Natural divisors and the brownian motion

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

E

UGENIJUS

M

ANSTAVI ˇ

CIUS

Natural divisors and the brownian motion

Journal de Théorie des Nombres de Bordeaux, tome

8, n

o

_{1 (1996),}

p. 159-171

<http://www.numdam.org/item?id=JTNB_1996__8_1_159_0>

© Université Bordeaux 1, 1996, tous droits réservés.

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

Natural divisors and the brownian motion

par EUGENIJUS

MANSTAVI010CIUS

RÉSUMÉ. On propose un modèle du mouvement Brownien relatif aux

di-viseurs d’un entier, et on établit la convergence faible de la mesure associée dans _l’espace D

_[0,1].

On obtient un résultat _analogue à celui obtenu _par

Erdös pour les diviseurs premiers

[6]

(cf.

[14]

pour une démonstration).

Ces résultats et les recherches de l’auteur

_[15]

étendent l’étude

_[9]

de la dis-tribution des diviseurs. Notre approche s’appuie sur les théorèmes limites fonctionnels en théorie des probabilités.

ABSTRACT. A model of the Brownian motion defined in terms of the natural divisors is proposed and weak convergence of the related measures in the

space D

[0,1]

is proved. An analogon of the Erdös arcsine law, known for

the _prime divisors

_[6]

_(see

_[14]

for the _proof), is obtained. These results

together with the author’s _{investigation}

_[15]

extend the _{systematic study} [9] of the distribution of natural divisors. Our _approach is based upon the

functional limit theorems of _{probability theory.}

1. Results.

The distribution of the

_prime

factors of an

_integer

determines that of the natural divisors. Therefore statements known for the

_primes

often have their

_{counterparts.}

That was

perfectly

demonstrated

by

R.R. Hall and

G. Tenenbaum in

_monograph

_[9]

for the normal orders of the k-th

_prime

factor

_pk(m)

and the k-th natural factor of m E N. Let

_{vx (... )}

denote the uniform

_probability

measure on the set

_{{I?"’ ? [x]l,}

Lu =

log

max

_lu,

_el,

and

Lk

=

L(Lk-1 ) .

Then _one has

Manuscrit _reçu le 10 octobre 1994

The results of this _paper have been obtained _during the author’s _stay at the University

of Bordeaux I _{financially supported by} Commission of the _European Communities in the framework of the program "S &#x26; T Cooperation with Central and Eastern European

Countries". The final version was _prepared under the partial support of International Science Foundation.

and

Here and in what follows

_{W(m) and r(m)}

denote the number of all

differ-ent

_prime

and natural factors

_{respectively.}

_Moreover,

after the

_change

of 6

to 2013~ these limits

(even

with liminf instead of

limsup)

are

_equal

to one. The same

_duality

also holds for the

_sharpened

estimates when the terms

êý2kL2k

and êV2(log2

k ) L3 k

are substituted

_by

some

_asymptotic

expan-sions

_{(c.f. [15]).}

In the _paper

_[15]

we extended the above mentioned results into the Strassen functional

_form,

that

_is,

both of them were included into

more

general

relations for

arithmetically

defined stochastic _processes.

Pro-ceeding

along

this way we can look for the

duality

in the functional limit theorems for arithmetic processes. The most of such the processes so far considered are defined in terms of additive functions. It means that the pro-cesses are related to the

_prime

divisors

_(see

_{[1], [3], [4], [8], [12], [13],}

_[16],

[19], [20]

and

_other).

We cannot

_give

_any reference

_concerning

weak con-vergence of processes defined in terms of the natural divisors. But several results do indicate such a direction of

_{possible investigations.}

Let

_T(m,

_u)

=

cardfd

E N :

dim,

d

u}.

It is known

[9], [18]

that

where # denotes weak convergence of distribution

functions,

is some

purely

dicrete distribution

_function,

oo. This relation can be

interpreted

as

_referring

to the increments of the arithmetic process

Yx

:=

Y~ (m, t)

=

having trajectories

in the

space

D:=D[0,1]

of

real-valued functions on

[o,1]

which are

_{right-continuous}

_{and have left-hand} limits.

_Moreover,

the remarkable

_asymptotic

formula

which has been obtained in

_[5],

shows the behaviour of the

_expectation

of the process.

In this paper we look for a

_counterpart

of the _process

where

_{cv(m, v)}

:=

card{p 2013

prime :

v},

which "simulates" the Brownian motion W =

W (t)

given

_{on some}

probability

To be more

_precise,

we will use some notations and

_concepts

introduced in the book

_[2].

Let

_p(.,.)

denote the supremum metrics in the space

D,

D

be the Borel

_a-algebra

in D with

_{respect to p,}

and v~

o

1/;;1

stand for the

measure on D defined

by

E

_A)

where A E D. Put p for the Wiener

measure P o

W -1.

In 1970 P.

Billingsley

[3]

proved

the

following

result. THEOREM B. The sequence of measures v, o

_weakly

_{converges to} the

Wiener measure it as z - oo.

More

_general

results can be found in the above cited _papers on the functional limit theorems.

Having

in mind that

_{statistically}

_T(m)

behaves like the function

_20(m),

where

_Q(m)

denotes the number of

_prime

factors of m E N counted

ac-cording

to their

_{multiplicity,}

we will consider the

_limiting

distribution of the process

with _respect to the

_probability

measure v., as z - oo. Denote _p, = v~ o

X-.

In what follows

_{the limiting}

_passage x --~ oo is not

_explicitly

indicated.

THEOREM 1. The _sequence of measures _pz

_weakly

_converges to the Wiener measure _p.

The corollaries

_presented

below follow from the relation

valid for each

u- ost everywhere

continuous functional

_p:D-

R.

_They

describe new features of the _sequence

dk(rn).

COROLLARY 1. For each fixed 0 s t

1,

Hence if s = 0 and t =

1,

we have the central limit theorem for the additive function

_log2

_T(m)

_belonging

to the Kubilius class H

_(see

_[9]).

But if 0 t

_1,

then is

_only

subadditive. That shows new direction of

_{possible investigations,}

e.g. to extend the

_{probabilistic}

number

COROLLARY 2. For each 0 t ~

_1,

and for u &#x3E;

_0,

The last assertion can be

compared

with the above mentioned estimates

of the law of iterated

_logarithm

_{([9], [15])}

illustrated

_by

_(1).

Let in what follows

_As(u)

be extended to R

_{by equalities}

_As(u)

= 0

when u 0 and

_As(u)

= 1 when u &#x3E; 1.

COROLLARY 3. We have

Now,

since the

_points

t =

_tj

=

L2x

_are the

_jumps

of the

tra-jectories

considered,

calculating

the

_Lebesgue

measure in the last relation we obtain certain sums of the

quantities

Even the estimate

₍₁₎

is not sufhcient to

_simplify

the sums of the ratios

_(2).

The

_following

statement of P. Erd6s

_[6]

indicates that a more

_simple

form of the arcsine law _may be valid for the natural divisors.

THEOREM E. We have

Proof.

see

[14].

The

_counterpart

of this Erd6s theorem is our next result. Let

I+

denote

THEOREM 2. We have

The

_proof

is similar to that

_given

in

_[14]

for Theorem E.

2. Proof of Theorem 1.

The

_trajectories

_{of X, (m, t)}

will be

_{approximated "for}

almost all m"

_by

t) .

By

the

_general

_theory

_[2]

the assertion of Theorem 1 will follow from Theorem B and the estimate

for each c &#x3E; 0.

To prove

(4),

at first we observe that for each

_t,

0 t

_1,

where =

card{d

E N :

dlm, p(d)

v}

and p(d)

denotes the maximal

prime

divisor of d. On

_prime

numbers the additive functions

_w(.,v)

and

log2

8(., v)

coincide,

hence

_(see

_[12])

for each 6 &#x3E; 0.

_Thus,

the relations

₍₅₎

and

₍₆₎

_yield

the

_required

upper

estimate of in terms of

_t).

The lower estimation is based upon ideas

suggested

in Exercises to

Chap-ter 1 of the book

_[9].

If

_{T(m, u, v)}

:=

card(d : dim,

d &#x3E; _u,

_p(d)

_v},

then we have

[9]

uniformly

in 1 u, v x with some

_positive

c &#x3E; 0. The last estimate follows from the

_{following inequalities}

where A =

c/Lv

with some c &#x3E; 0.

Now we divide the interval

(0,1~

into N :=

~(L2x)/(L3x)z~

equal

_parts

of the

_{length 7}

:=

N-1.

For the values

of y

:= we choose the

checkpoints

- -

--, - 1___- _ __

When _y E

_[uk,uA,+,],

from

₍₇₎

we obtain

provided

that k &#x3E; 1.

_Further,

in virtue of

₍₈₎

and

_Z+,

According

to

₍₉₎

this

_implies

that

_{for y}

E

_{[Uk, Uk+ll}

_uniformly

in 1 k

N-1

for almost all m x.

To estimate the second difference for almost all m, we will use Ruzsa’s

[17]

following

result.

LEMMA 1. Let

_{1, ... , s,}

be real-valued additive functions. There exist a

probability

_space

~5~, ~, P~,

independent

random variables

6(j),

p x,

defined on it

byP(6(j)

_0,

such

for

_axbitrary

_aj E R and v &#x3E;

_0,

with an absolute constant in the

_symbol

~.

Thus,

applying

Lemma 1 we obtain

Now

_~p,

p

x,

denote

_independent

random variables defined

_by

_P(~p

=

1)

= =

0)

=

1/p.

After

elementary

estimation of their

expectations

and variances from the

_{exponential inequalities}

_(see

_~11~,

_p.254)

we derive

for each - &#x3E; 0. Hence and from

₍₁₀₎

we conclude that

uniformly

1 for almost all m.

The

_{relation 7}

= and Theorem B

imply

Moreover,

for

_{sufficiently large x,}

which

_by

the law of

_large

numbers for additive functions

_[10]

tends to zero.

So,

the estimate

₍₁₁₎

remains valid

_uniformly

in 0 t 1. That

_completes

the lower estimation in

_(4).

Theorem 1 is

_proved.

3. Proof of Theorem 2.

For

_arbitrary

k &#x3E; 2 we

_{put ni}

Ji

= where 1 i k.

For convenience in the space ,C of distribution functions we shall use the

Levy

metrics defined

_by

In what follows the

symbol

o(l)

may

depend

on some

_parameters

which sometimes will be

_indicated,

while the other

_{symbol «}

will contain absolute

constants.

LEMMA 2. Let

Then the relation

₍₃₎

is

_equivalent

to

_{A(V~, As) --&#x3E;}

0.

Proof.

In the double sum of

(12) j

runs over the natural numbers

_belonging

to the interval

_{(1, 21-2X].}

To estimate the sum over

2~2’ j :5

r (m),

we observe that

_by

the law of

_large

numbers for the additive function

_{1092 r(m)}

(see

[10])

we have

log2 r(m) -

(L2x)3~4

for all but

o(x)

numbers m x. Hence for these numbers

Now from the definition of the

_Levy

metrics we obtain

~.(U~, V~)

- 0.

Lemma 2 is

_proved.

Observe,

since the distribution function

_As(u)

is

_continuous,

the conver-gence

A(Vae, As) -4

0

implies

uniform convergence in u E R.

LEMMA 3. Let We have

uniformly in j

such that

_{1092 j}

E

_Ja,

for each i =

_{2, ... ,}

k and 6 _{&#x3E; 0.}

Then

_by

₍₁₎

we obtain

vx (m g

=

0(1),

and

further,

uniformly in j,

log2 j

E

_Ja,

_{provided x}

is

_{sufhciently large.}

But

_according

to Theorem 1 and the definition of the process

Xz(m, t),

At the last

_stage

we have used the well-known

Levy

and

_Kolmogorov

in-equalities

(see

[11)).

Lemma 3 is

_proved.

LEMMA 4. We have

Proof. Suppose

1092 j E Ji

and

and

Hence and

_by

Lemma 3

provided

that 2 i 1~. Theorem 1

_implies

the one-dimensional limit

relation

Thus,

from

₍₁₃₎

we obtain

uniformly in j,

1092 j

E

_{Ji ,}

for each 2 i k and 6 &#x3E; 0. Now

Choosing

6

= k-1/3

we

_complete

the

_proof

of Lemma 4.

LEMMA 5. Let

_{V, (u)}

be defined in Lemma 1 and

Then .

for each 6 &#x3E; 0. Now if

then

Since = ₊

0(1)),

Lemma 5 is

proved.

The main

_{probabilistic ingreedient}

is the

_following

result of P. Erd6s and M. Kac.

LEMMA 6.

_{Let Yi,}

i &#x3E;

_1,

be

_{independent, normally}

distributed random variables such that

EY$

= 0 and

E y2 i

= i. If

then 0 as k -~ oo.

Proof see

[7].

Proof of

Theorem 2. Denote

_by

_{xu(y, .. -, yk)}

the characteristic function of the set

For the distribution function

_Wxk

defined in Lemma 5 we have

Further,

we

substitute Yi J---+

in the

integral

on the

_right-hand

side. Since the functional limit result

_presented

in Theorem 1

_implies

weak convergence of the k-dimensional

distributions,

we obtain

uniformly

in u E R. Hence and

_by

Lemma 6

According

to Lemmas 5 and 2 the last

_{equality yields}

the assertion of

Theorem 2.

Finally,

what about the _process

Yx?

We

_expect

that the

_following

con-jecture

is true.

PROPOSITION. Let

ÐI

be the

_u-algebra

in D of the Borel sets

_{generated by}

the Skorokhod

topology

(see

[2]

for the

_definition).

There exits a

_probability

measure

_Q

on

_Dl

such that vx o

Y.-I

weakly

_converges to

_Q.

Acknowledgment.

For the warm

_hospitality

_during

our

_stay

at the

Uni-versity

of Bordeaux I we remain indebted to Prof. M.

_{Mend6s-France,}

to

Dr. M.

_Balazard,

and to other

_colleagues.

REFERENCES

[1]

G.J. Babu, Probabilistic methods in the _{theory of} arithmetic functions, Ph.D.

dis-sertation, The Indian Statistical Institute, Calcutta, (1973).

[2]

P. _{Billingsley, Convergence of Probability Measures, Wiley} &#x26; Sons, New _York, (1968).

[3] P. _Billingsley, Additive functions and Brownian _motion, Notices Amer. Math. Soc. 17 _{(1970), 1050.}

[4]

P. _Billingsley, The _{probability theory of} additive arithmetic _functions, The Annals of Probab. 2 No 5 _(1974), 749-791.

[5] J.-M. Deshouillers, F. Dress &#x26; G. _Tenenbaum, Lois de _répartition des _{diviseurs, 1,} Acta Arithm 34 _(1979), 273-285.

[6]

P. _Erdös, On the distribution _{of prime divisors, Aequationes} Math. 2

_(1969),

177-183.

[7]

P. Erdös, M. _Kac, On the number of positive sums of independent random variables,

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[8]

B. _Grigelionis, R. Mikulevi010Dius, On the functional limit theorems of the probabilistic

number _theory, Lithuanian Math.J. 24 No 2 _{(1984), 72-81, (Russian).}

[9]

R.R. _Hall, G. _{Tenenbaum, Divisors, Cambridge University} Press _{Cambridge (1988).}

[10]

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[11]

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[12]

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E. Manstavi010Dius, Functional _approach in the divisor distribution problems, Acta Math. Hungarica 66 No 3 _(1995), 343-359.

[16]

W. Philipp, Arithmetic functions and Brownian motion, Proc. _Sympos. Pure Math. 24 _(1973), 233-246.

[17] I.Z. _{Ruzsa, Effective} results in _{probabilistic} number _{theory, In,} Théorie élémentaire

et analytique des nombres ed. _{J.Coquet, 107-130, Dépt.} Math. Univ. _{Valenciennes,.}

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Eugenijus MANSTAVICIUS

Department of Probability Theory

and Number Theory

Vilnius _University

Naugarduko str.24 2006 Vilnius, Lithuania