The distribution of the values of a rational function modulo a big prime

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

LEXANDRU

Z

AHARESCU

The distribution of the values of a rational

function modulo a big prime

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 863-872.

<http://www.numdam.org/item?id=JTNB_2003__15_3_863_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 distribution of the values of

a

rational

function modulo

a

big prime

par ALEXANDRU ZAHARESCU

RÉSUMÉ.

Étant

donnés un

_grand

nombre

_premier

p et une fonc-tion rationelle

_r(X)

définie sur

_Fp

=

Z/pZ,

_on évalue la

_grandeur

de l’ensemble

_{x

~

_{Fp :}

&#x3E; +

_1)},

où et +

₁₎

sont les

_{plus petits}

_{représentants}

de

_r(x)

et

_r(x

+

₁₎

dans Z modulo

pZ.

ABSTRACT. Given a

_large

_prime

number p and a rational function

r(X)

defined over

_Fp

=

Z/pZ,

_we

investigate

the size of the set

{x

~

_{Fp :}

&#x3E;

_+1)},

where and +

₁₎

denote the least

_{positive representatives}

_{of r(x)}

and

_r(x+1)

in Z modulo

_pZ.

1. Introduction

Several

_problems

on the distribution of

points

satisfying

various

con-gruence constraints have been

investigated recently.

Given a

_large

_prime

number _p, for _any a E

_{{1, 2, ... , p -1}}

let a E

_{{1,2,... p -1}}

be such that a a - 1

_(mod

_P).

A

_question

raised

_by

D.H. Lehmer

_(see

_Guy

_[4,

Problem

F12])

asks to _say

_something

nontrivial about the

_number,

call it

_N(~),

of those a for which a and a are of

opposite parity.

The

problem

was studied

by

Wenpeng

Zhang

in

_{[8], [9]}

and

_[10]

who

_proved

that

and then

_generalized

₍₁₎

to the case when _p is

replaced by

_{any odd number}

q. In

[2]

it is obtained a

generalization

of

(1),

in which the

pair

(a,

a)

is

replaced

by

a

_point

_lying

on a more

_general

irreducible curve defined mod

p.

Zhang

also studied the

problem

of the distribution of

distances la - aj,

where _{a, a} run over the set of

_integers

_{in 11, ... , n -1 ~}

which are

relatively

prime

to n. He

proved

in

[11]

that for _any

_integer

n &#x3E; 2 and _any 0 J 1 one has

where

_cp(n)

is the Euler function and

_d(n)

denotes the number of divisors of

n. In

[12]

_Zhiyong

_Zheng

_investigated

the same

_problem,

with

(a, a)

replaced

by

a

_pair

(x, y)

satisfying

a more

general

_congruence.

Precisely,

let _p be a

prime

number and let

_{f (x, y)}

be a

_polynomial

with

_integer

coefficients of

total

_{degree d &#x3E;}

_2,

_absolutely

irreducible modulo _p. Then it is

_proved

in

[12]

that for _any 0 8 1 one has:

A

_{generalization}

of this

problem,

where the

_pair

_{(x, y)}

is

_{replaced by}

a

point

lying

on an irreducible curve in a

higher

dimensional affine _space over

the field

_Fp

=

Z/pZ,

has been obtained in

[3].

There are different _ways to measure the randomness of the

distribu-tion of a

_given

set. B. Z. Moroz showed in

_[5]

that the _squares

_(or

the l-th powers, if 1 divides p -

1)

are

_randomly

distributed among the values

{ip( f (0)), ... ,

1))}

of a fixed irreducible

_polynomial

f (X)

in

_Z[X]

modulo a

_prime

_p, as _{p -} oo

(here

_ip

stands for the reduction modulo

p).

In the

_present

_paper we

study

what

happens

with the order of residue

classes

_{mod p}

when

_they

are transformed

_through

a rational function

r(X)

E

.

IFp (X ).

For _{any y} E

_Fp

denote

_{by j (y)}

the least

_positive

_{representative}

_{of y}

in Z modulo

_pZ.

To any rational function

r(X)

E

_1Fp(X)

we associate the

map f :

Fp ~ {0,1, ... ,

p -

1} given by

r(x)

= j (r (x))

if x E

_Fp

is not a

pole

of

_r(X),

and

_r(x)

= 0 if x is a

pole

of

r(X).

As the

_degree

of

r(X)

will be assumed to be small in terms of _p in what

_follows,

the contribu-tion of the

_poles

of

_r(X)

in our

_asymptotic

results will be

_negligible.

If we count those x E

_IF~

for which

_r(a?

+

₁₎

_T(x),

_respectively

those x for

which

_r(x

+

₁₎

&#x3E;

r(x),

there should be no bias towards any one of these

inequalities.

In other words one would

_expect

that for about half of the

elements x E

_{Fp, r(x}

+

₁₎

is

_larger

than

_r(x)

and for about half of the

elements x E

_{Fp, T(x}

+

₁₎

is smaller than

In order to handle the above

_problem,

we fix nonzero

_{positive integers}

_a, b

and

_study

the distribution of the set E For any real

number t consider the set

_{M (a,}

_{b, p,}

_r,

_t)

_

{~

_E

IFp :

p}

and denote

_by

_D(a,

_{b, p,}

r,

t)

the number of elements of

M (a,

r,

t).

Our aim is to

_provide

an

_asymptotic

formula for

D(a,

_{b, p,}

_r,

t).

We now introduce a function

G(t,

_a,

b)

which will

_play

an

_important

role

in the estimation of

_{D (a,}

_{b, p,}

r,

t) .

where W =

min{0, b - a}

and Z =

max{0, b - a}.

We will _prove the

following

Theorem 1.1. For _any

_positive

_integers

_a,

_b,

_d,

_any

_prime

number _{p, any} real number t and _arcy rational

_function

_r(X)

=

9 X~

which is not a linear

polynomial,

with

_f,g

E

_deg

_f,degg

d,

one has

As a _consequence of Theorem 1.1 we show that the

inequality

f (x)

&#x3E;

f (x

+

₁₎

holds indeed for about half of the values of x in

Fp.

Corollary

1.2. Let _p be a

_prime

_number,

d a

_{positive integer}

and let

r(X)

= be a rational

function

which is not a linear

polynomial,

with

f , g

E and

_{deg f , deg g}

d. Then one has

As another

application

of Theorem 1.1 we obtain an

asymptotic

result

for all the even moments of the distance between

_r(x

+

₁₎

and

_r(x).

Corollary

1.3. Let k be a

positive integer

and let _p,

d,

r(X)

be as in the

statements

_{of Corollary}

1. Then we have

In

_particular,

for k = 1 one has

2. Proof of Theorem 1.1

We will need the

_following

_lemma,

which is a _consequence of the Riemann

Hypothesis

for curves defined over a finite field

(see [7],

[6], [1]).

Lemma 2.1.

_{Let p}

be a

_prime

number and

_IFp

the

_field

with _p elements.

Let o

be a nontrivial character

of

the additive _group

of

IEP

and let

R(X)

be a nonconstant rdtional

function.

Then

--r

where the

_{poles of}

_R(X)

are excluded

from

the

summation,

and the

im-plicit

O-constaret

_depends

at most on the

_{degrees of}

the numerator and

denominator

_of

_F(X).

Let _{now p} be a

_{prime number,}

let _a,

_b,

d be

_{positive integers}

less

_{than p,}

let t be a real number and let

r(X)

=

9 X ,

r(X)

not a linear

_polynomial,

with

f (X), g(X)

E

_{Fp [X] ,}

_{deg f (X), degg(X)}

d. For _{any y, z} E

_{0,1,...

,~

-1 }

we set

Then we _may write

D (a, b, p, r, t)

in the form

Next,

we write

D (a,

_{b, p,}

_r,

_t)

in terms of

_exponential

sums mod _p. Denote

21Tiw

~ ~

where

and

Note that for m = _n = 0 one has

Next,

we claim that if

(m, n) ~ (0, 0)

then the rational function

h(X) =

mr(X)

+

_nr(X

+

₁₎

E is nonconstant.

_Indeed,

if n = 0 then

7n ~

0

and

_h(X)

=

mr(X)

is nonconstant

by

the

hypotheses

from the _statement

of the theorem. The same conclusion holds if m = 0 and n

,~

0. Let now

0, n ~

0 and assume that

for some c E

_lFp.

_Suppose

first that

_r(X)

is not a

polynomial

and choose a root a E

_Fp

of the denominator of

_r(X),

where

_Fp

denotes the

_algebraic

closure of

_{Fp .}

Since a is a

_pole

of

r(X),

from

(9)

it follows that a is also a

pole

of

_r(X

+

_1),

that is a + 1 is a

pole

of

r(X).

_{By repeating}

the above

reasoning

with a

_replaced

_by

a + 1 we see that

a+2,Q!+3,...,Q:+p20131

are

poles

of

r(X).

This forces

degg(X)

to be &#x3E; _p, so d &#x3E; _p, in which case

(3)

becomes trivial. Let us _suppose now that

r(X)

is a

polynomial,

_say

with _{ao, ... , al E}

_~

0. Then

_by

the

_hypotheses

of Theorem 1.1 it follows that 1 &#x3E; 2.

_Looking

at the coefficient of

Xi

in

₍₉₎

we deduce that m + n = 0 in

Fp.

But

then,

the coefficients of

XI-1

on the left side of

(9)

equals

lnal,

which is nonzero in

_IFP,

_{contradicting}

_(9).

This _proves our claim

that

_h(X)

is nonconstant in

_{Fp (X).}

_By

Lemma 2.1 it follows that

for any

(m, n) #

(0, 0).

Next,

we

_proceed

to evaluate the coefficients

_A(m,

_n).

We calculate

ex-plicitly

J?(0,0)

and

_provide

_upper bounds for

_n)

for

_{(m, n) ~ (0, 0).}

There are four cases.

By

the definition of

_H(y,

_z)

it follows that for each y E

_{{0,1, ...}

p -

1}

we

have a sum of

ep (nz)

with z

_running

over a subinterval of

{0,1, ... ,~ - 1},

that is a sum of a

_{geometric progression}

with ratio

ep(n).

The absolute

value of such a sum is and

consequently

where denotes the distance to the nearest

_integer.

II.

_{m ~}

_0,

n = 0.

Similarly,

as in case

_I,

we have

III.

_{m ~ 0, n ~}

0. We need the

_following

lemma.

Proof.

One has

Note that

Also,

Lastly,

if h + ku is not a

multiple

of _p, then

We also have the bound

which is valid for _any

_{h, k}

and u.

Putting

the above bounds

together,

Lemma 2.2 follows.

We now return to the estimation of

_n).

_Writing

as a sum of b sums

according

to the residue

of y

modulo

b,

one arrives at sums as in Lemma

_2.2,

with h =

mb, k

= n, u = _a. It follows that

...

IV. _{m, n} = 0.

By definition,

we have

Let D be the set of real

_points

from the _square

_[0,p)

x

_[0,p)

which lie below the line bz - ay =

_tp.

Then

equals

the number of

integer points

(y, z)

from D. Therefore

An _easy

_computation

shows that

_Area(D)

_equals

_p2G(t,

a,

b)

with

G(t,

a,

b)

defined as in the

Introduction,

while the

length

of the

boundary

aD is

4p.

By

(5)

we know that

where

One has

By

(11)

and

₍₁₀₎

we have

Similarly

one has

In order to estimate

_D3

we first use

(10)

and

(13)

to obtain

...

while the second double sum is

Hence

_{D3 =}

_{0 a,b,d}

_log2

_{p .}

_Putting

all these

_together,

Theorem 1.1

follows.

3. Proof of the Corollaries For the

_proof

of the first

_Corollary,

let us notice that

Here W = Z = 0 and so

Thus

which _proves

_Corollary

1.2.

In order to _prove

_Corollary

1.3 note that

This

_equals

where for _any t we denote

D (t)

=

D ( 1,1, p, r, t) .

From Theorem 1.1 it

Since

₍

for _{any m,} we derive

From the definition of G we see that

Using

the fact that for _any

_{positive integer}

r one has

if r is even and mr = 0 if r is

_odd,

the statement of

_Corollary

1.3 follows after a

straightforward

_computation.

References

[1] E. _BOMBIERI, On _exponential sums in _{finite fields.} Amer. J. of Math. 88 _(1966), 71-105.

[2] C. _COBELI, A. _ZAHARESCU, Generalization _of a _{problem of} Lehmer. _Manuscripta Math. 104

no. 3 _(2001), 301-307.

[3] C. _COBELI, A. _ZAHARESCU, On the distribution _of the _Fp-points on an _affine curve in r dimensions. Acta Arith. 99 no. 4 _(2001), 321-329.

[4] R.K. _GUY, Unsolved Problems in Number _{Theory. Springer-Verlag,} New York - _{Berlin, 1981,}

(second edition _1994).

[5] B. Z. _MOROZ, The distribution of power residues and non-residues. Vestnik LGU, 16 no.

19 _(1961), 164-169.

[6] G. I. _{PEREL’MUTER,} On certain character sums. _Uspechi Matem. _Nauk, 18 _(1963), 145-149.

[7] A. _WEIL, On some _exponential sums. Proc Nat. Acad. Sci. U.S.A. 34 _(1948), 204-207.

[8] W. _ZHANG, On a _{problem of} D. H. Lehmer and its _{generalization. Compositio} Math. 86

no. 3 _(1993), 307-316.

[9] W. _{ZHANG, A problem of} D. H. Lehmer and its _{generalization} II. Compositio Math. 91

no. 1 _(1994), 47-56.

[10] W. _ZHANG, On the _difference between a D. H. Lehmer number and its inverse modulo q.

Acta Arith. 68 no. 3 _(1994), 255-263.

[11] W. ZHANG, On the distribution _of inverses modulo n. J. Number _Theory 61 no. 2 _(1996), 301-310.

[12] Z. ZHENG, The distribution _of Zeros _of an Irreducible Curve over a Finite Field. J. Number Theory 59 no. 1 _(1996), 106-118.

Alexandru ZAHARESCU

Department of Mathematics

University of Illinois at _{Urbana-Champaign}

1409 W. Green _Street, Urbana, IL, 61801, USA E-mail: zaharescGmath.uiuc.edu