A Non-Abelian Analogue

of the Least Character Nonresidue

Kambiz Mahmoudian

A t hesis submitted in conformity wit h the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

@Copyright by Kambiz Mahmoudian 1999

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ABSTRACT

A Non-Abelian Analogue of the Least Character hTonresidue

Kambiz Mahmoudian

Depart ment of Mathematics, University of Toronto

Ph.D. Thesis, 1999

Let E / K be a Galois extension of number fields with group G, and let x be a

character of G. For a prime p of K, denote by op the conjugacy class of fiobenius

automorphisms at p. This thesis presents upper bounds on the prime p of srnaest

norm for which x(aP) # ~ ( 1 ) under the assumption of Artin's conjecture (AC).

To state the main results more precisely, let us set

For x ïrreducible and non-Abeiian we prove (Theorem 4.4) that

where A, is (essentially) the Artin conductor of X, and c > O is an absolute effective

constant. As well as AC, here we are assuming that the exceptional zero of &(s )

does not exist. We &O establish (Theorem 4.3) the explicit relationship between a

zero-fiee region for (s) L ( s , X) and the size of p(x) -

Another result (Corollary 5.6) states that for x not containing the trivial character,

the assumption of the Generalized Riemann Hypot hesis implies that

Finally, a version of the Deuring-Heilbronn phenornenon for non-Abelian Artin

. - ll

L-function L(s, X, EIK) has been proved. This result (Theorem 3.4) states that a

zero of L(s , X ) very close to s = 1 has the effect of pushing other zeros further away

£rom the Iine a = 1.

Acknowledgement s

1 would like to express my deepest gratitude to Prof. V. Kumar Murty who has not

only been a great teacher, advisor, and supervisor, but also with his caring personality

has played a major role in my mathematical (and general) growth. 1 am very proud

of, and gratefd for, being one of his students.

1 am ako sincerely gratefd to Prof. James Arthur and Prof. Fiona Murnaghan

for their kind advice and help before 1 fell in love with arithmetic.

1 would Iike to thank my best friend, Amir Akbas; whose fnendship and advice

bas been invaluable.

My sincere thanks to Ms. Ida Bulat, the great Graduate Secretary of the Mathe-

matics Department. She has been just an angel.

The scholarship kom the Iranian Ministry of Culture and Higher Education, and

also from the Mathematics Department here is also gratefully acknowledged.

My thanks also to my good hiends Fariborz, Heng, Karl, Maki, Vijay, to name a

few, for their kindness.

Finally, 1 express my sincere gratitude to Ms. Lucile Lo, the new Administrative

Assistant of the department, who kindly accepted, and professionally performed the

typing of t his t hesis.

Contents

1 The Basics and the Main Results 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Frobenius class 1

. . . . . . . . . . . . . . . . . . . 1.2 Representations of the Galois group 2

. . . . . . . . . . . . . . . . . . . 1.3 Ramification under representations 3

. . . . . . . . . . . . . . . . 1.4 Artin L-functions and Artin's conjecture 4

. . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The huictional equation 6

1.6 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The main results 8

2 Zeros of Artin L-firnctions 11

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The trivial zeros 11

. . . . . . . . . . . . . . . . . . 2.2 The nontrivial zeros and their density 12

3 Zero-free regions 21

CONTENTS CONTENTS

. . . . . . . . . . . . . . . 3.1 Zero-free regions and the exceptional zero 22

. . . . . . 3.2 The Deuring-Heilbronn phenornenon for Artin Gfunctions 23

4 The Least Character Nonresidue 33

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction 33

. . . . . . . . . . . . . . . . 4-2 Bounds on the Ieast character nomesidue 34

5 Proof of the Main Theorems 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The general set-up 38

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The lemmas 41

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Proof of Theorem 4.3 46

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Proof of Theorem 4.4 56

Chapter 1

The Basics and the Main Results

In this chapter we briefly r e c d the construction of Artin Lfunctions, as well as

their basic properties including the functional equation. This will provide us the

opportunity to introduce our notations, set our conventions (which will be applicable

throughout the whole thesis), and ha l l y state the main results.

1.1 Frobenius class

Let K be a number field. We employ the following notations:

OK = the ring of integers of K,

n~ = [K : Q] = [OK : Z] = the degree of K/Q,

dK = the absolute value of the discriminant of K,

CK = the set of (nonzero) prime ide& of K.

Let E be a finite Galois extension of K.

G = Gal (E /K) = the Galois goup of E I K ,

1

1.2 Representations of the Galois group The Basics and the ~ M a h Results

n = [E : K] = IGI = nE/nK.

Let p E CK. In E, p decomposes as pOE = (ql . . - qg)=, q i E C E . A prime q E C E

divides p (i.e. belongs to the h i t e set Q = Iqi)) ifE q n OK = P. If qlp, then the

finite field OK/p is na turdy imbedded in OElq. G acts transitively on Q. The

decomposition group D, and the inertia group Iq are defined by

D, = { ~ E G 1 x = y (modq) * ~ ( x ) =a(y) (mods), X , Y E & } ,

I, = { ~ E G 1 a ( x ) = x (mod q), x € 0 ~ ) -

We have e = IIqI, and e is c d e d the ramification index of p (in E). If e = 1, p is said

to be u~amif ied in E. We have n = e fg, where f = [O& : &/pl-

There is an exact sequence of finite groups

The Galois group Ga1 (oElq / oKIP) is cyclic with a distinguished generator (o de-

fined by p(r) = xNp , where Np = IoK/pl. Any preimage of 9 in D, is c d e d a

Frobenius element at q and is denoted by I,. If p is unramified in E, oq is uaiquely

defined; otherwise it is only defined mod Iq. If q' E Q, then there is a E G with

a(q) = q', and we have D,: = aD,a-', O,, = aoqa-'. If p is unramified in E, then

the set {a, 1 q E Q) is a conjugacy class in G. We denote it by op and call it the

Frobenius class at p.

1.2 Representations of the Galois group

The inner product of any two (not necessarily class) functionç and .tl, on G is defined 1

by (pl$) = - p(g)@(g). If p and + are characters of G, with + irreducible, then IGI ~ E D

The Basics and the Main Results 1 - 3 RwTiifica tion under moresen tatjons

(pl$) equals the multiplicity of occurrence of + in (o. The trivial character of G is

denoted by lc.

Let x be a character of G. Then (XI lc) equals the number of times that the trivial

character occurs in X. If p : G + GL(V) is the representation whose character is X,

then ~ ( 1 ) = dim V. As the character x determines the representation p uniquely, the

attributes of p are sometimes associated with X. In particular by "kerx" we mean

ker p, a normal subgroup of G.

1.3 Ramification under representations

Let x be a character of G, and p E CK. Fix a q E C E above p. If Iq is included in

ker X, then p is said to be unramified under X, or simply X-unramified. (As ker x is a

normal subgroup, the condition is independent of the choice of q above p.) Otherwise

In order to distinguish the X-rarnified primes from others, we have to introduce

the Artin conductor fx associated with X. This is an ideal of K given by

where ram(p, X) is a nomegative integer, zero for all but a finite number of p. The

exact description of ram(p, X) is as fobws. Let I, = Io > Il > I2 2 be the higher

r a d c a t i o n groups- Let V be the underlying representation space for X, and V'j the

subspace of V fixed pointwise by I j . Then

1.4 Artin L-fùnctions and Artin 's con iect ure The Basics and the Main Results

1.4 Artin L-functions and Artin's conjecture

Let p : G -t GL(V) be a linear representation of G = Gal(E/K) , with character X.

The Artin L-function L(s, X) = L(s , X, EIK) is constructed as follows.

Let p E CK be unramined under X. Then, although the Frobenius class op is

defined only mod 4, the conjugacy class of p ( q ) in GL(V) is well-defined. Therefme

the following function of the complex variable s = a + it is well-dehed:

where N p = NK/Qp, and 1 is the identity element in GL(V).

For general p E CK, LP(sl X ) is defked as follows. Although oq is only defined

mod I, , the restriction p ( q ) 1 ,,q is weU-dehed. We set

Having defined Lp(s , X) for dl p E Ex, we now introduce the Artin Gfunction

(associated with the Galois extension EIK, and the character X) for Res = 0 > 1,

This is an analytic function on the half-plane CT > 1.

Art in L-functions behave nicely under operations on the representation. More

The Basics and the Main Resdts 1.4 Artin L-functions and Artin's conjecture

precisely, for all characters xi, ~2 of G

where H is a subgroup of G, x a character of H, and EH the subfield of E h e d by

H.

Artin L-functions have meromorphic continuation to the whole C. Here is how.

First note that if ~ ( 1 ) = 1, then by Artin's reciprocity L(s, X) is the same as a Hecke

L-function for a ray class chaxacter, and so L(s, X) has an analytic continuation to

the whole @, except possibly a pole at s = 1. Now, for a general character x of G,

Brauer induction theorem says that there exist subgroups Hi, characters xi of Hi

with ~ ~ ( 1 ) = 1, and integers mi (possibly negative !) , such that

Therefore

This gives the meromorphic continuation of the general Artin L h c t i o n L(s, X) .

Artin's conjecture is that actually the Artin G h c t i o n L(s, X) is analytic every-

where, except at s = 1 where it has a pole of order (X 1 Io).

Artin's conjecture is known to be true in some cases. For example, Artin himc;elf

showed [l] that when G is a subgroup of S4, his conjecture holds. Also, Artin's

conjecture has plausible consequences. For instance, it implies Dedekind's conjecture

which says that for any (not necessarily Galois) extension E / K of number fields,

CE(s)/CK(s) is entire. Dedekind's conjecture is known to be true in some cases. For

1.5 The fbctiond eauation The Basics and the Main Results

example if E / K is Galois, or if Ë/K is solvable, where Ë is the normal closure of E

over K, then CE(s) &(s) is known to be entire (see [19, Chap. 21).

For ail these reasons Artin's conjecture is a very plausible açsumption. We will

base o u work in this thesis on the assumption of Artin's conjecture. As we just

need to know that the particular Artin Gfunction we are dealing with is analytic (at

s # l), we can get unconditional results by restricting our attention to the extensions

for which Artin's conjecture is already known. There seems to be the possibility of

reducing the general extensions to those for which Artin's conjecture is known, and

so another way of getting unconditional results.

1.5 The functional equation

In order to describe the functional equation which the Artin Gfunction L(s, X, E I K )

satisfies, we have to introduce Euler factors at infinite primes of K tao. Let v be such

a prime. We set

where a and b are respectively the dimensions of the +l and -1 eigenspaces of

p(complex conjugation), and so a + b = ~ ( 1 ) Now we set

v infinite prime of K

The Basics and the Main Results 1.6 Conventions -

The last factor we need to introduce is

The functional equation is

1.6 Convent ions

(a) A constant c > O is said to be absolute, if it does not depend on any parameter

present in the case. Also c > O is said to be effective, if we can assign an explicit

value to it. Any constant appearing in this thesis will be absolute and effective.

Also any constant implied by the signs «, » and O(-) will be absolute and

effective.

(b) By Minkowski's discriminant bound, log dK » n~ (with implied constant ac-

tually > 1)) if K # Q. Therefore

1.7 The main results The Basics and the Main Results

The expression log A, + c n K ~ ( l ) , which appears in our work kequently, is

t herefore « log A,-

Also, if K # Q, then dK 2 3, logA, 2 log3, and so loglog A, > 1/11.

We therefore assume K # Q. A l l the results are valid for (the easier) case of

K = Q with a slight modification: replace log A, by log A, + n.yx(1) = log A, + ~ ( 1 ) -

1.7 The main results

Let E/ K be a Galois extension of number fields, wit h group G. Let x be a character of

G. A prime p E CK is said to be "x-nonresidue" if it is X-unramified and op 6 ker X .

Note that, although the Frobenius class op is defined only mod Iq (where I, is the

inertia group at a prime q E X E above p), but as p is assumed to be X-unramified (Le.

1, 5 ker x), the condition a, $ ker x is well-defined. Also note that the condition

op 6 ker x is equivalent t O x (op) # ~(1) .

Observe also that there exists a prime p E CK which iç X-nomesidue iff x is not a

multiple of the trivial character (Chebotarev density theorem is needed to see this).

Let x be a character which is not a multiple of the trivial one. By "the least

X-nomesidue" we mean

p(x) = min {Np 1 p E CK X-nonresidue, and of degree one over Q ) .

Note that, as the minimum is taken over primes of degree one, the least X-nonresidue

p(x) is actually a rational prime.

Our main objective in this thesis is to find upper bounds on the least character

nonresidue. In aJl of the results Artin's conjecture is assumed. The highlights of the

thesis are as follows.

The Basics and the Main Results 1.7 The main results

1. For irreducible non-Abelian X, we prove (Theorem 4.4)

where c is an absolute effective constant. Here we are assuming that CK(s) does

not have the exceptional zero (see Definition 3.3).

2. For x not containing the trivial character, we prove (Theorem 4.3)

where c is an absolute effective constant, -1 a! 5 1 is a parameter, âr is given

and we are assuming that Cx(s) L(sl X, E I K ) does not vanish in the region

1 -

where cl is absolute and given by Proposition 3.1. This result is a generalization

of [Il, Theorem 1.21 which treats the Abelian (Hecke) character nonresidues.

3. For x not containkg the trivial character, it is proved (Corollary 5.6) that the

assumption of the Generalized Riemann Hypothesis implies that

with c > O absolute effective.

1.7 The main resdts The Basics and the Main Resdts

4. We also prove a version of the Deuring-Heilbronn phenomenon for non-Abelian

Artin Lfunctions. More precisely, let x be irreducible, and let po be any (real

or cornplex, trivial or nontrivial) zero of L(s, x). Then any other zero p = CT +it

of L(s, X) is proved to satis&

c4 log - C T < l - C 3

Il-polA

A 1

where A = A (t, X) = ~ ( 1 ) (log A, + n~x(1) log(lt1 + 2 ) ) , and c3 , c4 axe abso-

hte , effective, positive constants.

This means that a zero of L(s, x), which is very close to s = 1, has the effect

of pushing other zeros further away fiom the line a = 1. This phenomenon was

f is t observed by Deuring and Heilbronn in the context of Dirichlet Lfunctions.

In [Il] a version of the Deuring-Heilbronn phenomenon for k ( s ) has been proved.

Theorem 3.4 generalizes that resdt to non-Abelian Artin Gfunctions.

Although we have not used Theorem 3.4 elsewhere in the thesis, the result is of

obvious interest in the field.

Chapter 2

Zeros of Art in L-functions

In this chapter we investigate the location and density of the zeros of the Artin L-

hinction L(s, X) = L(s,x, EIK) .

Here, as well as in the rest of the thesis, we will use notations strictly as introduced

in Chapter 1 (e-g. EIK is a Galois extension of number fields, x a character of the

Galois group G, etc.).

2.1 The trivial zeros

Following a tradition set by Riemann, the real and imaginary parts of our complex

variable s will be denoted by a and t, i.e. s = a + it. The Euler product of L(s, x), which defines it for (T > 1, shows that L(s, X) # O

for a > 2. Shen the hinctional equation implies that any zero of L(s, X) with O < O

is located at a pole of the gamma factor y(s, x). These zeros of L(s, x ) , which are

said to be trivial, are therefore as follows.

Let nl and 2n2 be respectively the number of real and complex imbeddings of K.

The trivial zeros of L(s, X) are

2.2 The nontrivial zeros and their density Zeros of Artin L-functions

s = O with multiplicity nia + n2x(1) - ( ~ 1 1 ~ ) ,

s = -2k, k > 1, with multiplicity nla + n2x(1),

s = 1 - 2k, k 2 1, with multiplicity nib + nzx( l ) -

Notice that the sum of the orders of (possible) zeros at s = O and s = -1 is

nKx(l) - (xllc), and hence if K # 0, or is not a multiple of lc, then L(s,x)

vanishes at s = O or s = -1.

2.2 The nontrivid zeros and their density

Rom the basic fact orization

where x runs through all irreducible characters of G, we see that the assumption

of Artin's conjecture implies that (each) L(s, X) divides b ( s ) . On the other hand

CE(s) is known to be nonzero on the line a = 1, and therefore L(s, X) is analytic and

nonzero on a = 1, except a pole of order ( ~ 1 1 ~ ) at s = 1. The functiond equation

then implies that L(s, X) is analytic and nonzero on o = O as well, except a zero of

order nia + n2x(1) - (XI lG) at s = 0.

The zeros p = @ + iy of L(s, X) with O < ,LI < 1, are said to be nontrivid.

The Generalized Riemann Hypothesis (GRH) for h t i n G h c t i o n s is the state-

ment that any nontrivial zero p = + +y of L(s, X) has p = 112.

While we will assume Artin's conjecture in ail of our results, we wiU not assume the

GRH (except in one secondary result). By a conditional (resp. unconditional) result,

we mean one that assumes (resp. does not assume) the GRH. By this terminology, a

result assuming Artin's conjecture will be called unconditional as long as the GRH is

Zeros of Artin L -hctions 2.2 The nontrivial zeros aod their density

not assumed.

For real t , and O < r < 2, we set

We are going to estimate these two density hinctions. The followïng lemma is a

generalization for Artin L-functions of simila- estimates for Hecke Ghrnctions (see

[12, Lemma 5.41, and [Il, Lemma 2-21).

Lemma 2.1 Assuming Artin's conjecture for L(s, x), we have the follouring upper

bounds:

PROOF: (i) [This part haç been proved in [20, pp.262-2641. For the sake of corn-

pleteness here we reproduce the proof.] Artin's conjecture Mplies that the h c t i o n

is entire. It is also known to be of order one, and it does not vanish at s = O. It

therefore has a Hadamard factorization

where a(~), P ( x ) are constants, and p runs through d mas of (s(s - l ) ) ( x l l c ) ~ ( s l x), i.e. precisely the nontrivial zeros of L(s, X) .

2.2 The nontrivial zeros and their demity Zeros of Artin L-functions

By ditferentiating (1.1) and (2.1) we get

where p runs through the nontrivial zeros of L(s, x).

From the functional equation we get

At s = 112 t his gives

Choosing s = 1/2, we consider the real part of the two sides of (2.3). If p is a

nontrivial zero of L(s, x), then so is 1 - p, and the contributions of these two zeros 1

in the s u m Re cancel each other. Hence P i l 2 - P

(2 -3) t herefore implies

Zeros of Artin L-fùnctions 2.2 The nontrivial zeros and th& density

In t h equality we set s = 2 f i t . For each p = ,f? t iy with [y - tl 5 1 we have

1 Re - - 2 - P 2 113 .

( 2 + it) - p (2 - + (t - r)2

Hence (2.6) gives

At We are goïng to cornpiete the proof of (i) by bounding Re ,(2 + it, x), and for

1 L

this we need bounds on +i t ,~ ) I and I y (2 + i t , ~ ) / For 0 > 1 we have Y

The first equality is obtained £rom the Euler product of L(s, x), and by "X (OF)" we

mean the following. For unramified primes there is no ambiguity. If p is ramified in E

then we choose a prime 4 E CE above p, and fbc a Frobenius element oq. By x (op) we really mean

1 x (0:) = - C x (cfa) ..

I r , 1 a,,

2.2 The nontrivial zeros and their densiiy Zeros of Artin L-functions

Fkom this modification it e a d y follows that for o > 1

The second inequality in (2.8),

is [12, Lemma 3-21.

E'rom (2.8) we obtain

Y To bound -, first note that if 1s + kl 2 for ail integers k 2 0, then Y

1 g ( 2 + it, X)

(This is (12, Lemma 6.11.) Rom the definition of the gamma factor 7(s, X) and (2.13)

it follows that for s with 1s + kl 2 (for d integers k > O ) we have

« ~ K X W

In particular

-(2 + it, X) I Y « nKx(1) log(lt1 + 2) -

Zeros of Artin L -functions 2.2 The nontrivid zeros and their density

Finally ( 1 4 , (2.7), (2.12), (2.15) cooperate to give

The proof of (i) is complete.

(ü) R o m (2.2) we get

+ c ( l - s - p ( 3 + i t ) - p - -(s,x) y + -(3 YI +it,x) - P l > " 7'

(Here s = (T + it and 3 + it have equal imaginary parts .) From (2.14), (2.16) it follows

that for $ $ o 5 3

1 s - p ( 3 f i t ) - p (3 + it) - p 1 .

The two sums on the right side can be bounded as follows:

2.2 The nontrivial zeros and th& density Zeros of Artin L-functions

valid for 5 O 5 3.

To finish the proof of (ii), we need to bound /;/ (the bound given by (2.8) is not

sharp enough). Fkom (2.2), (2.5), and (2.20) applied to L(s, 1, K / K ) = &(s ) , we

obtain

where PK runs through the nontrivial zeros of CK (s) . If 1 < 0 < 3, then Re 1

> s - p K

O, and therefore

Zeros of Artin L-functioas 2.2 The nontrivial zeros and their deasity

The desired bound on 1 1 is naw wit hin reach:

Now, (2.20), (2.23) imply

We s u m (2.24) over t + j , -3 5 j % 3, to get

On the other hand

Finally, let O < r < 2. We choose CT = l f r , and set k = # { p 1 1s - pl 5 2r). Note

2.2 The nontrivial zeros and their densi@ Zeros of Artin L-functions

that rn&,t) 5 k. From (2.26) we have

which together with (2.25) implies

Chapter 3

Zero-free regions

We continue our program of studying the zeros of Artin Lfunctions under the as-

sumption of Artin's conjecture.

In order to obtain strong results about density of primes (with certain properties),

one needs to know t hat for the nontrivial zeros p = p i- +y of (various types of) G

functions, 1 - ,û is not too small, especially for s m d y. In other words, one needs to

know that the zeros are away from the line o = 1, especially fkom the point s = 1.

In this chapter our objective is to present some "zero-fiee region" results, which

will be used later. In the h s t section we quote a couple of results from [18]. In the

second section we will prove a result which shows that if there is any zero too close to

s = 1, then the other zeros are fairly away £rom the Line a = 1. Any result of this type

is known as an instance of the "Deuring-Heilbronn phenomenon." In [Il] a version

of the Deuring-Heilbronn phenomenon for CK(s) has been proved. We generalize i t

to Artin L-functions.

3.1 Zero-fiee regions and the exceptional zero Zero-fiee regions

3.1 Zero-free regions and the exceptional zero

The following two propositions are [18, Corollary 3-21 and [18, Proposition 3-71, re-

spectively (also see [ I l , Lemma 2.31).

Proposition 3.1 There is an absolute, effective constant ci > O such that the fol-

lovzng holds:

Assume Artin's conjecture for EIK. Ifx zs irredacible, then L(s, X) has at most

one zero zn the region

Proposition 3.2 There is an absolute, effective constant cz > O such that the fol-

lowzng holds:

Asswne Artin's conjecture for EIK. If x zs irreducible, with ~ ( 1 ) > 1 (or

~ ( 1 ) = 1 but X2 # lG), then L(s, X) does not uanish in the region

If ~ ( 1 ) = 1 and xZ = Ic, then L(s, X) has at most one zen, in this region. Such

a zero is necessan'ly real.

Definition 3.3 (The exceptional zero of L(s, x))

Proposition 3.1 says that in the rectangular neighborhood of radius of ~ ( 1 ) 1% A,

s = 1, L(s, x), with irreducible X, has at most one zero. If such a zero exists, we c d

it the exceptional zero of L(s, x).

Zerdcee regions 3.2 The Deuring-Heü b r o ~ plienomenon for Artin L - b c tions

3.2 The Deuring-Heilbronn phenomenon for Artin

L-funct ions

If the exceptional zero of L(s, X) exists, it can be a major obstacle for obtaining strong

results about density of (or existence of small) primes (of specSc types). There ase

two ways of dealing with this problem: we c m assume it does not exkt (after all, even

the GRH may be true!), or, if one insists on getting absolutely unconditional resdts,

one can show that if the exceptional zero does exist, then it "pushes" other zeros away

from the Line o = 1. This is the phenomenon first observed by Deuring and Heilbronn

in the context of Dirichlet Gfunctions. In [Il] a version of the Deuring-Heilbronn

phenomenon for &(s) has been proved. In the following theorem we generalize that

result to Artin L-functions.

Theorem 3.4 (The Deuring-Heilbronn phenomenon for Artin Gfunctions)

There are absolute, efective, positive constants CS, y such that the following holds:

Assume Artin's conjecture for E / K . If x zs irreduczble, and L ( s , X) has a zero

po (real or cornplex), then any other zero p = 0 + it of L(s, X) satisfies

Before we present the proof of Theorem 3.4, we have to pave the way by means

of a few lemmas.

Lemma 3.5 W e have

3.2 The Deuring-Heilbronn phenornenon for Artin L-functions Zero-tiee regions

< A2$i) ($ AXE -

PROOF: (i) Let V be the underlying representation space for X. Let V' be the dual

space of V. It would s d c e to show that for any subgroup H of G, dim V H = dim VH.

But the bilinear form

VH X vtH _t (C

(5, f * f (4

is easily seen to be non-degenerate on both sides, and hence dimvtH = dimvH.

2 x 1 ? (For the special case K = 0, this (ii) First we show that f x Z divides f,

has been proved in [21, Lemma 11. That proof dîrectly works for the general case as

well; for the sake of completeness we reproduce the proof-)

Let V be the underlying representation space for X. It would sufEce to show that

for any subgroup H of G

First observe that

d*vH = (xlH 1 1H)

Therefore (3.1) is equivalent t o

Zero-fiee regions 3.2 The Dezwhg-Heilbronn phenornenon for Artin ~-functions

where, in the sum, p runs through the nontrivial irreducible characters of X, and A.

and A, are nonnegative integers. Note that

Now, (3.2) can easily be seen to be equident to

The right side of (3.3) is negative only if for one p, A, = 1, ~ ( 1 ) = 1, and for all other

p, A, = O. In this case (3.3) reads -1 5 -1. In all other cases LKS 5 O 5 RHS.

Hence (3.3), and therefore (3.1), is proved. So our claim

is now established. In particulas

Therefore

(ii) is now proved.

3.2 The Ileuring-Heilbronn phenornenon for Arth L-funcitions Zerefiee regions

Lemma 3.6 Let u,v E @, &th lui, Ivl 2 1. Then for j 2 1

PROOF: We have

Finally, we quote [Il, Theorem 4.21 :

Then there exists jo, with 1 5 jo 5 24L, such that

We can now present

PROOF OF THEOREM 3.4: Let 9 be a character of G, not necessarily irreducible.

By our assumption of Artin's conjecture, (s - l ) ( v I 1 ~ ) ~ ( s , p) is entire. It is also of

order one, so we have the Hadamard factorkation

Zero-&ee regions 3.2 The Deuring-Heilbronn phenornenon for Artin L-functions

where r is the multiplicity of s = O as a zero of L(s, (o), and w nins through all the

zeros w # O of L(s, y), including the trivial ones. From (3.4) we obtain

By (2.10) on the other hand, we have

r a 2 - g L + L ) S - W W - - - S

valid for o > 1. Differentiating (3.5) and (3.6) 2 j - 1 times, we arrive at

1

( 2 j - l)! C C P ( q ) (1% &)(log ~p")~j- ' (Np) -ms

pECK m=l

Here w runs through a l l the zeros of L(p, X) (including the r trivial zeros w = 0).

(3.7) is valid for 0 > 1, j 2 1-

Now, let x be, as in the statement of Theorem 3.4, an irreducible character of G.

The case x = lG is equivalent to [Il, Theorem 5-11. Therefore we assume x # lG.

W e are going to use (3.7) with p = ( lG + x)(lG + z). Note that

3.2 The Deuring-Heilbronn phenornenon for Artin L-functions Zero-fiee regions

From (3.7) applied to (lG + x)( lG + X) we get

1

( 2 j - II! ECK C m=l C ((1~ + x ) ( ~ c + x ) ) ( ~ ~ ) ( ~ O ~ N P ) ( ~ O ~ N P ~ ) ~ ~ - '

where s = D + it, and z, are of the form (O - w ) - ~ or (s - w ) - ~ , with w a zero

of L (s, (lc + x)(lc + x) ) = k ( s ) L(s, x)L(s, X ) L(s, x x). (Remember that we are

assuming that p, is a zero of L(s, x), and therefore Po is a zero of L(s, X).)

The real part of the lefi side of (3.8) is nomegative, so if we choose o = 2 (and

so s = 2 + it),we obtain

(In the last inequality, we have used Lemma 3.6, and the choice CT = 2 guarantees the

assumption of the lemma.)

We are now ready to complete the proof of Theorem 3.4. Suppose that p =

,û + +y # po is a zero of L(s , x). In (3.8) we set t = 7, and apply Lemma 3.7 to the

left side of (3.9). In accordance with the notations used in Lemma 3.7, we have

Zero-&e regions 3.2 The Deuring-Heilbronn phenornenon for Artin L-functions

(3-2 is conilng fkom the fact that L(s, X) has a trivial zero at s = O or s = -1. We

have put 3-2 there to take care of the possibility of p = ,û being a trivial zero with

-0 large.) And therefore

where

w ninning through the zeros of L(s, x), and similarly for S(lc), S(X), and S(xX).

Since multiplicity of any trivial zero of L(s, X) is 5 nKx(l), the contribution of

the trivial zeros to S ( X ) is « nKx(l). The contribution of the nontrivial zeros is

estimated by using Lemma 2.l(i) giving

This, t oget her wit h Lemma 3.5, implies

(3.10) and (3.11) give us

3.2 The Deuring-Heilbronn phenornenon for Artin L-functions Zerefkee regions

Lemma 3.7 now says that there exists jo with 1 5 jo 5 24L, such that

This and (3.9) imply that

Since jo < L « A, for some c > O, jo < CA. Let also c' be the constaot implied by

<< in (3.12). Then (3.12) implies

and ha l ly this gives

1 We set CS = 2;, cd = 3. Theorem 3.4 is now proved. DI

Theorem 3.4 c m be complemented by the following result, which gives a lower

bound for the distance between the exceptional zero and s = 1.

Proposition 3.8 There are absolute, effective, positive constants CS, ce such that the

following holds:

A s s u m e Artin's conjecture for E / K . If x zs irreducible, and po is a zero of L(s, x),

then

(ii) if ~ ( 1 ) > 1 (or ~ ( 1 ) = 1, but x2 # lG), then 11 - pal 2 c6

x(113 1% A, '

Zero-fiee regions 3.2 The Deuring-Heilbronn phenornenon for Artin L-functions

PROOF: (i) There is always a trivial zero 2 -2, and therefore Theorem 3.4 implies

t hat

where A = A(0, X) « ~ ( 1 ) log A, = log A,- Assume (1 - po( < We will show

that for sufficiently large (but absolute) CS, we can reach a contradiction-

(3.13), with 11 - po 1 < KCS , implies

But for cg » 1, (3.14) contradicts the fact that A « log A,. This proves (i).

(ii) This follows immediately hom Proposition 3.2. O

Remark 3.9 Let us here make it more explicit how any zero of L(s, X) , with x irreducible, which is very close to s = 1, pushes other zeros further away hom c = 1.

Let po be a zero of L(s, X) with

where X > O is small. Then Theorem 3.4 implies that any other zero p = a + it of L(s, X) with Itl 5 1 satisfies

where c g l , are absolute, effective, positive constants.

3.2 The Deeuring-Heilbro~ phenornenon for Artin L-functions Zero-free regions

Now, if X satïsfies

then (3.16) implies that

Note that is the radius defining the exceptional zero. Cornparison of ~ ( 1 ) 1% A,

(3.15), (3.17), and (3.18) shows the effectivity of Theorem 3.4.

Chapter 4

The Least Character Nonresidue

[The notations used in this chapter are in W conformity with those introduced in

the h s t chapter. For convenience we recall the definitions of "character nonresidue",

and "the least character nonresidue" p(x) .]

4.1 Introduction

Definition 4.1 Let p E CK be X-unramified. We Say that p is X-nonresidue if

0, $ ker X.

Note that, although the Frobenius class q, is defined only mod1, (where 1, is the

inertia group at a prime q E C E above p), but since p is assumed to be X-unramified

(i.e. 1, 5 ker X) , the condition a, 4 ker x is well-defined. Also note that the condition

cp $ ker x is equivalent to ~(o,) # ~ ( 1 ) -

4.2 Bounds on the Ieast character nomesidue The Least Character Nonresidue

Definition 4.2 Let x be a character which is not a multiple of the trivial one. By

"the least X-nonresidue" we mean

P ( X ) = min{& p E CK X-nomesidue, and of degree one over Q) .

Note that, as the minimum is taken over primes of degree one, 'the least X-

nonresidue" p ( x ) is actually a rational prime.

Our goal in this chapter is to h d good upper bounds for p(x). This investigation

is one further step in a classicd trend:

(a) If we take K = Q, E = a quadratic extension of K, we get the problem of

hding upper bounds on the least quadratic nonresidue.

(b) If we take a general (Galois) extension E / K , but choose one dimensional X, we

get the problem of füiding upper bounds on the least character nonresidue in

the classical sense.

4.2 Bounds on the least character nonresidue

We are going to present two theorems giving bounds on the least character nonresidue.

The first one is a generalization of [14, Theorem 11, and [Il, Theorem 1-21, which are

dealing with the particular cases of {K = Q, G Abelian), and Abelian X, respectively.

In this result, the bound on the least character nonresidue depends on the size of a

hypothetical zero-free region around s = 1 for C&) L(s, X, EIK).

Our second theorem gives a bound on the least character nonresidue uncondition-

ally. Here we generalize the method of [Il, Theorem 1-11, which gives a bound on the

least prime in a conjugacy class.

Theorem 4.3 ir7tere is an absolute, e#ective constant c7 > O such that the follouing

The Least Character Nomesidue 4.2 Bounds on the least daracter n o m i d u e

holds:

Let -1 5 a 5 1 be a parameter, and set

Let x be a character o f G not containing the trivial one, and assume Artin's conjecture

for L(s , X) . If Ç&)L(s, X) dues not vanish in the region R(a, X) defined by

(where cl 2s the absolute constant given

C l Cl log log A, ' ~(l)~logA, by Proposition 3.1), and if CK(o) # O for

Theorem 4.4 There is an absolute, effective constant CS > O such that the following

holds:

Let x be an irreducible character of G with ~ ( 1 ) > 1. Assume Artin's conjecture for

E l K . If the exceptional zero of CK ( s ) does not exist, then

We will prove Theorem 4.3 and Theorem 4.4 in the next chapter. The rest of this

chapter wiJJ be devoted to a few corollaries and remarks.

Corollary 4.5 Let x be a nontrivial character of G with ~ ( 1 ) = 1- If CK(s)L(s, X)

4.2 Bounds on the Ieast character nomesidue The Least Character Nonresidue

does not vanish in the region

Cl 1-- Cl < o c 1 O # I t l < - (1 + Cl log log A,) ,

1% A, 1% A,

and i f the ezceptional zero of CK(s) does not exist, then

(where c7 is the absolute constant given by Theorem 4.3).

PROOF: Under the assumptions of the coroUary, all conditions of Theorem 4.3 are

met: L(s, X) is entire by Art in's reciprocity, and as log dK 5 log A,, the real segment

1-- < a < 1 covers the segment 1 - - < a < 1 (where we need to know 1% d~ 1% A,

CK does not vanish). (Note that the parameter ai plays no role in the case ~ ( 1 ) = 1.)

Therefore by Theorem 4.3, p ( x ) 5 4-

CorolIary 4.6 Let x be a cha~acter of G not containing the trivial one, and assume

Artin's conjecture for L(s, x). Assume also that the exceptional zero of CK(s) does

not exist.

(2) If f ~ ( s ) L(s, X) does not vanish in the region R(1, x), then

p(x) 5 ex(1)

(ii) If & ( s ) L(s , X) does not uanish in the region R(0, x), then

P(X) 5 A;:

The Least Character Nomesidue 4.2 Bounds on the least character nomesidue

(iii) If &(s)L(s ,x) does not vanish in the region R ( - l , ~ ) , then

(The region R(a, X) 2s d e f i e d in Theorem 4.3.)

PROOF: (i),(ii), (iii) are the special cases a = 1, a = O, a = -1 of Theorem 4.3

respectively- Note that, as log dK 5 - log the açsumption of nonexistence of the x(1)

exceptional zero of <&) is strong enough to satisfy the needs of all three cases.

Remarks 4.7 (a) Here we compare Theorem 4.3 with Theorem 4.4. The case cr = 1

in Theorem 4.3 (i.e. part (i) of Coro l lq 4.6) is closely related to Theorem 4.4. Yet,

neither of the two results implies the other: Theorem 4.4 does not imply the case

a = 1 in Theorem 4.3, because Theorem 4.4 has the restriction of irreducibifity of

x (as well as ~ ( 1 ) > l), while Theorem 4.3 is £ree in this respect. On the other

hand the case a = 1 in Theorem 4.3 does not imply Theorem 4.4 either, because

the hypothetical zero-free region R(1, X) is slightly taller than what we know hom

Proposition 3.1.

(b) Note that Corollazy 4.5 covers the missing case of Abelian x in Theorem 4.4.

(c) The condition in Theorem 4.3 that x does not contain the trivial character is

essential: e-g. for x = lc, there is NO X-nomesidue at all!

Chapter 5

Proof of the Main Theorems

In this chapter we prove Theorem 4.3 and Theorem 4.4. We will actually prove a

slightly stronger (and more complicated!) version of Theorem 4.3. This stronger

version, Theorem 5.5, will in particular enable us to prove Corollary 5.6, a result

assiim;ng the GRH.

5.1 The general set-up

In this section we set up the generalities needed for proving both Theorem 5.5, and

Theorem 4.4.

First we introduce a kernel function (which is the same as the one used in [Il],

[14], [17]). Let x, y be two parameters (to be chosen in terms of other data of the

case under consideration later) with 1 < x < y. The kernel function is

Proof of the Main Theorems 5.1 The general set-up

We define k by

Since all the cases are similar, we just prove the case x2 2 u 5 xy. We do this by

using the well-known discontinuous integral

where c > O (see [5, pp.104-1051). By (5.4) we have

5-1 The nenerd set-UR Proof of the Main Tbeorems

Similarly we see that

Therefore

1 y2 2 xy = -log- - - log -

U '11 21 U

(5.3) is now established.

Note that for ail u > O

Next, let E/K be a Galois extension of number fields, G the Galois group, and x a character of G. We set

From the Euler product (2.10) of L(s, X) we obtain

Findy, we describe the general o u t h e of the type of arguments used in proof of

results giving upper bounds on least primes of specified types.

First observe the h c t that the double sum in the right side of (5.7) is ac tudy a

finite s u : the only pairs (p, m) which have nonzero contribution in (5.7) are those

Proof of the hiain Theorems 5.2 The lemmas

with

and for any rational prime power pr, there are at most n~ primes p E CK for which

there exists an m with Npm = pr . O n the other hand, the integral (5.6) definhg I ( x )

can be written in terms of the zeros (and the possible pole at s = 1) of L(s, X ) by

Cauchy's theorem. Of course the sharpness of the information we get from this is

directly proportional to the depth of our knowledge about the location and density

of the zeros. The essence of all aguments is that, by choosing the parameters x, y

properly, the denial of the theorem would contradict (via (5.7)) the information about

I ( x ) gained through (5 -6) and zero-analysis. (To reach this contradiction we may need

to work with certain I (@) with # x as well.)

5.2 The lemmas

The present section is devoted to a few preparatory lemmas. The f%st two estimate

the contribution to the sum in the right side of (5.7) made by primes p E CK which are

X-ramified, and by pairs ( p , m) for which Npm is not a rational prime, respectively.

The last lemma relates I ( x ) to the zeros of L(s, x).

Lemma 5.1 We have

5.2 The lemmas Proof of the &fain Tbeorems

PROOF: We recd kom the section 1.3, that p E CK is X-ramified iE p divides the

Artin conductor f,. Using (5.5), we have the left side of the above

1 - < 2 ~ ( 1 ) ~ x 6% %) (1% A,) -

Lemma 5.2 We have

PECK mrl N p m not a rationai prime

PROOF: There are at most n~ primes p E CK dividing any rational prime p. Using

this fact , and (5.5), we have the left side of the above

Proof of the Main Theorems 5.2 The lemmas -- --

To justify the last inequality, we are going to show that

1 C P - ~ « - - x log x

P € ~ Q

We have

and the two sums on the right are estimated as follows:

This proves (5.8), which completes the proof of Lemma 5.2.

Lemrna 5.3 Let C* denote sum over those p E CK whzch are x-unramified and of P E C K

degree one over Q. We have

PROOF: This follows immediately from Lemma 5.1 and Lemma 5.2.

5-2 The Iemmas Proof of the Main Theorems

Lemma 5.4 Assuming Artin's conjecture for L (s, X) , we have

where p runs through al1 zeros of L(s, x).

PROOF: First note that k(s) is an entire hinction: the singulazity a t s = 1 is

removable, by definhg

Let B (T, N) be the (positively oriented) rectangle with vertices at 2 + iT, -N + iT, - N - iT, 2 - iT. Here T > 2, not equal to the absolute value of the irna,ggary part

of any zero of L(s, X) , and N = + a positive integer. We set

B y Cauchy's theorem

Therefore the lemma is proved if we show that

This is equivalent to showing that the contribution made by the two horizontal sides

and the left side of B(T, N) to the inteqal in (5.10) approaches zero as T, N -+ m.

Proof of the Main Theorems 5.2 The Iemmas

First s e need a gaod bound on ICI. We have the bound (2.8) on IL . - . .

The hct ional equation (1.2) (or (2.4)) translates that bound to a bound on

a < O as follows.

From (2.2), (2.3), (2.4) we get

From (2.8), (2.14), (5.12) we obtain

valid for a 5 -a with 1s +ml 2 a for all integers m 2 0.

From (5.13) it easily follows that as T, N approach idhi ty (with T # the absolute

value of the imaginary part of any zero of L(s, x), and N = $+ a positive integer),

the int egrals

all go to zero. It remains to show that the integrals

5.3 Proof of Theorem 4-3 Proof of the Main Theorems

&O go to zero when T -+ m. This easily follows fiom the fact that, by (2.20))

k (c+iT ) an easy estimation of da, for p with 17 - T 1 5 1, and recalling from

-L (c +iT) - p J. Lemma 2.1 that the rider of such p is

(see [12, Lemma 6-31). This completes the proof of Lemma 5.4.

5.3 Proof of Theorem 4.3

In this section, we state and prove Theorem 5.5, which is slightly more general than

Theorem 4.3. The parameter 6 in Theorem 5.5 below will enable us to prove Corol-

lary 5.6, which gives a bound on the least character nonresidue p ( ~ ) assuming the

G W .

Theorem 5.5 generalizes [I l , Theorem 1.21 both contextwise (from Abelian char-

acters to general characters) and contentwise (by introducing the parameter a).

Here we outline the proof. The essence of the argument is the comparison of I ( x )

with I(lG). We will show that under the assumptions of the theorem, ReI(x) is

srnall, while I(lc) is (positive and) large, due to the pole at s = 1 of &(s) If the

conclusion of the theorem were false, then I ( x ) and x(1)1(l6) would be closer to each

other than what they actually are. This contradiction will then prove the result.

Theorem 5.5 There zs an absolute, effective constant cg > O such that the followilag

holds:

Proof of the Main Theorems 5.3 Proof of Tbeorem 4.3

Let -1 5 a! 5 1 be a parameter, and set

Let x be a character of G not containing the trivial one, and assume Artin's conjecture

for L(s, x). Fznally, let 6 be another parameter related to a by

where CI is the absolute constant giuen by Proposition 3.1.

If CK(s)L(s, X) dues not uanish in the region Ri(6,x) defined by

Before we prove Theorem 5.5, we present the

PROOF OF THEOREM 4.3: In Theorem 5.5 we take

It is easily seen that

5.3 Proof O£ Theorem 4.3 Proof of the Main Theorems

and with the choice (5.14) for 6

where cg = + I'og(clcg)J. This proveç Theorem 4.3. Cl

where p runs over all zeros of L(s , x). For any real zero p, -k(p) is negative, and

t herefore

Our first goal is to bound Re I ( x ) , and by (5.16) it is sufEcient to find a bound for

C Ik(p) 1, where the sum is over nonred zeros, and in particular trivial zeros as well f74W as the exceptional zero (if it is real) have no contribution to it.

For aûy zero p = p + ir we have

Proof of the Main Theorems 5.3 Proof of Theorern 4.3

We divide the strip 1 - 26 5 o 5 1 - 6 into squares of side length 6 (we owe this

idea, which is better than integration, to [14, p.1661). By Lemma 2.i(ii), and (5.17),

we have the estimate

We sum (5.19) over m 2 O to get

We now divide the strip 1 - 6 < 0 < 1 into squares of side length 6. The

contribution of the zeros in the jth square, where j # O, again by Lemrna 2.1(ii) and

(5.17), is

Now, by the assumption of the theorem, we have L(s, X) + O for

5.3 Proof of Theorem 4.3 Proof of the Main Theorems

where

Therefore, (5.2 1) implies

Combining (5.16), (5.20), and (5.23) we finally get

We have therefore shown that Re I (x ) is "small" . The next step of the proof is to

obtain an estimate for I( lG). This time Lemma 5.4 gives

where p~ runs over a l l zeros of CK(s ) The contribution of the real zeros (both trivial

and nontrivial) is estimated as follows. First note that for any real zero p~ = pK, by

the assumption of the theorem, ,BK 5 1-6. Hence (5.17) and (5.20) (whose derivation

Proof of the Main Theorems 5.3 Proof of Theorem 4.3

did not use the fact that the sum was over nonreal zeros) imply that

PKEW PK trivial PK=BK o<@~sl-s

« n ~ x - ~ + x-266-2 + x-266- log dK .

The contribution of the nonreal zeros is estimated by (5.20) and (5.23), where J is

the same as the previous, and since ~ ( 1 ) log dK 5 log A,, fiom (5.25) and (5.26) we

obtain

Now, we assume that

This means that ~(o,) = ~ ( 1 ) for aJ X-unramified primes p E CK of degree one over

Q with Np 5 y2. Using the notation " Cu " of Lemma 5.3, we therefore have PECK

and hence Lemma 5.3 implies

and since I(IG) E R,

5.3 Proof of Theorem 4.3 Proof of the Main Theurems

Now, we choose x and y to be

where C is a sufEciently large (yet absolute and

y = x c i ,

effective) constant .

Let co be the sum of the constants implied by « in (5.24), (5.27), and (5.30). We

are going to show that

where the seven terms Ti, coming from

given in a moment. But (5.24), (5.2?),

contradiction will prove the theorem.

The terms Ti are

We recall that

the right sides of (5.24) and (5.30), will be

and (5.30) cooperatively deny (5.32). This

Proof of the Main Theorems 5.3 Proof of Theorem 4.3

and

hence

The hinction 6 c, (M)+ attains itç maximum at 6 = f. This fact, with X =

CX(1)& CL log A,, implies that on the intenal given by (5.33), x as given by (5.31)

is a decreasing function of 6, because

Therefore

CI z 2 -X(l)" log A, .

2c1

Also, note that on the interval (5.33), z2 > cg, and so y 5 z3. Hence

logy < 1 . log x

We are now ready to prove (5.32). First note that the left side of (5.32) is

~ ( i ) (Log :) = (log ~ ) ~ 6 - ~ x ( 1 ) -

5-3 Proof of Theorem 4.3 Proof of the Main Theorems

The terms Ti, 1 5 i 2 7, on the right side of (5.32) are estimated as follows.

By (5.22), J 2 1, hence

Therefore

Proof of the Main Theorems 5.3 Proof of Theorem 4.3

For TG, we use (5.34), and then (5.33) to get

I Y 4 4 Ta = ~ ( 1 ) (log Ax) -p log G < -(log C2 C ) P X ( l ) '-"(log A,) -'

Finally for Ti, using (5.34) and (5.35) we have

Note that the constants implied by « used in the above estimations for Ti, are aU

absolute and in particular independent of C. From these estimations for Ti, we see

that, with su£Eciently large Cl (5.32) is proved. The contradiction mentioned above

shows that the assumption (5.28), with the choice (EU), is false. Therefore

Setting cg = - cl we have completed the proof of Theorern 5.5.

Corollary 5.6 There is an absolute, effective constant clo > O such that the following

holds:

5.4 Proof of Theorern 4-4 Proof of the Main Theorems

Let x be a character of G not contazning the trivial one. Assuming both Artin's

conjecture and the GRH for CK (s) L(s, X) , we have

1 PROOF: Theorem 5.5 is applicable with a = -1, 6 = 5. Setting cl0 = $4, we pet

the corollary, 0

5.4 Proof of Theorem 4.4

In this section we prove Theorem 4.4. The method of the proof will be simüar to that

of Theorem 5.5. This time we are not assuming a zero-free region (except we assume

that the exceptional zero of k ( s ) does not exist), but we have the restriction that x is assumed to be irreducible of degree > 1.

PROOF OF THEOREM 4.4: Set

where, as usual, cl is the absolute constant given by Proposition 3.1. Proceeding as

in the proof of Theorem 5.5, we get to (5.24), with 6 = 6,, J = 1, except we have the

contribution of (the possible) exceptional zero po of L(s, X) (which is not known to

be real!) as well:

Proof of the Main Theorems 5.4 Proof of Theorern 4.4

Now we estimate 1 k(po) 1. We have

(The inequality in (5.38) cornes fiom the fact that

Rom (5.37), (5.38) we obtain

(Note that the "actual" coefficient of the term Ik(po) 1 in (5.37) is one, and ql is an

absolute effective constant .)

Next, we estimate I ( lG) . As we are assuming that the exceptional zero of G ( s )

does not exist, we get (5.27), with 6 = i fK, J = 1:

where the last term is estimating the contribution of the trivial zeros of CK(s). Rom

(5.36), (5.40) we get

5.4 Proof of Theorem 4.4 Proof of the Main Theorems

and therefore

with c22 absolute effective.

Now, we assume that

Again this means that ~(a,) = ~ ( 1 ) for ail X-unrarnified p E CK of degree one over

Q with Np 5 Y2, and therefore

and Lemma 5 -3 gives

and since l(lG) E R,

with c23 absolute effective.

Now, we choose x and y to be given by

2 log x = Cx(1) log A, , y = x ,

Proof of the Main Theorems 5.4 Proof of Theorem 4.4

where C is a sutticiently large (yet absolute and effective) constant.

We set c24 = ~ 2 1 + c22 + ~23. We are going to show that

1 1% Y Y) + (log :)2 . + ~ K X ( I ) ; ~ log ;

But (5.39), (5.41), and (5.44) deny (5.46). The contradiction proves the theorem.

Since we are assuming ~ ( 1 ) > 1, we have ~ ( 1 ) - 1 2 k x ( 1 ) - Therefore (5.46) is

proved if we show

Note that the left side of (5.47) is

The fbst term on the right side of (5.47) is therefore dominated by the left side. For

the other two terms on the right side we have

This proves (5.47), and therefore (5 -46). The above mentioned contradiction shows

that the assumption (5.42), with the choice (5.45), is falçe. Therefore

5.4 Proof of Theorem 4.4 Proof of the Main Theorems

Setting ca = 4C, we have established Theorem 4.4. 0

Remarks 5.7 (a) The restriction ~ ( 1 ) > 1 in Theorem 4.4 can be removed if we

add the assumption that the exceptional zero of L(s, X) does not exist either. This

is because the only time we used the assumption ~ ( 1 ) > 1 was when we needed to 2

dominate the term (log :) on the right side of (5.46), and that term is contributed

by the (possible) exceptional zero of L(s, x).

(b) The assumption of irreducibility of x was used in the proof of Theorem 4.4,

just to ensure the assumption of Proposition 3.1.

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