) ≥ 0.
di
Thus
X
X
D
z≤p≤exp((log x)3 )
ν(D)≥L/3
χD (p) log p
p
Chapter 4. Omega Theorems for L0 /L(1, χD )
≥−
42
X
X
D
z≤p≤exp((log x)3 )
ν(D)≤L/3
≥ −(log x)3
X
log p
p
1.
D
ν(D)≤L/3
However #{D ; ν(D) ≥ L/3} ≤ #{D} = 2L , and #{D ; ν(D) ≤ L/3} =
L
j
P
j≤L/3
≤
(1.9)L using Stirling’s formula. Then there exists D such that ν(D) ≥ L/3, and
X
z≤p≤exp((log x)3 )
1
χD (p) log p
≥ −(log x)3 (0.95)L ≥ −c3
.
p
log x
2
Now Lemma 10.3 from [6] implies that there are ≥ x 19 integers D as in the above
lemma. Since the number of moduli D ≤ x whose L-function has a Landau-Siegel zero is
O(log log x) (see, for example, Subsection 4.4.1 below), we deduce, using Lemma 4, that
for the remaining moduli thus constructed, we have
−
X log p
L0
c3
(1, χD ) ≥
−
+ O(1).
L
p
log x
p≤z
Thus,
−
L0
(1, χD ) ≥ log2 x + O(1).
L
This completes the proof of Theorem 16.
4.3
Conditional bounds
Proposition 7. Assume GRH. Let z ≥ 2 be a real number, and let P (z) =
Q
p = ez+o(z) .
p≤z
Let p = ±1 for each prime p ≤ z. Denote by P the vector (p )p≤z and let P (x, P ) denote
1
the set of primes q ≤ x such that pq = p for each p ≤ z. Then for z x 2 , we have
X
q∈P (x,P )
log q =
x
2π(z)
1
2
2
+ O x {log(xP (z))} ,
(4.3.1)
and
−
X
q∈P (x,P )
1
L0
x X p log p
x
(1, χq ) log q = π(z)
+ O(x 2 {log(xP (z))}3 ) + O( π(z) ). (4.3.2)
L
2
p
2
p≤z
Chapter 4. Omega Theorems for L0 /L(1, χD )
43
Proof. As in [6], the proof of (4.3.1) is similar to and easier than (4.3.2). For l|P (z), let
Q
l = p . Notice that for all q ≤ x we have
p|l



1 if q ∈ P (x, P )
l
l
=

q

l|P (z)
0 otherwise .
1
X
2π(z)
(4.3.3)
Now, using (4.1.1), Lemma 3, taking y = x2 P 2 (z) and using (4.3.3) we get under GRH
−
X
q∈P (x,P )
L0
(1, χq ) log q =
L
X
q∈P (x,P )
x log x
Φ(y) log q + O( √ )
y
which is equal to
X
(log q){
1 X
2π(z)
p
p<y
q∈P (x,P )
=
X χq (p) log p
log q
X log p X
p
p<y
q≤x
x log x
+ O(1)} + O( √ )
y
l χq (lp) + O(
l|P (z)
x
2π(z)
x log x
) + O( √ )
y
we then get
−
X
q∈P (x,P )
1 X X log p X
x
L0
(1, χq ) log q = π(z)
l
χq (lp) log q + O( π(z) )
L
2
p q<x
2
p<y
(4.3.4)
l|P (z)
If lp is a square then under GRH the inner sum over q in the RHS of (4.3.4) is
X
1
χq (lp) log q = x + O(x 2 (log x)2 ).
q≤x
Moreover since l|P (z), it is square free and we can write l = p in this case; the contribution
of the squares to the sum in the RHS of (4.3.4) is then
1
X
2π(z)
l|P (z)
=
l (
X
p<x2 (P (z))2
1
log p
)(x + O(x 2 (log x)2 ))
p
X log p
1
(
p
)(x + O(x 2 (log x)2 ))
π(z)
2
p
p≤z
=
1
1
x X p log p
+ O(x 2 (log x)2 ).
π(z)
2
p
p≤z
Chapter 4. Omega Theorems for L0 /L(1, χD )
44
If lp is not a square, in other terms if l 6= p, then the inner sum over q in the RHS of
(4.3.4) satisfies under GRH
X
1
χq (lp) log q x 2 (log(xP (z)))2 ,
q≤x
since χq (lp) =
lp
q
, and
.
lp
is a character of conductor lp or 4lp. Thus the contribution
of these non square terms to (4.3.4) is
1
1
x 2 (log(xP (z)))2
2π(z)
X
(
X
l|P (z) p≤x2 (P (z))2
log p
+ O(1))
p
1
x 2 (log(xP (z))3 .
We are now ready to prove Theorem 17.
4.3.1
Proof of Theorem 17
1
Let z be such that 2π(z) ≤ x 2 −3 , then using (4.3.1), (4.3.2) from Proposition 7, and the
fact that L0 /L(1, χq )| ≤ 2 log q, we get
X
−
q∈P (x,P )
X L0
X L0
L0
(1, χq ) log q =
− (1, χq ) +
− (1, χq )
L
L
L
q∈M
q ∈M
/
where
M := {q ∈ P (x, P ) , such that −
X p log p
L0
(1, χq ) ≥
− C}.
L
p
p≤z
This gives in turn that
−
X
q∈P (x,P )
X X p log p
X
L0
2
(1, χq ) log q ≤
2(log q) +
− C log q
L
p
p∈z
q∈M
q ∈M
/
and this in turn is
2
≤ |M |(log x) +
X X p log p
q∈P (x,P )
p∈z
p
− C log q.
Now using 4.3.1 and 4.3.2 we get
X
1
x X p log p
x
3
2
2 {log(xP (z))} )+O(
+O(x
)
≤
|M
|(log
x)
+(
2π(z) p≤z
p
2π(z)
q∈P (x,P )
log q)(
p log p
−C)
p
Chapter 4. Omega Theorems for L0 /L(1, χD )
45
Then
1
|M | ≥
Cx
x2
x
3
+
O(
{log(xP
(z))}
)
+
O(
).
2(log x)2 2π(z)
(log x)2
2π(z) (log x)2
We conclude that for a suitably large constant C > 0, there are
≥
Cx
2(log x)2 2π(z)
values of q ∈ P (x, P ) such that
−
X p log p
L0
(1, χq ) ≥
− C.
L
p
p≤z
Here we take z = δ log x log2 x, for some suitably small positive constant δ, and p = 1 for
1
all p ≤ z, to deduce that there are x 2 primes q ≤ x such that
L0
(1, χq ) ≤ − log2 x − log3 x + O(1).
L
This proves the first part of the theorem. As for the second part, take p = −1 for all
p ≤ z. Then with a few obvious modifications to the above argument, we get the result.
4.4
Moments of L0/L(1, χD )
We need to get an asymptotic formula for the moments
k
X ∗ L0
− (1, χD ) ;
L
0<βD≤Y
β = ±1, Y > 1 and k ≥ 1.
Here the asterisk on the sum indicates that we are summing over fundamental discriminants 0 < βD ≤ Y . To prove Theorem 18, we will need to estimate the above sum
through several steps.
4.4.1
Exceptional Zeros
Crucial to our discussion will be the treatment of possible zeros of L(s, χ) near s = 1
(or equivalently near s = 0, by the functional equation). By a theorem of Landau ([4],
Chapter 4. Omega Theorems for L0 /L(1, χD )
46
p.93-94), there is a positive absolute constant µ0 such that if χ1 and χ2 are distinct
real primitive characters to the moduli q1 and q2 respectively, and if the corresponding
L-functions have real zeros β1 and β2 , then
min(β1 , β2 ) < 1 −
µ0
.
log(q1 q2 )
Now let us set µ1 = µ0 /3 and consider the sequence of discriminants D1 , D2 , ... up to Y
(we can assume they are all positive) for which the corresponding L-functions have real
zeros β1 , β2 , ... satisfying
βi > 1 −
µ1
.
log Di
We define an exceptional zero to be as above. Then, we have
1−
µ0
µ1
< min(βi , βi+1 ) < 1 −
.
log Di
log(Di Di+1 )
This implies that
Di+1 > Di2 .
Hence, the number of Di ≤ Y is O(log log Y ), and with possibly one exception, for each
of them we have
1
log Y.
βi (1 − βi )
To see this, apply Landau’s theorem with one of the moduli being the Di that has the
largest real zero. The one exception is handled by Siegel’s theorem ([4], chapter 21). So
all of this gives a contribution of
X∗
X
0<βD≤Y exceptional ρD
1
ρD (ρD − 1)
(log Y )(log log Y ) + Y for the exceptional zeros. Now for the remaining zeros, we can assume all of them satisfy
Chapter 4. Omega Theorems for L0 /L(1, χD )
Re(ρ) ≤ 1 −
4.4.2
47
µ1
.
log D
The main term
We refer to the identity (4.1.1), with χ = χD . We define, for each integer k ≥ 0, the
arithmetic functions
Λ0 (n) =



1 (n = 1)


0 (n > 1) ,
Λk (n) =
X
Λ(n1 )...Λ(nk ) (for k > 0).
n=n1 ...nk
Then Λk (n) = 0 unless the number of prime factors of n is at most k and the sum of
exponents in the prime factorization of n is at least k, furthermore
r
Λk (p ) =
r−1
(log p)k , if 1 ≤ k ≤ r
k−1
and
Λk (n) ≤ (log n)k .
We will show that only
P
ΦχD (x) will contribute to the main term in the asymptotic
D
formula for the moments of L0 /L(1, χD ). Recall that a version of the Mean Value Theorem
for polynomials in two variables gives
|(z + ω)k − z k | ≤ k|ω|(|z| + |ω|)k−1 ,
where z, ω ∈ C. In our case, take z = −L0 /L(1, χD ), and ω = L0 /L(1, χD ) + ΦχD (x), to
get
0
0
0
k−1
0
(Φχ (x))k −(− L (1, χD )k ) ≤ k L (1, χD )+Φχ (x) L (1, χD )+ L (1, χD )+Φχ (x)
.
D
D
D
L
L
L
L
Chapter 4. Omega Theorems for L0 /L(1, χD )
48
But Lemma 2 from [17], p.268 gives for x ≥ D



D 0
0
L
(1, χD ) + L (1, χD ) + Φχ (x) D
L
L


(log D)2
for χD = χ1
for χD 6= χ1 .
Here χ1 is the possible exceptional character to Page theorem ([4] p.95). Thus



Dk
0
L
k
k
(Φχ (x)) − (− (1, χD ) ) k
D
L0

L

2(k−1)
(log D)
L (1, χD ) + ΦχD (x)
for χD = χ1
for χD 6= χ1 .
Furthermore, by (4.1.1) we have
X X xρ − 1
X∗ L0
1
N
(Y
)
log
x
(1, χD ) + Φχ (x) +O
.
D
L
x − 1 0<βD≤Y ρ ρ(1 − ρ)
x
0<βD≤Y
Let us now estimate
X∗
k
(ΦχD (x)) =
k
X
1
x
( − 1)Λ(n)χD (n)
(x − 1)k n<x n
X∗
0<βD≤Y
0<βD≤Y
X
X∗ D−1
1
=
(x − 1)k0<βD≤Y c=1
X
(
n1 ,...,nk <x
n1 ...nk ≡c mod D
x
x
− 1)...( − 1)Λ(n1 )...Λ(nk )χD (c).
n1
nk
Now let
λ(a) (c, x) =
1
(x − 1)a
X
n1 ,...,na <x
n1 ...na ≡c mod D
a
Y
x
( − 1)Λ(ni ),
ni
i=1
for a ≥ 1, and λ(0) (c, x) = 1, for c = 1, 0 for c > 1.
Similarly, let
1
L (N, x) =
(x − 1)a
(a)
a
Y
x
( − 1)Λ(ni ),
ni
n1 ,...,na <x i=1
X
n1 ...na =N
for a, N ≥ 1, and L(0) (N, x) = 1, for N = 1, 0 for N > 1. Then
[
λ(a) (c, x) =
(xa −c)
]
D
X
l=0
L(a) (c + lD, x).
(4.4.1)
Chapter 4. Omega Theorems for L0 /L(1, χD )
49
However, we will show that the terms with l > 0 are altogether negligible. Note that
L(a) (N, x) = 0 if N ≥ xa , and elsewhere
L(a) (N, x) ≤
1
N
X
Λ(n1 )...Λ(na ) ≤
n1 ,...,na <x
n1 ...na =N
1
Λa (N ).
N
Hence
(log x)a
(log N )a
< aa
.
N
N
a+1
Therefore the sum of terms with l > 0 in (4.4.1) is Oa (log x)
, which implies
L(a) (N, x) ≤
a+1
λ (c, x) = L (c, x) + Oa (log x)
,
(a)
(a)
and since λ(0) (c, x) = L(0) (c, x), the exponent of log x can be replaced by 0 if a = 0. But
L(a) (c, x) =
Λa (c)
(log D)a
+ O(
),
c
x
for x ≥ D as in [17], (4.2.6), p.265. Then we get
X∗
0<βD≤Y
X∗ D−1
X Λk (c)
N (Y )(log x)k
k+1
Φ (x) =
χD (c) + Ok ((log x) ) + Ok
.
c
x
0<βD≤Y c=1
(k)
However, exchanging the sum over c and the sum over D, this is equal to
Y
−1
X
c=1
Λk (c) X∗
N (Y )(log x)k
k+1
χD (c) + Ok ((log x) ) + Ok
,
c 0<βD≤Y
x
because the sum over l > 0 in (4.4.1) is negligible, so there is no harm in adding the sum
of these additional terms to our main sum, since its contribution will be absorbed into
the error term. It turns out that we only need to estimate the sum
Y
−1
X
c=1
Λk (c) X∗
χD (c),
c 0<βD≤Y
which is equal to
Y
−1
X
c=1
Λk (c)
fY (c).
c
(4.4.2)
Chapter 4. Omega Theorems for L0 /L(1, χD )
Here fY (n) =
X∗
50
χD (n). If n is a square, we have
0<βD≤Y
1
fY (n) = un Y + O(Y 2 d(n)),
d being the divisor function, and un =
3
π2
Q
(1 + p1 )−1 ([27], p.106). The contribution of
p|n
squares to (4.4.2) is then
√
[ Y −1]
X Λk (c2 )
1
(uc Y + O(Y 2 d(c))),
2
c
c=1
which in turn is equal to
1
Ck Y + O(Y 2 ).
Here Ck is a constant depending only on k, as in the statement of Theorem 18. As for
the contribution of non-squares we have by Cauchy-Schwartz inequality
Y −1
X0
c=1
Λk (c)
fY (c) c
YX
−1
0
c=1
|fY (c)|2
c
12 YX
−1
0
c=1
(Λk (c))2
c
12
,
where the prime on the summation indicates that we are summing over non-square c’s.
However, by (4.4.1) of [27] which states that
X0
|fY (n)|2 M Y (log M )4
1<n≤M
we conclude that the above is
Y
4.4.3
3
+
4
.
The sum over zeros
we separate exceptional and non-exceptional zeros. For all non-exceptional zeros, we use
Jutila’s zero-density estimate [19](see Theorem 20, Chapter 4) which implies that for
σ ≥ 56 ,
X∗
0<βD≤Y
7−6σ
N (σ, T, χD ) (Y T ) 6−4σ + ,
Chapter 4. Omega Theorems for L0 /L(1, χD )
51
where the sum is over fundamental discriminants, and > 0. Then
X
x
<ρ
1
Z
xσ dN (σ, T, χD ).
=−
5
6
|=ρ|≤T
<ρ≥ 56
This is
1 Z
= −x N (σ, T, χD ) +
1
σ
5
6
N (σ, T, χD )xσ log x dσ.
5
6
Summing this over all fundamental discriminants 0 < βD ≤ Y , we get
X∗ X
x
<ρ
5
6
x (Y T )
3
+
4
1
Z
7−6σ
xσ log x(Y T ) 6−4σ + dσ,
+
5
6
0<βD≤Y |σ|≤T
<ρ≥ 65
and this is
5
6
x (Y T )
3
+
4
+ (Y T )
3
+
4
1
Z
xσ log xdσ
5
6
5
3
3
x 6 (Y T ) 4 + + x(Y T ) 4 +
3
x(Y T ) 4 + .
We use this to estimate
X∗ X00
1
xρ
x − 10<βD<≤Y
ρ(1 − ρ)
|γ|≤T
where the double prime indicates that we sum only over non-exceptional zeros, again we
need only to concider <ρ ≥ 56 , γ being =ρ. We have
1
µ1
< <ρ < 1 −
2
log (Y T )
by Subsection 4.4.1. Hence,
X∗ X00
1
xρ
log(Y T ) X∗ X00 <ρ
x
x − 10<βD≤Y
ρ(1 − ρ)
x 0<βD≤Y
|γ|≤T
<ρ≥ 56
|γ|≤T
<ρ≥ 56
Chapter 4. Omega Theorems for L0 /L(1, χD )
52
3
(Y T ) 4 + log (Y T ).
As for the contribution of exceptional zeros, we refer to previous argument which says
that the total contribution of such zeros is:
X∗
0<βD≤Y
1
log Y log log Y + Y .
β1,D (1 − β1,D )
Now, gathering all error terms, we see that if x ≥ D for all 0 < βD ≤ Y , then
k
X∗ L0
− (1, χD ) = Ck Y + O(Y
L
0<βD≤Y
3
+
4
1
) + O((log x)k+1 ) + O(Y 2 )+
3
N (Y )(log x)k
)+O((log Y )2(k−1) (Y T ) 4 + log(Y T ))+O((log Y )2(k−1) (log Y log log Y +Y ))+
O(
x
log x
N (Y ) log Y T
N (Y )(log Y )2
O((log Y )2(k−1) N (Y )
)+O((log Y )2(k−1)
)+O((log Y )2(k−1) (
)).
x
x−1
T
Here the last two error terms come from
X∗
1
x − 10<βD<≤Y
X
non-exceptional
1
ρ(1 − ρ)
ρ
and
X∗ X
1
xρ
x − 10<βD<≤Y
ρ(1 − ρ)
|γ|>T
by using (5.4.4) and (5.4.6) from [17], which state that
X
non-exceptional
1
(log D)2
ρ(1 − ρ)
ρ
and
X
|γ|>T
We choose x = Y, and T = Y
1
9
xρ
x log(DT )
.
ρ(1 − ρ)
T
to finish the proof of Theroem 3.
Chapter 5
On Distribution functions
5.1
Introduction
In this chapter we study Bernoulli Distributions and their connection to the distribution
of the logarithmic derivatives of real Dirichlet L-Function, L0 /L(1, χD ), where χD is a
fundamental discriminant. It is based on previous papers of B. Jessen, and A. Wintner
[18], [38], and [39]. However we will consider a less general overview of their results given
the real situation we are working in.
5.2
Distribution Functions
Let R be the real line with x as variable point.
Definition 5. A distribution function in R is a completely additive(also known as countably additive), non-negative set function φ(E) defined for all Borel sets E in R with value
1 for E = R.
Notation. An integral with respect to φ is denoted by
Z
f (x)φ(dx)
E
53
54
Chapter 5. On Distribution functions
where the integral is to be understood in the Lebesgue-Radon (or Lebesgue-Stieltjes)
sense.
Remark 6. The Lebesgue-Stieltjes measure, also known as Lebesgue-Radon measure, is
the ordinary Lebesgue integral with respect to a measure associated with a function of
bounded variation on R. For more details on the definition see [1].
The notation for ordinary Lebesgue integral will be
Z
f (x)dx.
E
Definition 6. A continuity set E of φ is a set satisfying
φ(E 0 ) = φ(E 00 )
where E 0 is the set of all interior points of E, and E 00 is the closure of E.
Definition 7. A sequence of distribution functions φn is said to be convergent if there
exists a distribution function φ such that
φn (E) −→ φ(E)
for all continuity sets E of the limit function φ, which is unique in this case, and the
symbol
φn −→ φ
is justified only in this situation.
Proposition 8. The sequence of distribution functions φn is convergent to φ if and only
if
Z
∞
Z
∞
f (x)φn (dx) −→
−∞
f (x)φ(dx)
−∞
for all bounded continuous functions f (x) in R. Furthermore, if φn −→ φ then
Z ∞
Z ∞
f (x)φ(dx) ≤ lim inf
f (x)φn (dx)
−∞
−∞
holds for every non-negative, continuous function f (x) in R.
55
Chapter 5. On Distribution functions
Definition 8. If φ1 and φ2 are two distribution functions, then we can define their
convolution φ as follows
Z
∞
φ(E) = φ1 ∗ φ2 (E) :=
φ1 (E − x)φ2 (dx)
−∞
for every Borel set E. Here E − x denotes the set obtained from E after translation −x.
Definition 9. The spectrum S = S(φ) of a distribution function φ is the set of points x
in R for which φ(E) > 0 for any set containing x as an interior point. We note that S is
always a non-empty closed set.
Definition 10. The point spectrum P = P (φ) is the set of points x for which φ(x) > 0.
Here φ(x) := φ({x}), P is countable, and may be empty.
Using these notations we have
S(φ1 ∗ φ2 ) = S(φ1 ) + S(φ2 ),
and
P (φ1 ∗ φ2 ) = P (φ1 ) + P (φ2 ).
Definition 11. A distribution function will be called continuous if P (φ) is empty, and
discontinuous otherwise. It is said to be purely discontinuous if
φ(P (φ)) = 1.
φ is called singular if it is continuous and there is a Borel set E of measure zero for which
φ(E) = 1, and is called absolutely continuous if φ(E) = 0 for all Borel sets of measure
zero.
Proposition 9. ([18]) The distribution φ is absolutely continuous if and only if there
exists in R a Lebesgue integrable point function D(x) such that
Z
φ(E) =
D(x)dx
E
for any Borel set E. We call D(x) the density of φ.
56
Chapter 5. On Distribution functions
5.3
Fourier transforms
If φ is a distribution function in R, then we define its Fourier transform by
Z
∞
exp(ixy)φ(dx).
b :=
Λ(y, φ) := φ(y)
−∞
5.3.1
Properties of Fourier Transforms
Let φ, ψ and τ be distribution functions in R, then we have the following properties
1. φb is a uniformly continuous and bounded function.
b = 1.
2. The maximum value of φb is equal to φ(0)
b ≡ ψ(y)
b
3. If φ(y)
for all y, then φ = ψ.
4. The correspondence between distribution functions and their Fourier transforms is
a one-to-one correspondence.
b
5. Multiplication Rule: φ[
∗ ψ = φbψ.
6. Commutativity: φ ∗ ψ = ψ ∗ φ.
7. Associativity: φ ∗ (ψ ∗ τ ) = (φ ∗ ψ) ∗ τ .
8. δbc = exp(icy) and δb ≡ 1. Here δc (x) = δ(x − c).
9. If ψ(E) := φ(E − c) then ψ = φ ∗ δc , in particular φ = φ ∗ δ for every φ.
10. If φn −→ φ, and φn ∗ ψn −→ φ, then ψn −→ δ.
cn (y) −→ φ(y)
b
Proposition 10. If φn −→ φ then φ
in all intervals |y| ≤ a. Conversely,
cn is uniformly convergent in every interval |y| ≤ a,
if a sequence of Fourier transforms φ
then the limit function is also the Fourier transforms φb of a distribution function φ and
φn −→ φ.
57
Chapter 5. On Distribution functions
Corollary 2. The one-to-one correspondence between all distributions and their Fourier
transforms is in fact a continuous correspondence.
Proposition 11. If the integral
Z
∞
b
|y p |φ(y)dy
−∞
converges for some integer p ≥ 0, then φ is absolutely continuous and its density D(x)
can be determined by the inversion formula:
Z ∞
−1
b
D(x) = (2π)
exp(−ixy)φ(y)dy.
−∞
In this case, D(x) is continuous and converges to zero as |x| −→ ∞. When p > 0, it has
derivatives of order ≤ p. This holds in particular if there exists > 0 such that
b = O(|y|−(k+p+) ), for y large enough.
φ(y)
Lemma 7. (Riemann-Lebesgue Lemma) A necessary condition for the absolute continuity
of φ is that
b −→ 0
φ(y)
as |y| −→ ∞.
Corollary 3. If
b = O(e−A|y| )
φ(y)
as |y| −→ ∞
for some A > 0, then D(x) is regular analytic in R.
Proposition 12. If there exists an integer r > 0 such that
Z ∞
Mr (φ) :=
|x|r φ(dx)
−∞
b
is finite, then φ(y)
has continuous derivatives of order s ≤ r, which may be obtained from
b
the formula defining φ(y)
under the integral sign. These derivatives become at y = 0 the
moments
Z
∞
µs (φ) =
−∞
of φ of order s ≤ r multiplied by the factor is .
xs φ(dx)
Chapter 5. On Distribution functions
5.3.2
58
Some properties of Mr
1. The existence of Mr (φ) for one value of r implies its existence for smaller values of
r.
2. If Mr (φ1 ) and Mr (φ2 ) are finite and φ = φ1 ∗ φ2 , then Mr (φ) is also finite.
3. If M1 (φ) is finite, let c = c(φ) be the moment of first order, in other terms, c is
the center of gravity of the mass distribution determined by φ. Denote by φe the
distribution function defined by
e
φ(E)
= φ(E + c(φ)).
Then
e = 0 and Λ(y, φ) = eic(φ) Λ(y, φ).
e
c(φ)
e = S(φ) − c(φ).
4. The spectrum S(φ)
5. If M1 (φ1 ) and M1 (φ2 ) are finite and φ = φ1 ∗ φ2 , then
c(φ) = c(φ1 ) + c(φ2 ) and φe = φe1 ∗ φe2 .
6. If M2 (φ) is finite we have M2 (φ) = µ2 (φ) and
e + |c(φ)|2 .
M2 (φ) = M2 (φ)
e ≤ M2 (φ).
This implies M2 (φ)
e = M2 (φe1 ) + M2 (φe2 ).
7. If M2 (φe1 ) and M2 (φ2 ) are finite, and φ = φ1 ∗ φ2 , then M2 (φ)
Definition 12. We say that a point set, a set function, or a point function in R is of
radial symmetry if it is invariant under all rotations about the origin.
A distribution function is said to be of radial symmetry if and only if Λ(y, φ) is of radial
symmetry.
Chapter 5. On Distribution functions
5.3.3
59
Convergence of infinite convolutions
Definition 13. Let {φn }n be an infinite sequence of distribution functions. The infinite
convolution
φ1 ∗ φ2 ∗ . . .
is said to be convergent if for
Φn := ψ1 ∗ · · · ∗ φn
we have
Φn −→Φ as n −→ ∞
for some distribution function Φ. We can then write
Φ = φ1 ∗ φ2 ∗ . . . .
Proposition 13. The infinite convolution of distributions
φ1 ∗ φ2 ∗ . . .
converges if and only if the infinite product
φb1 ∗ φb2 ∗ . . .
converges uniformly in every interval |y| ≤ a. We have in such a case
(φ1 ∗\
φ2 ∗ . . . ) = φb1 ∗ φb2 ∗ . . .
Theorem 21. ([18], Theorem 1) A necessary and sufficient condition for the convergence
of the infinite convolution
φ1 ∗ φ2 ∗ . . .
is that for all positive integers r
ρn,r := φn+1 ∗ · · · ∗ φn+r −→ δ as n → ∞.
Chapter 5. On Distribution functions
60
Theorem 22. ([18], Theorem 2) If
Φ = φ1 ∗ φ2 ∗ . . .
is convergent, then so is
ρn := φn+1 ∗ φn+2 ∗ . . .
for every n and
ρn −→ δ as n → ∞.
Theorem 23. ([18], Theorem 3) If Φ = φ1 ∗ φ2 ∗ . . . is convergent then
S(Φ) = S(φ1 ) + S(φ2 ) + . . . .
Theorem 24. ([18], Theorem 4) If M2 (φn ) is finite for every n then the convergence of
the two series
c(φ1 ) + c(φ2 ) + . . . and M2 (φe1 ) + M2 (φe2 ) + . . .
implies the convergence of Φ = φ1 ∗ φ2 ∗ . . . . Furthermore, M2 (Φ) is finite; finally,
e = M2 (φe1 ) + M2 (φe2 ) + . . . .
c(Φ) = c(φ1 ) + c(φ2 ) + . . . and M2 (Φ)
Theorem 25. ([18], Theorem 5) If all spectra S(φn ) are contained in a fixed interval
|x| ≤ a, then the convergence of the two series
c(φ1 ) + c(φ2 ) + . . . and M2 (φe1 ) + M2 (φe2 ) + . . .
is necessary and sufficient for the convergence of Φ = φ1 ∗ φ2 ∗ . . . .
The following theorem follows immediately from the previous two theorems.
Theorem 26. ([18], Theorem 6) If M2 (φn ) is finite for every n, then the convergence of
the two series
|c(φ1 )| + |c2 (φ)| + . . . and M2 (φe1 ) + M2 (φe2 ) + . . .
Chapter 5. On Distribution functions
61
implies the absolute convergence of φ1 ∗ φ2 ∗ . . . . If all spectra S(φn ) are contained in a
fixed interval |x| ≤ a, then the converse is also true. Notice also that since M2 (φn ) =
fn ) + |c(φn )|2 , the convergence of the two series is equivalent to the convergence of
M2 (φ
the series
|c(φ1 )| + |c2 (φ)| + . . . and M2 (φ1 ) + M2 (φ2 ) + . . .
Chapter 6
Distribution of values of L0/L(σ, χD )
In this chapter we discuss the distribution of L0 /L(σ, χD ) . Here D is a fundamental
discriminant, and σ >
6.1
1
2
. The analytic preliminaries are presented in Chapter 5.
Introduction
The study of the value distribution theory of L0 /L(s, χ) , s = σ + it ∈ C , σ > 21 , χ
being a complex character, has been initiated by Ihara as in [10], following many previous
related papers. A more detailed work has then followed by Ihara and Matsumoto in [16].
We were motivated by their work to study the distribution of values of L0 /L(σ, χD ), where
σ is real > 12 , D a fundamental discriminant, and χD the real character attached to D.
In particular, we prove , under GRH, that for each σ there is a density function Qσ with
the property that for real numbers α ≤ β, we have
#{D fundamental discriminants, such that |D| ≤ Y, and α ≤
6
∼ √ Y
2
π 2π
Z
β
Qσ (x)dx.
α
62
L0
(σ, χD ) ≤ β}
L
Chapter 6. Distribution of values of L0 /L(σ, χD )
6.2
63
Admissible functions
In their paper [16], Ihara and Matsumoto define what is called an admissible arithmetic
function, and a uniformly admissible family of arithmetic functions. We consider a slight
generalization in which only real characters intervene.
Let s = σ + it , σ >
1
2
and D a fundamental discriminant.
Definition 14. An arithmetic function λ : N −→ C is called admissible if it satisfies the
following conditions:
1. λ(n) n
for any > 0.
2. For any D, let χD :=
.
D
be the real character attached to D, and consider the
Dirichlet series
gλ (s, χD ) :=
∞
X
χD (n)λ(n)
ns
n=1
.
Then gλ (s, χD ) extends to a holomorphic function on <s > 12 .
3. log |gλ (s, χD )| l(t)l(D) + l(t)2 on <s ≥
1
2
+ . Here l(x) := log(|x| + 3).
The holomorphic functions gλ (s, χD ) will be called the g-functions associated with λ.
Remark 7. By 1. gλ (s, χD ) converges absolutely for <s > 1 and is a holomorphic
function in this half plane.
Definition 15. If Λ is a family of admissible functions such that the implicit constant in
condition 1 and condition 3 of Definition 14 can be chosen independently of λ ∈ Λ, then
Λ is called a uniformly admissible family of arithmetic functions.
√
We note that in our work we consider the quadratic field Q( D), where D is a
fundamental discriminant being positive or negative. However, in their work [16], Ihara
and Matsumoto considered the case where the base field is either the rational number
Chapter 6. Distribution of values of L0 /L(σ, χD )
64
field, an imaginary quadratic field, or a function field over a finite field having only one
archimedean prime divisor. They studied
Avgχ (gλ (s, χ, f )gλ0 (s, χ, f )) :=
X
1
|Gf | f =f
χ
gλ (s, χ, f )
chi6=χ0
where f represents the conductor of the character χ, and gλ (s, χ, f ) :=
s
(I,f )=1 (χ(I)λ(I))/n ,
P
the sum being over all integral divisors I coprime with f . Then, they consider the limit
of this average as N (f ) −→ ∞, where f is prime. They proved
Theorem 27. ([16], Theorem 1) Let Λ be any uniformly admissible family of arithmetic
functions, and let λ, λ0 run over Λ. Fix any such that 0 < <
over the domain σ >
1
2
1
2
and let s = σ + it run
+ . In the number field case, we also fix T > 0 and impose that
|t| ≤ T . Then
1. For any integral divisor f , we have
Avgχ (gλ (s, χ, f )gλ0 (s, χ, f )) −
X
λ(I)λ0 (I)N (I)−2σ N (f )− 2 .
(I,f )=1
where I runs over integral divisors of the base field. In particular, the left hand side
tends to 0 uniformly as N (f ) −→ ∞ .
2. Let f run only over the prime divisors. Then
lim Avgχ (gλ (s, χ, f )gλ0 (s, χ, f )) =
N (f )→∞
X
λ(I)λ0 (I)N (I)−2σ ,
I
and the convergence is uniform w.r.t λ, λ0 , s. Moreover, the above average may be
replaced by that over all χ with the conductor fχ = f .
6.3
Statement of results
We propose to prove
Chapter 6. Distribution of values of L0 /L(σ, χD )
65
Theorem 28. Let Λ be any uniformly admissible family of arithmetic functions and let
λ run over Λ. Fix 0 < <
1
2
and T > 0 ; let s = σ + it run over the domain σ ≥
1
2
+
and |t| ≤ T . Then for any Y large enough we have
Q
∞
X
λ(n2 ) p|n (1 + p1 )−1
1
1 X∗
gλ (s, χD ) −
Y −8
2s
N (Y )
n
n=1
|D|≤Y
where
N (Y ) := #{|D| ≤ Y ; D is a fundamental discriminant}.
and
X∗
means we are summing over fundamental discriminants |D| ≤ Y .
In particular,
∞
X
λ(n2 )
1 X∗
gλ (s, χD ) =
lim
Y →∞ N (Y )
n=1
Q
|D|≤Y
p|n (1
n2s
+ p1 )−1
where the convergence to the limit is uniform.
In what follows, Qσ will denote the density function governing the distribution of values
of L0 /L(σ, χD ), D being a fundamental discriminant. This density function will be
constructed in the following sections. We propose to prove
Theorem 29. Let σ > 12 , and assume GRH. Then we have
1 X∗ L0
1
lim
Φ( (σ, χD )) = √
Y →∞ N (Y )
L
2π
|D|≤Y
Z
∞
Qσ (x)Φ(x)dx
(6.3.1)
−∞
holds for any bounded continuous function Φ on R. The equality (6.3.1) holds also when
Φ is the characteristic function of either a compact subset of R or the complement of such
a subset.
6.4
Preliminary results
In the proof of Theorem 28, we follow closely Ihara and Matsumoto ’s method in [16],
where they considered complex values of s. Their method was in particular useful to get
Chapter 6. Distribution of values of L0 /L(σ, χD )
66
the distribution of values of L0 /L(s, χ), where χ runs over all complex characters, mainly
because of the cancellation that occurs from orthogonality relations among complex
characters. However, when we consider real characters, we have less cancellation. This is
the case considered here, namely real characters attached to fundamental discriminants.
For this among other reasons to be stated later, we had to consider only real values of
σ. As for Theorem 29, it was motivated by results from Jessen-Wintner [18], and results
from Ihara-Matsumoto [16].
We state below the definitions and some results from [16] needed in the present chapter.
Remark 8. ([16], p.7) Let λ1 , . . . , λk ∈ Λ, we define their ∗-product by
X
(λ1 ∗ · · · ∗ λk )(n) =
λ1 (n1 ) . . . λk (nk ).
n=n1 ...nk
If λ1 , . . . , λk ∈ Λ are admissible, then λ1 ∗ · · · ∗ λk is admissible and it is associated
with the product
gλ1 . . . gλk .
Moreover, if Λ is a uniformly admissible family of arithmetic functions, then for any
k ≥ 1, Λk := {λ1 ∗ · · · ∗ λk ; λ1 , . . . , λk ∈ Λ} is again a uniformly admissible family of
arithmetic functions.
Proposition 14. ([16], Proposition 2.2.1)
i) For <s ≥
1
2
+ and X > 1, one can write g := gλ (s, χD ) as
g = g+ − g−
where g+ , g− are two holomorphic functions satisfying
where
R
(c)
1
g+ := g+ (s, χD , X) :=
2πi
R
means (<w=c) , and
Z
1
g− := g− (s, χD , X) :=
2πi
Γ(ω)g(s + ω, χD )X ω dω
(c)
Z
(0 −)
Γ(ω)g(s + ω)X ω dω
Chapter 6. Distribution of values of L0 /L(σ, χD )
67
such that c > max(0, 1 − σ) and 0 < 0 < .
ii) g+ has the Dirichlet series
X
g+ =
χD (n)λ(n) exp(
(n,D)=1
−n −s
)n
X
which is absolutely convergent for any fundamental discriminant D, and s ∈ C.
Proof. This is proved by the Mellin-inverse transform, and by moving the line of integration
from (c) to (0 − ), getting a pole at ω = 0 coming from the Γ-function.
Proposition 15. ([16], Proposition 2.2.3)
Let σ ≥
1
2
+ , Λ an admissible family.
1. For all 0 > 0 and fundamental discriminants D, we have
1
0
|g+ (s, χD , X)| X 2 + − .
2. For all 0 < 0 < 4 , T > 0 and |=s| ≤ T , we have
0
|g− (s, χD , X)| 0 ,T (DX) X − .
Let ψz1 ,z2 denote the quasi-character of the additive group of C
i
ψz1 ,z2 (ω) = exp (z1 ω + z2 ω)
2
and consider the Dirichlet series
fs (z1 , z2 ) =
M
X
λz1 (I)λz2 (I)N (I)−2s
(<s > 1/2).
I
Theorem 30. ([16], Theorem 3) Assume |P∞ | = 1, and assume also GRH in the number
field case. Then
0
fσ (z1 , z2 )
lim
Avgfχ =f ψz1 ,z2 (L (s, χ, f )/L(s, χ, f )) = M
f prime
N (f )→∞
uniformly on |z1 |, |z2 | ≤ R and for s = σ + it with σ ≥ 1/2 + , and |t| ≤ T in the number
field case.
Chapter 6. Distribution of values of L0 /L(σ, χD )
68
Here P∞ represents the set of all archimedean primes of K. This very theorem, was the
motivation behind Theorem 28. Furthermore, if we let Mσ (z) be the M -function associated
with L0 /L(s, χ, f ) (this has been constructed by Ihara in [10]), then Ihara-Matsumoto
proved the following theorem.
Theorem 31. ([16], theorem 4) Let σ = <s > 1/2, assume |P∞ | = 1, and GRH in the
number field case. Then
Z
0
lim Avgfχ =f Φ(L (s, χ, f )/L(s, χ, f )) =
f prime
N (f )→∞
Mσ (ω)Φ(ω)|dω|
C
holds for any continuous function Φ on C with at most exponential growth, i.e when
Φ(ω) ea|ω| holds for some a > 0. The equality holds also when Φ is the characteristic
function of either a compact subset of C or the complement of such a subset.
6.5
Proof of Theorem 28
Proof. Fix 0 < < 12 ; throughout the section, the symbol will depend on . Let
s = σ + it ; σ ≥ 12 + ; X ≥ 1 and Λ a uniformly admissible family, λ ∈ Λ; g = gλ . Write
g = g+ − g−
as in Proposition 14 and let
g± (χD ) := g± (s, χD )
and
1 X∗
g± (s, χD ).
Y →∞ N (Y )
AvgD (g± (χD )) := lim
|D|≤Y
We have
g+ (χD ) =
X λ(n)χD (n)
n
exp(−
).
ns
X
(n,D)=1
Let
S=
1 X∗ X λ(n)χD (n)
n
exp(− )
s
N (Y )
n
X
n
|D|≤Y
Chapter 6. Distribution of values of L0 /L(σ, χD )
69
and this is in turn equal to
1 X fY (n)λ(n)
n
exp(− )
s
N (Y ) n
n
X
where
fY (n) :=
X∗
χD (n) and N (Y ) = #{D fundamental discriminants ; |D| ≤ Y }.
|D|≤Y
Our aim is to estimate S as a function of Y . Whenever n is a square, fY (n) will be
large. We will show that the main term in the growth of S comes from these values of n .
Contribution of squares. For n square, we have by [6]
1
fY (n) = cn Y + O(Y 2 d(n))
where
cn =
6 Y
1
(1 + )−1 .
2
π
p
p|n
Thus the contribution of squares to S is
Y
∞
X
λ(n2 )cn
n=1
n2s
X
|λ(n2 )| exp(− Xn )
1
n
exp(− ) + O Y 2
d(n) .
2σ
X
n
n
This is equal to
∞
2
6 X λ(n )
Y
π 2 n=1
Q
p|n (1
n2s
+ p1 )−1
1
+ O(max(Y 2 ,
Y
)).
X
Notice here that
N (Y ) :=
X∗
1=
|D|≤Y
X∗
χD (1) = fY (1) =
|D|≤Y
1
6
Y + O(Y 2 ).
2
π
Contribution of non-squares.
As for non squares, their contribution to S is
X
1
X0 λ(n)fY (n)
0
n
n 2
2
exp(− ) |fY (n)| exp(− )
s
n
X
X
n
n
Chapter 6. Distribution of values of L0 /L(σ, χD )
by Cauchy-Schwartz inequality. Here
P0
70
means we are summing over non-square n. This
contribution is then
1
1
Y 2 X 2 (log X)2 .
However, by partial summation, and the fact that ([27], Chapter 5)
X0
|fY (n)|2 N Y (log N )4
1<n≤N
we conclude that by Proposition 15 in addition to the above
Q
X
λ(n2 ) p|n (1 + p1 )−1
1 X∗
g(χD ) −
Y −8
2s
N (Y )
n
n
|D|≤Y
If we choose 0 <
6.6
4
1
2
and X = Y
. This finishes the proof of Theorem 28.
Toward a Density Function
As in [16], write
X
∞
xt
exp
=
Gr (x)tr ,
1−t
r=0
explicitly, G0 (x) = 1 and for r ≥ 1
r
X
1 r−1 k
Gr (x) =
x .
k! k − 1
k=1
Theorem 32. ([16], Theorem 2) For each z ∈ C, and n =
tive arithmetic function λz (n) by
λz (n) :=
Y
λz (pαp )
p
and
λz (pαp ) = Gαp (−
In particular λz (1) = 1 . Then
iz
log p).
2
Q
p
pαp , define the multiplica-
Chapter 6. Distribution of values of L0 /L(σ, χD )
71
1. The family {λz }|z|≤R satisfies condition 1 of Definition 14 uniformly. In other terms,
the implicit constant in condition 1 of Definition 14 depends only on 0 and R.
2. If we assume GRH, the family {λz }|z|≤R also satisfies conditions 2, 3 of Definition
14 and is a uniformly admissible family of arithmetic functions. The associated
g-function is given by
iz L0
gλz (s, χ) = exp
(s, χ) .
2 L
Remark 9. In particular, n could be 1, and the αp = 0 , for all p. Also, λz (1) = 1
because Go (x) = 1.
Now, put
ga (σ, χD ) := gλ2a (σ, χD )
put also
ψa (ω) := exp(iaω)
and
fσ (a) :=
M
∞
X
λ2a (n2 )
n=1
Q
p|n (1
n2σ
+ p1 )−1
.
Corollary 4. For > 0, a ∈ R and R > 0, we have
0
1 X∗
L
fσ (a)
ψa
(σ, χD ) = M
N (Y )
L
|D|≤Y
uniformly on |a| ≤ R and σ ≥
1
2
+ .
Proof. Clearly ga (σ, χD ) = ψa
corollary.
L /L(σ, χD ) . We then use Theorem 28 to get the
0
Chapter 6. Distribution of values of L0 /L(σ, χD )
6.6.1
72
Construction of Mσ (x)
fσ (x) is given by
The Euler product of M
fσ (x) =
M
∞
∞
X
X
λ2x (n2 ) Y
1 −1 Y
λ2x (p2r )
1 −1
(1 + ) =
1+
(1 + )
n2σ
p
p2rσ
p
n=1
p
r=1
p|n
∞
Y
1 −1 X G2r (−ix log p)
=
1 + (1 + )
p
p2rσ
r=1
p
∞
Y
1 −1 X G2r (−ix log p)
=
1 + (1 + ) (
− 1)
p
p2r
p
r=0
Y 1
p
−ix log p
p
ix log p
+
exp( σ
)+
exp( σ
) .
=
p + 1 2(p + 1)
p −1
2(p + 1)
p +1
p
Now let for each p,
Mσ,p (y) :=
p
p
1
δ(y) +
δap +
δ−b ,
p+1
2(p + 1)
2(p + 1) p
and
g
M
σ,p (y) :=
where ap =
log p
pσ −1
1
p
−ix log p
p
ix log p
+
exp( σ
)+
exp( σ
)
p + 1 2(p + 1)
p −1
2(p + 1)
p +1
, bp =
log p
,
pσ +1
δc (x) := δ(x − c). Then clearly,
Z
∞
d
M
σ,p (y) =
g
eixy Mσ,p (dx) = M
σ,p (y)
−∞
is the Fourier transform of Mσ,p in terms of distributions. Now let
Mσ (x) := ∗p Mσ,p (y)
Then by [18], Theorem 6, the above convolution of distribution converges and Mσ exists
as a distribution function.
6.6.2
Construction of Qσ (x)
We have
fσ (x)| =
|M
Y 1
p
−ix log p
p
ix log p
|
+
exp( σ
)+
exp( σ
) |
p + 1 2(p + 1)
p −1
2(p + 1)
p +1
p
Chapter 6. Distribution of values of L0 /L(σ, χD )
73
Y 1
p
x log p
x log p
x log p
x log p
+
(| cos( σ
) + cos( σ
)| + | sin( σ
) − sin( σ
)|)
p + 1 2(p + 1)
p +1
p −1
p −1
p +1
p
Y 1
p
x log p
x log p
x log p x log p
x log p x log p
+
(| cos( σ
)|+| cos( σ
)|+|2 sin( σ
−
) cos( σ
+
)|)
p + 1 2(p + 1)
p +1
p −1
p − 1 pσ + 1
p − 1 pσ + 1
p
Y 1
p
x log p
x log p
2x log p
x log p x log p
+
(| cos( σ
)|+| cos( σ
)|+|2 sin( 2σ
) cos( σ
+
)|)
p + 1 2(p + 1)
p +1
p −1
p −1
p − 1 pσ + 1
p
Y 1
p
x log p
x log p
2x log p
+
(| cos( σ
)| + | cos( σ
)| + |2 sin( 2σ
)|) .
p + 1 2(p + 1)
p +1
p −1
p −1
p
For any > 0, and x sufficiently large, we look at those p such that
x log p
x log p
and σ
∈ [1 − , 2 + ]
σ
p +1
p −1
p
p
which implies both | cos( xpσlog
)| and | cos( xpσlog
)| are < 23 , and also
+1
−1
log p
in turn implies that 0 < sin( 2x
)<
p2σ −1
4
,
pσ +1
(6.6.1)
2x log p
p2σ −1
<
4
.
pσ +1
which
for pσ ≥ 5. We try to investigate how often
such p occurs. Let K(x) be the number of primes p satisfying (6.6.1). We have
1
x log p
1
x log x ≤ pσ ≤ x log x =⇒ 1 − ≤
≤ 2 + .
2σ
σ
pσ
Thus
1
1
K(x) ≥ π(( x log x)1/σ ) − π(( x log x)1/σ ) ≥ αx1/σ .
σ
2σ
Here α is a positive constant dependent on σ and π(x) represents the number of primes
p ≤ x. Furthermore we have for those primes pσ ≥ 15, satisfying (6.6.1)
g
|M
σ,p (x)| ≤
1
p
4
8
+
( + σ
)
p + 1 2(p + 1) 3 p + 1
≤
47
<1
48
We then conclude that
fσ (x) ( 47 )K(x) exp(−(log( 48 ))K(x)) exp(−Cσ x1/σ )
M
48
47
where Cσ > 0 is a constant dependent on σ.
fσ (x) = Oσ (exp(−Cσ x σ1 )), however since
Thus we proved that for x > 0, M
fσ (x) = M
fσ (−x)
M
Chapter 6. Distribution of values of L0 /L(σ, χD )
74
we conclude that the above bound is also valid for x < 0, in other terms
fσ (x) = Oσ (exp(−Cσ |x| σ1 )).
M
This implies that
Z
∞
fσ (x)dx
|xk |M
−∞
converges for all k ≥ 0, which implies by Chapter 5 that Mσ (x) is an absolutely continuous
cσ (x) = M
fσ (x), the Fourier transform being here in the sense of
distribution (since M
distribution) and its density function Nσ (x) is given by
1
Nσ (x) =
2π
Z
∞
fσ (y)dy.
exp(−ixy)M
−∞
Define now the function
Qσ (x) =
√
2πNσ (x)
then we can see that Qσ (x) is continuous, approaches zero as |x| −→ ∞ and possesses
continuous derivatives of all order. In other terms, it is regular analytic in the whole real
line. This follows from the definition of Qσ and from Chapter 5. The motivation behind
the definition of Qσ is that
^
cσ (x) = M
Q
σ (x).
Here the Fourier transform is as in [16]. We may notice as well that Qσ (x) is a good
density function on R in the sense of Ihara-Matsumoto [16], which we will cover in detail
in the next subsection.
6.6.3
Fourier transform of distributions and connection with
Fourier transform of functions
We will need the following terminology from [16]. Let Rd = {x = (x1 , . . . , xn ) ; xi ∈ R},
d
d ≥ 1, and let |dx| = (dx1 , . . . , dxd )/(2π) 2 be the self-dual Haar measure with respect
Chapter 6. Distribution of values of L0 /L(σ, χD )
75
to the self-dual pairing ei<x,y> of R . For any function f belonging to L1 , we define its
Fourier transform fb, and the inverse Fourier transform fˇ by
Z
Z
i<x,y>
b
ˇ
f (x) = f (y)e
|dy| , f (x) = f (y)e−i<x,y> |dy|.
Let Λ := Λ(Rd ) denote the space of all f ∈ L1 ∩ L∞ such that fˇ also belongs to L1 ∩ L∞
ˇ
and that fb = f .
A good density function on Rd , will mean any non-negative real valued continuous function
M (x) that belongs to Λ and satisfies
Z
M (x)|dx| = 1
For a finite set X with a measure ω and total measure ω(X) = 1, let X ∗ = (X, ω), and let
Z
X
AvgX ∗ φ := φω :=
ω(χ)φ(χ).
χ∈X
Consider now any pair X ∗∗ = (X ∗ , l) of an X ∗ = (X, ω) and a mapping l : X −→ Rd .
Then for a sequence {Xn∗∗ }n of Xn∗∗ = (Xn∗ , ln ) and a test function φ on Rd , consider the
condition
Z
lim AvgXn∗ (φ ◦ ln )) =
M (x)φ(x)|dx| .
n→∞
(6.6.2)
Proposition 16.
Z
∞
Qσ (a) =
fσ (x)e−iax |dx|.
M
−∞
fσ . In particular,
is the Fourier inverse of M
fσ (0) = 1
M
Proof. The first statement follows directly from properties of Nσ and the fact that
√
Qσ (x) = 2πNσ (x). Now
d
g
M
σ,p (a) = Mσ,p (a)
for all primes p, then
Y
p
d
M
σ,p (a) =
Y
p
g
M
σ,p (a)
Chapter 6. Distribution of values of L0 /L(σ, χD )
76
in other terms
Y
\
fσ (a)
Mσ,p (a) = M
p
which means
cσ (a) = M
fσ (a).
M
In particular,
Z
∞
cσ (0) = M
fσ (0).
Mσ (dx) = M
1=
−∞
Remark 10. Notice that
1
Qσ (a) = √
2π
Z
∞
fσ (x)eiax |dx|
M
−∞
However
fσ (x) =
M
∞
X
λ2x (n2 ) Y
1
(1 + )
2σ
n
p
n=1
p|n
thus
fσ (x) =
M
∞
X
1
λ−2x (n2 ) Y
(1
+
)
2σ
n
p
n=1
p|n
this implies that
fσ (x) = M
fσ (−x).
M
It follows that
1
Qσ (a) = √
2π
Z
∞
fσ (−x)eiax |dx|
M
−∞
which is
1
√
2π
Z
∞
fσ (x)e−iax |dx|.
M
−∞
Thus
Qσ (a) = Qσ (a).
This is one way to prove that Qσ (x) is real. Another way to see it is from the fact that by
construction of Mσ as an infinite product of distributions, one can see directly that Mσ (x)
is real, thus its density function Nσ (x) is real as well. This implies that Qσ (x) is real.
Chapter 6. Distribution of values of L0 /L(σ, χD )
77
Now using Lemma A from [16], to be stated below, we conclude that for any bounded
continuous function φ : R −→ C one has
0
Z ∞
L
1
1 X∗
φ
(σ, χD ) = √
Qσ (x)φ(x)dx .
lim
Y →∞ N (Y )
L
2π −∞
|D|≤Y
This holds also when Φ is the characteristic function of either a compact subset of R or
the complement of such a subset. We have then proved Theorem 29.
Remark 11. When we replace σ by s = σ + it ∈ C , t 6= 0, it will no longer be the case
that
^
f
M
σ (x) = Mσ (−x)
and
Qσ (a) = Qσ (a).
This is due to the fact that
λa (s) 6= λa (−s)
if s is not real, thus it will be difficult to establish the necessary connection between Qs (a)
fs (a), in such a way that Lemma A from [16] is applicable.
and M
Lemma 8. ([16], Lemma A)
Let M (x) be any good density function on Rd , and {Xn∗∗ }n≥1 be a sequence of pairs
Xn∗∗ = (Xn∗ , ln ) of a finite measure space Xn∗ and a mapping ln : Xn −→ Rd .
1. Suppose that condition 6.6.2 holds for any additive characters φ = ψ (y) : x 7−→
ei<x,y> and that the convergence is uniform in the wider sense with respect to the
parameter y ∈ Rd . Then condition 6.6.2 holds for any function belonging to Λ. In
particular, it holds for any compactly supported C ∞ -function.
2. Suppose condition 6.6.2 holds for all compactly supported C ∞ -function on Rd . Then:
(a) it holds for any bounded continuous function Φ.
Chapter 6. Distribution of values of L0 /L(σ, χD )
78
(b) it holds for any continuous function Φ satisfying
|Φ(x)| ≤ φ0 (|x|),
if there exists a continuous monotone non-decreasing function φ0 (r) > 0 of r > 0
satisfying limr→∞ φ0 (r) = ∞ and
Z
M (x)φ0 (|x|)|dx| < ∞,
and
AvgXn∗ (φ0 ◦ |ln |)2 1;
(c) it holds when φ is the characteristic function of either a compact subset of
Rd or the complement of such a subset.
Using the above lemma (case d = 1), we conclude the proof of Theorem 29.
Proof. Take
Xn∗ = (Xn , ωn )
where
Xn = {χD ,
|D| ≤ n}
and let
ωn (χ) =
1
|Xn |
and
L0
ln (χD ) = (σ, χD )
L
take also
M = Qσ
then clearly this is a good density function. Conditions 1) and 2 a,c) of Lemma 8 are
satisfied.
Chapter 6. Distribution of values of L0 /L(σ, χD )
79
Remark 12. Notice that
N (Y ) := #{Fundamental discriminants D; |D| ≤ Y } ∼
6
π2
we then conclude that for φ := 1[α,β] , the characteristic function of the interval [α, β], we
have
#{fundamental discriminants |D| ≤ Y , such that α ≤ L0 /L(σ, χD ) ≤ β}
6
∼ √ Y
π 2 2π
6.7
Z
β
Qσ (x)dx.
α
Average of L0/L(σ, χD ), σ > 1/2
As a motivation, we propose in this section to estimate the average over fundamental
discriminants of L0 /L(σ, χD ). We will show that the average is a constant under GRH,
for any
1
2
< σ ≤ 1, and we try to investigate for what σ’s the average is still valid
unconditionally. We propose to prove
Theorem 33. Assume GRH. For Y > 1 and σ > 1/2, we have
X∗ L0
(σ, χD ) = cσ Y + O(Y
L
7−6σ
6−4σ
)
0≤|D|≤Y
for some non-zero constant cσ depending only on σ.
Theorem 34. Unconditionally, for Y > 1, and σ satisfying
|σ − ρ| >
1
∀|D| < Y, ∀ρ,
log D
we have
X
0<|D|≤Y
cσ being a non-zero constant.
L0
(σ, χD ) = cσ Y + O(Y
L
5−2σ
4
(log Y )2 )
Chapter 6. Distribution of values of L0 /L(σ, χD )
6.7.1
80
The main Lemma
Lemma 9. For a real primitive character χ to the modulus q and an X ≥ 1, we have
unconditionally the identity
∞
X
X
χ(n)Λ(n)
L0
−σ
(σ, χ) = −
exp(−n/X)
+
(1
−
a)Γ(−σ)X
−
Γ(ρ − σ)X ρ−σ
σ
L
n
n=1
ρ
1
−
2πi
Z
Here
X
(−σ−)
L0
π 1 Γ0 1 − σ − s + a Γ0 σ + s + a
(1−σ−s, χ)−log ( )+
(
)+ (
) Γ(s)X s ds.
L
q 2 Γ
2
Γ
2
is taken over all non trivial zeros of L(σ + s, χ), and a = 0 or 1 according as
ρ
whether χ is even or odd.
Proof. We have for any > 0
X χ(n)Λ(n)
L0
(σ + s, χ)Γ(s)X s ds = −
exp(−n/X)
(6.7.1)
nσ
(1−σ+) L
n
R
by the Mellin-inverse transform, where () F (s)ds means that we are integrating over
1
2πi
Z
Re(s) = . Now, by moving the line of integration from (1 − σ + ) to (−σ − ) we pick
up one simple pole at s = 0, with residue L0 /L(σ, χ), simple poles at s = ρ − σ for each
non-trivial zero ρ with residue Γ(ρ − σ)X ρ−σ , and one simple pole at s = −σ, with residue
Γ(−σ)X −σ , if a = 0. Then the above equation becomes
X
Γ(ρ − σ)X
ρ
1
+
2πi
Z
(−σ−)
ρ−σ
L0
+ (σ, χ) + (1 − a)Γ(−σ)X −σ
L
X χ(n)Λ(n)
L0
(σ + s, χ)Γ(s)X s ds = −
exp(−n/X).
L
nσ
n
Now apply the functional equation of L0 /L, deduced from taking the logarithmic derivative
of the functional equation of L-function
1 Γ0 s + a
L0
π 1 Γ0 1 − s + a
L0
(
) + (s, χ) = log −
(
) − (1 − s, χ)
2Γ
2
L
q
2Γ
2
L
Chapter 6. Distribution of values of L0 /L(σ, χD )
81
to get
X
Γ(ρ − σ)X ρ−σ +
ρ
1
−
2πi
Z
(−σ−)
L0
(σ, χ) + (1 − a)Γ(−σ)X −σ
L
L0
π 1 Γ0 1 − σ − s + a Γ0 σ + s + a
(1−σ−s, χ)−log ( )+
(
)+ (
) Γ(s)X s ds.
L
q 2 Γ
2
Γ
2
=−
X χ(n)Λ(n)
n
nσ
exp(−n/X).
Now rearrange terms to get the result of the Lemma.
6.7.2
The main term
Notice first that
log q
π
1 Γ0 σ + s + a
Γ0 1 − σ − s + a
s
− log ( ) +
(
)+ (
) Γ(s)X ds σ+ .
q
2 Γ
2
Γ
2
X
(−σ−)
Z
1
2πi
By Stirling’s formula. On the other hand,
1
2πi
Z
(−σ−)
X Λ(n) 1
L0
(1 − σ − s)Γ(s)X s ds L
n1+ X σ+
n
Z
|Γ(s)|ds (−σ−)
1
X σ+
.
Corollary 5. For Y > 1, we have unconditionally
X
0<|D|≤Y
X X Λ(n)χD (n)
L0
n XX
N (Y ) log Y
(σ, χD ) = −
exp(−
)−
Γ(ρ−σ)X ρ−σ +O(
).
σ
L
n
X
Xσ
n
ρ
D
0<|D|≤Y
We can now prove Theorem 33 .
Proof. Define
S := −
X X Λ(n)χD (n)
n
exp(− )
σ
n
X
n
0<|D|≤Y
and let fY (n) =
P
0<|D|≤Y
D
n
. If n is a square then
1
fY (n) = cn Y + O(Y 2 d(n)),
Chapter 6. Distribution of values of L0 /L(σ, χD )
d being the divisor function, and cn =
Y
X
n
cn
6
π2
Q
p|n
82
(1 + p1 )−1 . The contribution of squares in S is
X Λ(n2 )
1
1
Λ(n2 )
2
2
exp(−n
/X)
+
O(Y
exp(−n2 /X)) = Cσ Y + O(Y 2 )
2σ
2σ
n
n
n
for some positive constant Cσ . As for the non-squares, the contribution to S is
X
12 X 2 12
X0
0
0 Λ (n)
n
Λ(n)
n
fY2 (n) exp (− )
,
fY (n) σ exp(− ) n
X
X
n2σ
n
n
n
by Cauchy’s inequality, where the prime on the summation indicates that we are summing
over non-square n’s.
Now apply (4.1) from [27], p. 106, and partial summation to get
X0
fY2 (n) exp (−
n
n
) XY (log X)4 .
X
Thus,
1
1
S = Cσ Y + O(Y 2 X 2 (log X)2 ).
6.7.3
Estimation of the sum over zeros under GRH
We need to estimate
PP
D
Γ(ρ−σ)X ρ−σ . If we assume the GRH, we will not have to worry
ρ
about possible poles for the Γ function. Now, Stirling’s formula implies that ∃ T0 > 0
such that for all ρ with |=ρ| > T0 , one has
π
|Γ(ρ − σ)| exp− 2 =ρ 1
.
T0 (1 + (=ρ)2 )
1
Then the contribution of these zeros is O(Y 2 log Y ), if T0 is chosen large enough. See [D],
p.102. As for the zeros in
B := {ρ ; |=ρ| ≤ T0 }
there exists an M > 0 such that |Γ(ρ − σ)| < M for all ρ ∈ B, and the contribution of
P P 1 −σ
zeros in B is then O(
X 2 ), which in turn is
D
ρ
X
7−6σ
N (σ, T, χD ) (Y T ) 6−4σ +
D
by Jutila’s zero density estimate (see Theorem 20, Chapter 4). Now, take X = Y
the first theorem.
1
2
to get
Chapter 6. Distribution of values of L0 /L(σ, χD )
6.7.4
83
Estimation of the sum over zeros, unconditionally
Here we will need to look at those σ that only satisfy the following
|σ − ρ| >
PP
We need to estimate
ρ
D
1
∀|D < |Y, ∀ρ.
log D
Γ(ρ − σ)X ρ−σ . The main trouble here comes from the fact
that Γ(ρ − σ) might have poles close to σ, so we can write
Γ(ρ − σ)X
−X ρ−σ
=
+ O(X ρ−σ ).
ρ−σ
ρ−σ
Thus it is enough to estimate the contribution of
XX
X ρ−σ
ρ
D
this is because
1
< log Y
|ρ − σ|
by the choice of σ. But,
XX
D
1
XZ
ρ
X =−
ρ
X δ dN (δ, T, χD ).
0
D
By partial summation this becomes
−
X
1
Z
X
X N (δ, T, χD ) +
δ
0
D
1
N (δ, T, χD )X δ log Xdδ
0
D
and this is equal to
1
Z
X
N (0, T, χD ) +
X
0
D
N (δ, T, χD )X δ log Xdδ.
D
But
X
N (0, T, χD ) = 2
D
X
D
1
N ( , T, χD ) (Y T )1+ ,
2
and
Z
0
1
2
X
D
1
N (δ, T, χD )X δ log Xdδ (Y T )1+ X 2 log X,
Chapter 6. Distribution of values of L0 /L(σ, χD )
84
by Jutila’s zero density estimate (see Theorem 20, Chapter 4), where > 0. On the other
hand,
1
Z
1
2
X
1
Z
δ
7−6σ
(Y T ) 6−4σ + X δ log Xdδ,
N (δ, T, χD )X log Xdδ 1
2
D
which is
1
Z
Y
7−6σ
++ 2δ
6−4σ
log Y dδ
1
2
1
where X = Y 2 , and T = T0 . Now consider the function of δ
f (δ) =
δ
7 − 6σ
++ .
6 − 4σ
2
We have
f 0 (δ) =
1
2
1
−
< 0 , ∀δ ∈ ( , 1).
2
2 (3 − 2δ)
2
Thus f is a decreasing function, and f (δ) < f ( 12 ) = 1 + 14 + .
Hence
XX
D
1
σ
σ
1
X ρ−σ Y 1+ 4 − 2 + = O(Y 1−( 2 − 4 ) log Y ).
ρ
Now we get the second theorem.
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