A
NTANAS
L
AURIN ˇ
CIKAS
On the distribution of complex-valued
multiplicative functions
Journal de Théorie des Nombres de Bordeaux, tome
8, n
o1 (1996),
p. 183-203
<http://www.numdam.org/item?id=JTNB_1996__8_1_183_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
On the Distribution of
complex-valued
multiplicative
functionspar ANTANAS
LAURIN010CIKAS
In honour of Professor H. Delange on his 80th birthday
RÉSUMÉ. Soient g1 (m), g2 (m) deux fonctions multiplicatives a valeurs com-plexes. Dans un article on donne les conditions nécessaires et suffisantes de la convergence en un certain sens de
1/n Nn ((g1(m), g2(m))
~ A), A ~B(C2
),quand n ~ ~.
ABSTRACT. Let
gj(m), j
= 1, 2, be complex-valued multiplicativefunc-tions. In the paper the necessary and sufficient conditions are indicated for the convergence in some sense of probability measure
1/n card{0
~ m ~ n : (g1(m), g2(m)) ~ A}, A ~B(C2),
as n ~ ~.Let
R,
C,
N and Z denote the sets of all realnumbers,
of allcomplex
numbers,
of all natural numbers and all realinteger
numbers,
respectively.
Afunction g :
N ~C,
notidentically
zero, is calledmultiplicative
iffor all natural numbers m, n
relatively
prime
to each other. Themultiplica-tive functions
play
animportant
role in numbertheory,
therefore its values distribution was studiedby
many authors. Inprobabilistic
numberthe-ory the behaviour of
multiplicative
functionsusually
is describedby limit
theorems in sense of weak convergence of
probability
measures. Most ofthe works of this kind are devoted to real-valued
multiplicative
functions.Manuscrit recu le 20 octobre 1993
*This paper has been written during the author’s stay at the University Bordeaux I in the framework of the program "S&T Cooperation with Central and Eastern European
H.
Delange
was the first tobegin
to useprobabilistic
methods tostudy
the distribution of
complex-valued multiplicative
functions. In[1]
he ob-tained thefollowing
result(Theorem A).
LetNn (... )
denote a number ofm n,
m e N,
satisfying
the conditions instead ofdots,
and letDenote
by
p aprime
numberand,
for agiven complex-valued multiplicative
functiong(m),
define the functionsHere
arg g(m)
is defined to within the addition of aninteger multiple
of27r.
Let stand for the class of Borel sets of the space S.
THEOREM A. In order that the sequence of
probability
measuresconverges "6troitement" to the
probability
measure not concentrated atz =
0,
it is necessary and sufficient that thefoflowing
conditions be satisfied:10
the series11 , ’"
converge;
2°
there exists at least one m E N such that the seriesfor all m E N and aU u E R.
In
probabilistic
numbertheory
the multidimensional theorems are known for additive functions as well as for real-valuedmultiplicative functions,
see, for
example
[2]-[6].
The aim of this paper is topresent
a limit theorem which describe thejoint
distribution of fewcomplex-valued multiplicative
functions =
1,
..., r. To do this we can define the
probability
measure
where and consider its convergence in some sense to
probability
measure on as n - oo. Thisproblem
is rathercomplicated,
and,
forsimplicity,
we shall limit ourselves to the case r = 2only.
To state our
theorem,
we will need some notions from[7].
Let P andPn
beprobability
measures on((C2,ti(~)).
We say thatPn
convergesweakly
in sence
O
to P oo ifPn
convergesweakly
to P and in additionHere Pj
denote themarginal
distributions ofP,
that isLet,
forsimplicity,
0’ = 0 for all z E C.Suppose
that1, j
=l, 2.
THEOREM. In order that the
probability
measureconverge
weakly
in sencee2
to aprobability
measure whichmarginal
dis-tributions are not concentrated on the circleI
z1=
a, a >0,
it is necessary and sufficient that thefollowing
conditions be satisfied:1 °
the series ~ andj
=l, 2
converge;
converge, or
for all
For the
proof
of the theorem we will use the characteristic transforms ofprobability
measures on introduced in[7].
In our case r = 2the functions
I-are called the characteristic transforms of
probability
measure P onprovided
that theintegrands
areequal
to zerowhen r j =
0for
some j - 1, 2.
Here r j=1
-argzj,
tj
EJR, kj
EZ, j =
1, 2.
We shall
require
thefollowing
continuity
theorems forprobability
measureson
LEMMA 1. Let be a sequence of
probability
measures on(C2 ,B(CJ))
and let
1, 2,
be a sequence ofcorre-sponding
characteristic transforms.(j=l,2,
no0 201300
for all
tl, t2 E
kl, k2 E 7Z ,
and that the functions1, 2,
are continuous atthe points t ;
=0; ti
=0,
t2 =0,
respectively.
Then there exists a
probability
measure P on«([:2, fi(~ ))
such thatPn
converges
weakly
in sense~
to P 00. In this case =1, 2,
are the characteristic transforms of the measure .~’. LEMMA 2. LetP, Pi,
P2,
...be probability
measures on andW2;j
(tj,
=1, 2,
w2(tl, ~2) ~i) ~2)?’"
denote their characteristictrans-forms, respectively.
Proof.
Lemmas 1 and 2 are the case r = 2 of Theorems 2 and3,
respectively
from
[7].
Also we shall use some results on the mean-value
of the
multiplicative
functionsg(m),
if this limit exists.LEMMA 3. Let
g(m)
be amultiplicative
functionsatisfying ~
g(m) 1:!~,
1. IfM(g)
exists and is not zero, then the seriesconverges.
Lemma is the
Delange
theorem,
for theproof
see[8], [15~.
LEMMA 4. In order that the mean-value for the
multiplicative
functiongem),
I~
1,
exists and be zero, it is necessary and sufficient that one of thefollowing
conditions be satisfied:for every real u;
20
there exists a realnumber uo
such that theseries ’
converges and
Proof.
Lemma is acorollary
of the fundamental work of G. Halisz[9].
Itsstatement is
given
in[10]-[12].
LEMMA 5. Let
g(m)
=I
g(m)
1:5
1,
be amultiplicative
function,
and the seriesuniformly
intj
forProof.
Lemma is aspecial
case of result from[13].
Let
-for all
ki , k2
E7G, ~
I ki I +
k2
10
0.By
Lemma 4 this isequivalent
to the condition- - I’ll-,, , __
for all u E
R,
since ifand
then we have that
too, but
Let us
investigate
the set of thosetriples
1 k1
+ ~
10
0, t
E for whichBy
a remark above this set is notempty
if andonly
ifLEMMA 6. Let the relation
(2)
be satisfied and Thenthere exist ql, and t E
1~,
or there existand
tl, t2 E
R such thatif and
only
ifor if and
only
ifProof.
Let A be a set of thosepairs
(kl,k2)
for which(1)
is valid. Ifthen, clearly,
(-kl, -k2)
EA,
since Rez = Rez and(1)
remains true with -t instead of t.
Obviously
(0,0)
EA,
because in this case one can take t = 0. Now let(k1, k2), (k1,
~2 )
EA,
that isfor some real numbers
t~
andt~~ .
Then in virtue of theinequality
Consequently,
A is asubgroup
of thegroup Z2.
From thegeneral theory
of groups it follows that there exists(ql, q2)
such that each element(k1,k2)
EA is
presented
asor there exist
(al, a2), (bl, b2)
satisfying
(3)
such that each element(kl, k2)
EA is of the form
It is well known that for each
(ki , k2)
E A there existsexactly
one real number usatisfying
(4).
We will find anexpression
for this u. First let us consider the case(6).
Suppose
that u = tcorresponds
to(ql, q2). Applying
the
inequality
(5) 1 m
times,
weeasily
find thatso in this case u = mt. Now let
(kl, k2)is
decribedby
(7).
Suppose
thatu =
tl,
u = t2correspond
to(a,, a2)
and(bl, b2),
respectively. Then,
reasoning similarly
asabove,
we obtain thatso in this case u =
mltl
+m2t2 .
Proof
of Theorem.Sufficiency.
Denoteby
1,2,
Wn(tl,
t2,
k1,
k2)
the characteristic transforms of the measurePn.
Then we have thattl, t2 E
R, ki, k2
~ Z.We will
begin
ourinvestigation
fromObviously,
in case of the latter function we can limit ourselves 0only.
Let the series ,
converge.
Using
the condition10
andreasoning similarly
as in[10],
p.224-227,
we can prove the existence ofsuch q;
E N that the seriesconverge if and
only
if qj Forthese kj
we will examine the convergence of the seriesThe condition
1°
shows thatuniformly
intj, 1 tj
T. Here and further B is a number(not
always
thesame)
boundedby
a constant.Thus,
by (5),
using
the convergenceof the series
(8),
we obtain that the series(9)
also convergesuniformly
inas n -+ oo,
uniformly
in T . It is easy to seethat,
forp >
po,From
this,
(11),
using
thehypotheses
of thetheorem,
we obtain thatas ?~ 2013~ oo,
uniformly
intj, 1 tj
~
T,
for every T >0,
and for allintegers
0. Therefore the functions
=1, 2,
are continuousat tj
=0.
Consequently,
from Lemma 1 we have that themarginal
distributionsl, 2,
of the measurePn
convergeweakly
in sense C to some measurePj
on(C,B(C))
as n - oo. It remains to prove thatPj
are notconcentrated on the circle for
some a; >
0. This can be obtainedin the
following
manner. Since1#
1, j
=1, 2,
it is well known[16]
that under thehypothesis
1°
of the theorem a real-valuedmultiplicative
functionI
possessesnon-degenerate
limitdistribution,
that is theprobability
measureas n --> oo, converges
weakly
(and
at x =0)
to someprobability
measureQj
not concentrated at x =
a~, 0. If we suppose that
Pj
are concentrated on the circlez
~=
aj then it follows thatQj
aredegenerate
at thepoint
x =Thus,
we have obtained a contradiction which shows that =1, 2,
are not concentrated on the circle.In the case when it can be
proved using
a similar method as in[10]
thatfor a~ll u E R. Let Cl, C2, ... be some
positive
constants.Then,
using
theidentity
and
taking
into account(10)
and(13),
we havefor all
tj
E R and all u E R.Consequently,
by
Lemma 4 it follows in this case thatas n --+ oo, for aJl tj E R.
Iffor all
ki , k2 E Z ,
I +
~
0,
and all u ER,
then,
taking
k2 = 0
and after this0,
we find thatfor all
kj
E N and all u E R.Therefore, reasoning exactly
as in the casekj
consideredabove,
we obtain thatas
The function
already
has been considered above.Conse-quently,
we see from(12), (14)
and(15)
that in all casesas n - oo, for all
tj
E Rand kj
EZ,
and the functionW j ( t j ,
0)
is continuousat tj = 0, j = 1, 2.
Now we will examine the function When
J~1
=k2 =
0we have that
and to prove the existence of limit function it suffices to
verify
thehypoth-esis of Lemma 5. In view of
(5)
and(10)
uniformly
inT, j
=1, 2.
Thus Lemma 5yields
thatas n --~ oo,
uniformly
inT, j
=1, 2.
It is clearthat,
forTherefore
(18)
and thehypotheses
of the theoremyield
(19)
uniformly
inT, j =
1, 2,
andevidently
the limit functionW (t 1 , t2, 0, 0)
is continuous at tl =0, t2
= 0.Now we will
investigate
thegeneral
case of the function Wn(tl,
k2).
First assume there exist N such that the seriesconverge.
Then,
clearly,
and whence in view of
(5)
for each m E N. From this and Lemma 4 we have that the relation
(2)
issatisfied,
and therefore Lemma 6gives
or
By
Lemma 6 ifthen
for all u E R. For these
ki,
k2
we haveby
(17)
thattends to
infinity
when x tends toinfinity
for alltj
E1, 2,
and all u E R. In consequence,by
Lemma 4 for thesek2
as n -- oo, for all E
R, =1,2.
It remains to
study
morecomplicated
cases when(ki, k2)
ispresented
asor
First suppose that
(20)
is true and consider(k1, k2) having
the form(23).
Note that in this case t = 0. Then from(4)
it follows thatfor all m e N. Here ap =
qlarggl(p)
+q2 aXgg2 (p),
and the dash indicatesthat the sum is extended over those p for which
91(P)92(P) #
0. We mustconverges, too. To prove this we
apply
theDelange
method used in[10],
[14].
Let,
for x >2,
m, ml , m2 EN,
and
Then,
for x2 >2,
by
(25).
Thus,
has a finite limit as x --> oo. From the conver-gence of the seriesand
(20)
we deduce thatFrom the definition of it follows that
has a finite limit as z - oo, from this and
(29)
we find thatThe
equality
(27)
gives
Consequently,
theproperties
of and(30), (31)
yield
so
by
induction the series(26)
converges for all m e N. From this and(25)
we have that the seriesconverges for all
ki, k2
satisfying (23).
Now we assume that
(21)
is valid and consider(k1,
k2)
having
the form(24).
Let usput
Then from
(4)
and we must prove that the series
converges, too. The convergence of the series
(28)
and(21)
imply
the convergence of the seriesFrom
this,
using
again
the convergence of the series(28)
and theequality
converges. Let us put, for
and
Then,
for ~2 >2,
in view of(33)
Thus,
the limitexists. Now we take
m2 =
M02, 0.Clearly,
we can suppose that mo 1 > 0. Then(36)
gives
the formulaSince the series
(35)
converges, the limitexists. Therefore
(37)-(39)
yield
Then in virtue of
(36)
and therefore
by
(37), (40)
and(41)
for all
m E N, and,
consequently,
for all m E Z.Reasoning similarly,
we can show thatfor all m E Z. Now it is easy to see that
(37),
(42)
and(43)
yield
the relationfor all ml , m2 E
Z,
and this shows that the series(34)
converges for auml, m2 E Z.
Thus,
we haveproved
that in considered case the serieswhere
is
convergent.
Now we return to The convergence of the series
(32)
and
(17)
imply
for all
tl,t2 E
R andkl,k2
satisfying (23) (if
it takeplace).
Therefore, by
Lemma 5Since,
for p > po,Hence and from
(45)
we have that the limitexists.
The case of convergence of the series
(44)
is consideredsimilarly,
and weagain
obtain the relation(46).
If the seriesdiverges
for allkl, k2 E
Z,
I ki
I
+I
J~2
~~
0,
and all u ER,
thenclearly,
the relation(22)
is valid. In consequence, thesufficiency
of the theoremfollows from
(16), (19), (22),
(46)
and Lemma 1.Necessity.
Suppose
that there exists aprobability
measure Pon
((C2 , ~3 (~ ) )
withmarginal
distributions not concentrated on a circle such thatPn
convergesweakly
in senseC2
to P as n - 00.By
Lemma 2the characteristic transforms of
Pn
converge to those of P as n - oo. Inwhere Wj
(tj,
are the characteristic transforms ofP~ .
From this we obtain that themultiplicative
functionI 9j(m)
possesses
anon-degenerate
limit distribution. Then it follows from[16]
that thehypothesis
10
of the theorem is valid.Moreover
Suppose
that wj(0,
0. Let wj(0,
ko~ )
#
0. Thenby
Lemma 3 theseries
J--
-converges. From this we deduce that
for all m e N.
Consequently,
Lemma 4 shows thatfor all
ki, k2
E7~, ~ I
kl
I + ~
I
1~2
1#
0- If --+ 0 for all n-+ookl, k2 E
7~, ~ I ki
1 +
k2 1#
0,
then Lemma 4 for theseimplies
for all u E R.
This
completes
theproof
of the theorem.Acknowledgment.
The author expresses hisdeep gratitude
to Prof. M.Mend6s-France,
Dr. M. Balazard and theircolleagues
for constant attentionto his
stay
at theUniversity
Bordeaux I. He also would like to thank theREFERENCES
[1]
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[2]
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H. Delange, A remark on multiplicative functions, Bull. London Math. Soc., 2 (1970),pp. 183-185.
[15]
G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres,Publications de l’Institut Élie Cartan, Université de Nancy I, 1990.
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B. V. Levin, N. M. Timofeev, S. T. Tuliaganov, Distribution of values of multiplicativefunctions, Liet. matem. rink., 13, No 1 (1973), pp. 87-100 (in Russian).
Antanas LAURINCIKAS
Department of Mathematics Vilnius University
Naugarduko 24