The correction factor in Artin's primitive root conjecture

P

ETER

S

TEVENHAGEN

The correction factor in Artin’s primitive root conjecture

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 383-391

<http://www.numdam.org/item?id=JTNB_2003__15_1_383_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The

correction factor in

Artin’s

_primitive

root

_conjecture

par PETER STEVENHAGEN

RÉSUMÉ. En

_1927,

E. Artin

_proposait

une densité

conjecturale

pour l’ensemble des nombres

premiers

p pour

lesquels

un entier

donné g

est une racine

primitive

modulo _p. Des calculs effectués en 1957 par D. H. et E. Lehmer ont mis en évidence des écarts

inat-tendus par rapport à cette densité

conjecturale,

incitant Artin à _y introduire un facteur correctif. La

_conjecture

ainsi modifiée a été

prouvée

par C.

Hooley

en 1967 conditionnellement à

l’hypothèse

de Riemann

_{généralisée.}

Cet article a _pour

_sujet

deux

développements

récents relatifs

à ce facteur correctif. Le

_premier

est de nature

_historique,

et se

rapporte à des

lettres,

récemment découvertes à

Berkeley

dans les archives des

Lehmer,

entre Artin et les Lehmer durant les années 1957-58. Le second concerne une nouvelle

interprétation

du

fac-teur

_correctif,

due à H. W.

_Lenstra,

P. Moree et

_{l’auteur, qui}

_peut

s’appliquer

à de nombreuses

_{généralisations}

au

_{problème originel}

d’Artin.

ABSTRACT. In

_1927,

E. Artin

proposed

a

conjectural

_density

for

the set of

_primes

p for which a

_{given integer g}

is a

_primitive

root

modulo p. After computer calculations in 1957

_by

D. H. and E. Lehmer showed

_{unexpected deviations,}

Artin introduced a

correc-tion factor to

_explain

these

discrepancies.

The modified

_conjecture

was

_proved

_by

Hooley

in 1967 under

_assumption

of the

generalized

Riemann

hypothesis.

This _paper discusses two recent

_developments

with _respect to

the correction factor. The first is of historical nature, and is based

on letters between Artin and the Lehmers from 1957-58 that were

discovered in the Lehmer archives in

_Berkeley

in December 2000. The second concerns a new

_{interpretation}

of the correction factor

in terms of local contributions

_by

H. W.

_Lenstra,

P. Moree and the author that is well-suited to _{deal with many}

_{generalizations}

In a conversation with Hasse that took

_place

in

_September

_1927,

Emil Artin

conjectured

that the set

_~9

of

_primes

_p for which a

_{given integer g}

E Z that

is not an exact _power is a

_primitive

root modulo _p is

_infinite,

and that it has a natural

_density

_equal

to Artin’s constant A =

I1q

.3739558. In

_particular,

the

_density

should be

_independent

of _g.

The heuristical

_argument

_underlying

Artin’s

_conjecture

is

_simple

and well-known. A

_prime

number p

{

2g

is in

_Eg

if and

only

if there is no

_prime

number _q

_dividing

_{p - 1} such

_{that g}

is a

_q-th

_power modulo _p.

Writing

~q

for a

_{primitive q-th}

root of

_unity

and

_{Kq =}

_Q((q,

for the

_splitting

field of

_{xq - 9}

over

_Q,

this condition means that for no

_{prime q}

_p,

the

_prime

p

splits

completely

in the number field

_Kq.

It follows from the

Chebotarev

_density

theorem

_(or

one of its 19th

century

predecessors,

see

[9])

that for fixed _q, the set of

_primes

_p that do not

_split

_completely

in

Kq

has

_density

assume the

splitting

conditions in the

l

various fields

_.K

to be

_{’independent’,}

it seems reasonable to

_expect

_Eg

to

have

_density

_I1q

_{K 1.~ ) ~}

_{If g is}

not an exact _{power in}

_Z,

one has

[Kq :

Q]

- q (q -

1)

for all

_primes

_q and

we

obtain Artin’s constant. For

arbitrary g =,A

:f:1,

there is a

_largest

_{integer h}

E Z for

_{which g}

is an h-th

power in

Z,

and the

conjectured density

becomes the rational

multiple

of Artin’s constant A = Note that

Ah

vanishes if and

_only

if h is _even,

i.e.,

if g

is a _square; in this case it is clear

_{that g}

does not

_generate

_Fp

for

a

_single

odd

_prime

_p.

2.

_History

of the

_conjecture

In the

_1930s,

there were various efforts to _prove Artin’s

_conjecture.

Deal-ing

with

_splitting

conditions in an infinite number of fields

Kq

proved

to be too hard for Hasse and his student Bilharz. In

_1935,

Hasse learned via

Davenport

that Erd6s believed to have a

_proof,

and he turned to the anal-ogous

problem

for the field

k(X)

of rational functions over a finite field k.

Erd6s’s

_’proof’

turned out to

_depend

on the

_generalized

Riemann

hypothe-sis,

and on a statement in the distribution of

_prime

numbers he was unable

to

_justify.

Hasse and Bilharz were successful in

proving

Artin’s

_conjecture

for

_{I~(X ),}

and even for

arbitrary

function fields in one variable over k. In

these easier _cases, the Riemann

_hypothesis

was known to hold or was

_being

established at the same time

by

A. Weil.

The

_{original conjecture}

remained _open, and was discussed

_by

Hasse in his 1950 textbook

_[3]

and in a 1952 _paper

[4].

In

_1957,

Derrick H. Lehmer and

his wife Emma

_numerically

tested Artin’s

_conjecture

for various values of

g on a

_computer

and sent a table with their

findings

to Artin in Princeton. This is the

_beginning

of a short and

amusing

exchange

of letters

[2]

that were

discovered in December 2000 in the Lehmer archives in the Bancroft

_library

in

_Berkeley.

In a letter dated December

_19,

_1957,

Artin writes to Emma Lehmer that he had tested his

_conjecture

at the time he made it

_using

tables of

_Kraitchik,

_{and goes} on to

_explain

in detail the heuristic

_argument

given

in our introduction. As for the references to the literature Emma Lehmer had asked

_for,

Artin has ’no idea’ but ’Hasse who is very much

interested should

_know’,

and Hasse’s mail address follows. D. H. Lehmer

replied

on New Year’s Eve of 1957 that he and his wife had

thought

that

Artin had obtained his constant from the

_expression

A

_{= ¿p}

_prime

P-1

I

which counts for each _p the fraction of elements that are

_generators

of the

finite field

_F~,

and he asks for a reference for the

density

statement for

the

_primes

_splitting

in

_Kq.

Artin’s

_January

6 reaction consisted of three

densely

handwritten _pages

_starting

as follows:

Dear Professor Lehmer:

Since you are interested in the _density of _primes connected with the factorisation of _polynomials I would like to stress the fact that the root

of these _{questions belongs} to

_algebraic

number _theory and should be viewed from this _point of view. _{Any interpretation} in terms of

elemen-tary number _theory hides very essential insights into the nature of the questions. If you have the patience to _study the

_following

_explanations

I should think that _{you will agree. As} a matter of fact one knows the

most _{general density} law of this nature.

A crash course in

_algebraic

number

_{theory leading}

_{up to} the Chebotarev

density

theorem follows.

_By

the end of the

_month,

on

_January

_{28, 1958,}

a

fourth letter follows. It is

_{interesting enough}

to be

_quoted

in

_full,

as it is the birth certificate of the correction factor in Artin’s

_conjecture.

Dear Mrs Lehmer!

Your run _{away 5 started} me _{thinking again} and I can tell _you the rea-son. Please return to _my first letter. Let us first consider the _primitive

root 2

_(that’s

all I did when I made my

conjecture).

Consider the field Kq =

_{R((q ,}

_{which has degree}

_{q(q - 1)}

_{and accounts} for the factor

1 - in the formula

_((q

_{primitive q-th} root of

_unity).

There is no

difficulty showing that a finite number of the _{probabilities} are

indepen-dent. We need however an infinite number and that is _extremely hard.

That a finite number is _independent means _merely that the fields _Kq have no common _{subfield # R,} that the Galois group of a _compositum

is a direct _product.

But _replace now 2 _by a _{prime p ==} 1 mod 4. Then I was _extremely

hasty. Namely K2 =

R(-1, vFp)

=

R(Jfi)

is a subfield of

R((p)

hence of _Kp.

_(This

is the _{only dependence} that occurs between the

_Kq,

but

prime splits then _{automatically} in the subfield Since another condition was not to _split in K2 =

R(Jfi) ,

a _prime which satisfies this

will _already not _split in _Kp. In _elementary terms: if p is quadratic non-residue of a _prime then X p - p cannot split into linear factors modulo

that _prime.

This means _merely that the factor

_{1 - p~p 1~}

in the _product has to

be _dropped, for _p = _{5 that means} that the _expected value is by the

factor

_20/19

_larger. I turn to your results. In the column 16 - 103 you

give as _expected value 696 which is very _good for the primitive root

2. For 5 it has to be _{replaced by} 733 which is close _enough to 744. In the column 23 - 103 the _expected value is 959 and is _good for _2, for 5 it should be 1010 which is _{again very good. For p - 3 mod} 4 _nothing

has to be

_changed

and for _composite numbers I did not work out the change. In the column 20 ~ 103 you get for 5 the value _890,

_right

on the

dot of the _expected value 890.

So I was careless but the machine _caught up with me.

Cordially, E. Artin.

The Lehmers mention Artin’s correction for

_{primes 9 ==}

1 mod 4 without attribution in their 1963 paper

[6],

but do not appear to have

digested

Artin’s crash course. Even Hasse

provides

a correction factor in his 1964 edition of

_[3]

that is incorrect

_{if g -}

1 mod 4 is not

_prime.

_Lang

and

Tate,

who were Artin’s students in Princeton in

1958,

list the correction factor

for

_arbitrary

_{non-square g with d} = 1 mod 4 in their 1965

preface

to Artin’s collected works

_[1]

as

Thus,

_Eg

should have

_density

_{E(d) .}

_{Ah for g}

_satisfying

d =

disc(Q(q§)) *

1 mod 4. If

K2

= is of _even

discriminant,

no correction factor is

necessary and the

conjectured density simply equals

the value

Ah

from

(1.1).

In

_1967,

_Hooley

_[5]

_proved

Artin’s corrected

_conjecture

under the

as-sumption

of the

_generalized

Riemann

_hypothesis.

The

_hypothesis

is used to obtain sufficient control of the error terms in the

_density

statements for the sets of

_primes

that

_{split completely}

in the fields

_Kq.

So

_far,

there is not a

_single

value

of g

for which one can show

_{unconditionally}

that E is

infinite.

-.

3. The correction factor

Hooley

attributes the correction factor

_(2.1)

to

_Heilbronn,

and derives it

from the inclusion-exclusion

_expression

the

_density

in terms of the

_degrees

of the

_splitting

fields

Kn

of over

_Q.

If the field

K2 =

_{Q( y0)}

is not

_quadratic

of odd

_{discriminant,}

then n -

_{~Kn : Q]}

is a

_{multiplicative}

function and the

expression

reduces to the

_{product Ah}

given

in

_(1.1).

If

_K2

is

_quadratic

of odd discriminant

d,

then

_{[Kn : Q]}

is no

longer multiplicative

as it

Q]

for all

squarefree

n that are divisible

_by

d. In this _case, a ’rather harder’ calculation

[5,

p.

219]

leads

to the correction factor

_E(d)

from

_(2.1).

We note

_{that, just}

like

_{Ah itself,}

the factor

_E(d)

has a

_{‘multiplicative}

structure’ of the form

_{E(d) =}

1 +

_Hq

_prime

_Eq,

with ’local factors’

_Eq

that

can be described in terms of

_~q

as

The local factors 1 -

_Q]

_going

into

Ah

have a clear

_{interpretation:}

they

represent

the fraction of the

_{primes having}

the

_’right’

local behavior in

_Kq.

The local factors

_Eq

from

_(3.1)

also admit an

_{interpretation}

of a

local

_nature,

as we will show now.

The

requirement

we

_impose

on our

_primes

_p in order to

_guarantee

that

g is a

primitive

root modulo _p is that _p does not

_{split completely}

in the number field

_Kq

for all

_primes

_{q p.} In other

_words,

the Frobenius

_symbol

of p

in

_Gq

=

Gal(Kq/Q)

has to lie in the set

_Sq

=

Gq B {1}

of non-trivial elements for these _q; this

_happens

for a fraction =

1-1/(Kq :

Q]

of

_primes

_p when we look at a

_{single prime}

_q. If the fields

_Kq

are

linearly

independent

over

_Q,

then for _any

_squarefree

number _n, the Galois group

Gn

=

Gal(Kn/Q)

is

simply

the direct

product

Gq

of the Galois _groups at _q, and we want the

_primes

_p that have Frobenius element in

_Sn

=

Sq.

Such _p

clearly

form a set of

_density

_Letting

n tend to

_infinity

as in

[7]

and

_{assuming GRH,}

we are left with a set of

primes

p

having density Ah

as in

(1.1).

As Artin

_observed,

the

_only

case where the fields

_Kq

are not

_linearly

independent

over

_Q

occurs when

K2

=

Q(..¡g)

is

_quadratic

of odd discrim-inant d.

_Indeed,

_K2

is then a subfield of

Q(~’d),

hence a subfield of

Kn

for all

_squarefree

values of n that are divisible

_by

d. This leads to a

differ-ence in the

_computation

of the fraction of

_’good’

Frobenius elements inside

G~,

=

Gal(Kn/Q)

for all

squarefree

_n that are divisible

_by

2d.

_Indeed,

suppose d is

odd,

and fix such an n. Then we have an exact _sequence

Here X

=

(Xq)qln

is the

_quadratic

character on G =

flql,,

Gq

having

as its

2-component

the

_isomorphism

X2 :

G2 ~

~~1~,

and

having

components

~q at odd

primes

q defined

by

XK2 = Xq. In other

words,

Xq is the non-trivial

_quadratic

character

_{modulo q}

at the

_primes

_{and xq}

= 1 _at

flqjn

8q

_rlqln

Gq

’good’

elements,

i.e.,

8n

= of

Gn .

The _{correction factor}

E {d )

arises as the factor

_by

which

differs from the Cuncorrected value’ =

Clearly,

we have

_#G

= _{2 .}

_#Gn-

We can relate

_#S

to

_using

a

character sum.

Observe

first that the

_{function 1(1}

+

_{x) :}

G ~

_{~0,1 }}

is the characteristic function of

_G~

on G.

Let

G -~

{O, 1}

be the characteristic

function of ~S’ =

ITqln

Sq

on G. is the characteristic function

of

_,S’n

on

G,

so we have

As the functions

_{1/;, X}

and

_#S

on G =

IIqln

Gq

are

simply products

of

functions _Xq and

_#Sq

defined on the

_components

Gq,

we can rewrite

this as

From

_(3.2)

we see that the

_quotient

of and

#S/#G

is indeed of

the form 1 +

_nqln

_Eq.

Note that this

_quotient

does not

_change

if we

replace

n

by

_any other

squarefree multiple

of 2d: it is constant if n tends to

infinity.

As

_{oq : Gq - {0,1}}

is the characteristic function of

_SQ

on

_Gq,

we have

i.e.,

Eq

is the _average value

_{of xQ}

on At all ’non-critical’

_primes

_q

f

2d,

we find

_{Eq =}

1 as _~Q is then the trivial character on

_Gq.

At the

primes

ql2d,

we are

summing

a non-trivial character _xq over the set

_,5’q

=

Gq ~ ~ 1 ~,

which

_yields

We find =

-11([Kq :

Q~ - 1)

in accordance with

_(3.1).

This

’explains’

the

_{multiplicative}

structure of the correction factor

_E(d).

4. Generalizations

As indicated in

_[7],

Artin’s

_original

_conjecture

on

_primitive

roots admits

many variants for which

Hooley’s proof

(under

GRH)

essentially

goes

through unchanged.

The main

_problem

becomes the

_{explicit computation}

of the correction factor that transforms the

_{product 6}

=

nq

of

given

works in

_{great generality,}

and in this final section we will formulate

a rather

general

result

along

these lines.

In the

_{original conjecture,}

we determined the

density

of the set

_Eg

of

primes

p with the

property

that for all

primes q

p, the Frobenius element

of _p in

_Gq

=

Gal(Kq/Q)

is in the subset

Sq

C

_Gq

of non-trivial elements. Here we have

Q(Cq,

for all q.

For our

generalization,

we allow

_{Kq and Sq}

to be different from the choice above for

_finitely

many

primes

q.

Thus,

for _every

_prime

q, we assume

that

_Kq

is an extension of

(a(~Q,

that is finite and normal over

Q

and

contained in the field

_Q((qoo,

obtained

_{by adjoining}

all _q-power roots

of 9

to

_Q.

We write

_Gq

=

Gal(Kq/Q),

and take for

Sq

any subset

of Gq

that is stable under

_conjugation.

For all but

_finitely

_{many q,} we _suppose that

we have

_Kq

=

Q((q,

9 g)

and

_Sq

=

Gq B

~1~.

The modified

_question

_now

becomes: what is the

_density

of the set E of

_primes

_{p t g}

with the

_property

that for _every

_prime

_{q p,} the Frobenius element

_{of p}

in

_Gq

=

Gal(Kq/Q)

is in

_Sq?

The modification we have

given

_{may appear} to be

only

slight,

but it

already

covers a number of

important

generalizations.

For

instance,

it

enables us to deal with

_primes

p that lie in a

prescribed

residue class or for

which g

generates

a

subgroup

of

given

index in

(Z/pZ)*.

The situation we are in is still _{very similar} to that of the

original

con-jecture.

As the fields

_QV9)

for odd

_primes

_q are

_{linearly disjoint}

over

Q,

the

only possible dependency

between the fields

Kq

occurs when

K2

contains a

_quadratic

field F of odd discriminant

d,

which is then

con-tained in

_Q((Idl).

As in the case of Artin’s

conjecture,

this leads to two rather different situations. If

_K2

does not contain a

_quadratic

field F of

odd

_{discriminant,}

then all fields

_Kq

are

_{linearly independent,}

and we find as in

[7]

that under

GRH,

the

density

of E

equals

the

product

of all local densities

_By

our

_assumptions,

this is a rational

multiple

of Artin’s constant. It vanishes if and

_only

if

_,Sq

is

_empty

for

some _q.

If

_K~

does contain a

_quadratic

field ~’ of odd discriminant

_d,

we take

X2 = XF the

quadratic

character of conductor

d,

and define Xq for the

primes

q &#x3E; 2

_by

xF - Xq. In

particular,

Xq is trivial for all q

~’

2d.

We let

-be the average value of Xq on

_,S’q.

_Clearly,

we have

_Eq

= 1

_{for q}

{

2d. Now

with 6 the

_product

of all local densities defined above.

_Thus,

the correction factor 1 +

_~9

_Eq

has

_exactly

the same

_shape

for these

_{generalizations}

as it

has for the

_original

Artin

_problem.

We finish this section

_by

a concrete

_example

that shows how the character

sum method is

applied

in

_{practice. Suppose}

that g

E Z not an exact _power,

and that F = has odd discriminant d. We _{want to} determine the

density

(under GRH)

of the set E of

_primes

_{P s} a mod

f

for

which g

is a

primitive

root.

Let

_f

= factorization of _our modulus

f .

Then _{we can}

make the standard choice

_Kq

=

Q(q,

and Sq

=

Gal(Kq/Q) B

{1}

at all

_primes

_{q f}

_{f .}

At these q, the local

density

has the standard value 1 -

_{1/(q2 - q)}

from

_(1.1).

At

_qlf,

we take

_Kq

=

{ý9)

and let the set

_Sq

of

_’good’

Frobenius elements be the intersection of the

_{’congruence}

set’ of elements

_raising

the root of

_unity

_(qeq

to its ath power and the ’Artin set’ of elements that are

non-trivial on the subfield C If we have

a ~

1 mod _q, then the first set is contained in the second and the local

_{density simply equals}

If we have a = 1 mod _q then it becomes

(1 - llq) -

as

a fraction

1/q

of the elements in the first set is outside the second. This

yields

a

product

of all local densities. As

_K2

contains the

_quadratic

field F = of

odd discriminant

_d,

there is a correction factor to be taken into account. At the

_primes

_ql2d

that do not divide

_f,

the local correction factor

_equals

Eq

= -1/(q2 -

q - 1)

as in

(3.1).

We have

E2

= -1 as _X2

equals -1

on

S2.

For

_{q~ gcd(d, f )}

the character _{X9 has} the constant value

_{Xq (a) =}

₍₂

on

_S9,

so we find

_Eq

=

(~).

This leads to the

_global

correction factor

and we find

_{(under GRH)}

that E has

density

E ’ 6 with E as in

(4.2)

and

6 as in

(4.1).

As the

_{preceding example illustrates,}

the character sum method reduces the calculation of the correction factor to a number of

_fairly

mechanical

directly

covered

_by

the

_{generalization}

considered in this

_section,

such as the case in which one

replaces

the base field

Q by

an

arbitrary

number field.

We refer the reader to

_[8]

for a fuller treatment.

Acknowledgements. I thank Pieter Moree for his _{encouragement} in _getting hold of the letters in

_[2]

and for _providing some of the historical details in section 2, and the Bancroft _Library in _Berkeley for its courteous _help in _accessing the Lehmer archives.

References

[1] E. ARTIN, Collected papers. ed. S. _Lang, J. T. Tate, Addison-Wesley, 1965.

[2] E. ARTIN, D.H. LEHMER, E. _{LEHMER, Correspondence} 1957-58. Archives of D.H. Lehmer,

Bancroft _{Library, Berkeley.}

[3] H. HASSE, Vorlesungen über Zahlentheorie, Akademie-Verlag, 1950

[4] H. HASSE, Über die Artinsche _Vermuting und verwandte Dichtefragen. Ann. Acad. Sci. Fen-nicae Ser. A. I. Math. _Phys. 116 _(1952).

[5] C. HOOLEY, On Artin’s conjecture. J. Reine Angew. Math. 225 _(1967), 209-220.

[6] D. H. _LEHMER, E. _{LEHMER, Heuristics,} anyone?. Studies in Mathematical _Analysis and Related Topics, Stanford University Press, 1962.

[7] H. W. _{LENSTRA, JR,} On Artin’s _conjecture and Euclid’s _algorithm in _{global fields.} Invent. Math. 42 _(1977), 201-224

[8] H. W. LENSTRA, JR, P. _MOREE, P. _STEVENHAGEN, Character sums _{for primitive} root

densi-ties, in _preparation.

[9] P. STEVENHAGEN, H. W. _{LENSTRA, JR,} Chebotarëv and his density theorem. Math. Intellig.

18 _(1996), 26-37. Peter STEVENHAGEN Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden The Netherlands pshomath . leidenuniv , nl