×ÅÁÛØÅÂÑÊÈÉ ÑÁÎÐÍÈÊ
Òîì 10 Âûïóñê 1 (2009)
ÓÄÊ 519.14
THE JOINT DISTRIBUTION OF MULTIPLICATIVE
FUNCTIONS
A. Laurincikas
Àííîòàöèÿ
In the paper, the existence of a joint limit distribution for a real multiplicative and a complexvalued multiplicative functions is considered. For one
class of multiplicative functions, the necessary and sucient conditions for
the existence of a limit distribution are obtained.
Introduction. Denote by N, N0 , Z, R and C the sets of all positive integers,
nonnegative integers, integers, real and complex numbers, respectively. A function
g : N → C is called multiplicative if g ̸≡ 0, m ∈ N, and g(mn) = g(m)g(n) for
all m, n ∈ N, (m, n) = 1. Clearly, we have that g(1) = 1. A function f : N → C
is said to be additive if f (mn) = f (m) + f (n) for all m, n ∈ N, (m, n) = 1. This
denition implies f (1) = 0. The main problem of the probabilistic number theory
are asymptotic distribution laws for additive and multiplicative functions. We recall
some results on multiplicative functions. Let, for n ∈ N,
1
#{1 ≤ m ≤ n : ...},
n
where in place of dots a condition satised by m is to be written. As usual, by B(S)
denote the class of Borel sets of the space S , and by p a prime number. The rst
probabilistic result for multiplicative functions was obtained by P. Erdos in [6]. Let
Pn and P be two probability measures on (R, B(R)). We say that Pn converges m
weakly to P as n → ∞ if Pn converges weakly to P and limn→∞ Pn ({0}) = P ({0}).
In the case P ({0}) = 1, the latter condition is ommited. We use the notation
{
u if |u| ≤ 1,
||u|| =
1 if |u| > 1.
νn (...) =
Òåîðåìà
ty measure
1. ([6]). Let g(m) ≥ 0 be a multiplicative function. Then the probabili-
νn (g(m) ∈ A),
A ∈ B(R),
(1)
converges mweakly to a certain probability measure P on (R, B(R)), P ({0}) ̸= 1, as
n → ∞, if and only if the series
∑ ||g(p) − 1||2
∑ ||g(p) − 1||
,
(2)
p
p
p
p
converge.
A. LAURINCIKAS
42
The rst result on real multiplicative functions of an arbitrary sign belongs to A.
Bakstys [1]. We recall that a probability measure P on (R, B(R)) is called symmetric
if, for some a ∈ R,
P (−∞, a) = 1 − P (−∞, a].
2. ([1]). Let g(m) be a real multiplicative function. Then the probability measure (1) converges mweakly to a certain nonsymmetric probability measure
on (R, B(R)), as n → ∞, if and only if the series (2) and
Òåîðåìà
∑ 1
p
g(p)<0
converge, and there exists α ∈ N such that g(2α ) ̸= −1.
The problem of the existence of limit distribution for real multiplicative functions
was completely solved in [14]. Let, for a ∈ R and A ∈ B(R),
{
1 if a ∈ A,
Pa (A) =
0 if a ∈
/ A.
Òåîðåìà 3. ([14]). Let g(m) be a real multiplicative function. Then the probability measure (1) converges mweakly to a certain probability measure P on (R, B(R)),
P ̸= Pa for every a ∈ R, as n → ∞, if and only if the series
∑
g(p)̸=0
| log |g(p)||<1
log |g(p)|
,
p
∑ 1 log2 |g(p)|
,
p 1 + log2 |g(p)|
g(p)̸=0
∑ 1
p
g(p)=0
converge.
Now let g(m) be a complexvalued multiplicative function. Dene
{
g(p)
if g(p) ̸= 0,
|g(p)|
ug (p) =
0
if g(p) = 0,
and
{
log |g(p)| if e−1 ≤ |g(p)| ≤ e,
vg (p) =
1
if |g(p)| < e−1 or |g(p)| > e.
Let Pn and P be probability measures on (C, B(C)). We say that Pn converges
weakly in the sense of C to P as n → ∞ if Pn converges weakly to P as n → ∞,
and, additionally, limn→∞ Pn ({0}) = P ({0}).
4. ([4]). Let g(m) be a complexvalued multiplicative function. Then
the probability measure
νn (g(m) ∈ A), A ∈ B(C),
Òåîðåìà
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
43
converges weakly to a certain probability measure P on (C, B(C)), P ({0}) ̸= 1, as
n → ∞, if and only if the following two hypotheses hold:
10 The series
∑ vg (p)
∑ vg2 (p)
,
p
p
p
p
converge;
20 Either, for all m ∈ N and all t ∈ R,
−it
∑ 1 − Re um
g (p)p
= +∞,
p
p
or there exists at least one m ∈ N such that the series
∑ 1 − um
g (p)
p
p
converges.
The joint distribution of real multiplicative functions was discussed in [7]. We
will state the result for two real multiplicative functions g1 (m) and g2 (m). In [7] limit
theorems are stated in terms of distribution functions, however, it is not dicult to
use for them a language of probability measures. Let Pn and P be a probability
measures on (R2 , B(R2 )). We say that Pn converges mweakly to P as n → ∞, if
Pn converges weakly to P and
lim Pn (R × {0}) = P (R × {0}),
n→∞
lim Pn ({0} × R) = P ({0} × R),
n→∞
lim Pn ({0} × {0}) = P ({0} × {0}).
n→∞
Òåîðåìà 5. ([7]). Let g1 (m) and g2 (m) be real multiplicative functions. Then
the probability measure
νn ((g1 (m), g2 (m)) ∈ A),
A ∈ B(R2 ),
converges mweakly to a certain probability measure P on (R2 , B(R2 )), P (R × A) ̸=
Pa (A), P (A × R) ̸= Pb (A), A ∈ B(R2 ), for every a, b ∈ R, as n → ∞, if and only if
the series
∑
gj (p)̸=0
| log |gj (p)||<1
converge.
log |gj (p)|
,
p
j = 1, 2,
2
∑
∑ 1 log2 |gj (p)|
,
2
p
1
+
log
|g
(p)|
j
j=1
g(p)̸=0
2
∑
∑ 1
p
j=1
gj (p)=0
A. LAURINCIKAS
44
A twodimensional limit theorem for complexvalued multiplicative function was
proved in[9].
The aim of this paper is to prove a twodimensional limit theorem for a real and
a complexvalued multiplicative functions. Let, for brevity, X = R × C. Let Pn and
P be probability measures on (X, B(X)). We say that Pn converges mweakly in the
sense of X to P as n → ∞ if Pn converges weakly to P as n → ∞, and
lim Pn (R × {0}) = P (R × {0}),
n→∞
lim Pn ({0} × C) = P ({0} × C),
n→∞
lim Pn ({0} × {0}) = P ({0} × {0}).
n→∞
Dene PR (A) = P (A × C), A ∈ B(R), and PC (A) = P (R × A), A ∈ B(C).
6. . Let g1 (m) be a real and g2 (m) be a complexvalued multiplicative
functions such that the series
∑ 1
∑ 1
and
p
p
Òåîðåìà
g1 (p)>0
g1 (p)<0
do not diverge simultaneously. Then the probability measure
def
Pn (A) = νn ((g1 (m), g2 (m)) ∈ A),
A ∈ B(X),
converges mweakly in the sense of X to a certain probability measure P on
(X, B(X)), PR ̸= Pa for every a ∈ R and PC ({0}) ̸= 1, as n → ∞, if and only if the
following hypotheses are satised:
10
∑
∑ 1 log2 |g1 (p)|
∑ 1
log |g1 (p)|
,
,
p
p 1 + log2 |g1 (p)|
p
g1 (p)̸=0
| log |g1 (p)||<1
converge;
20 The series
g1 (p)̸=0
∑ vg (p)
2
,
p
p
g1 (p)=0
∑ vg2 (p)
2
p
p
converge;
30 Either, for all k ∈ N and all u ∈ R,
∑ 1 − Re ukg (p)p−iu
2
= +∞,
p
p
or there exists at least one k ∈ N such that the series
∑ 1 − ukg (p)
2
p
converges.
p
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
45
Characteristic transforms.
For the proof of Theorem 6 we will apply
the method of characteristic transforms. Therefore, we will recall denitions of
characteristic transforms of probability measures on (R, B(R)), (C, B(C)) and
(X, B(X)) as well as the correspondence between probability measures and their
characteristic transforms.
Let P be a probability measure on (R, B(R)). The functions
∫
vl (t) =
|x|it sgnl xdP,
t ∈ R, l = 0, 1,
R\{0}
are called the characteristic transforms of P . The measure P is uniquely determined
by its characteristic transforms.
1. . Let Pn be a probability measure on (R, B(R)) with its characteristic
transforms vln (t), l = 0, 1. Suppose that
Ëåììà
t ∈ R, l = 0, 1,
lim vln (t) = vl (t),
n→∞
where the functions vl (t), l = 0, 1, are continuous at t = 0. Then on (R, B(R)) there
exists a probability measure P such that Pn converges mweakly to P as n → ∞. In
this case, vl (t), l = 0, 1, are the characteristic transforms of the measure P.
Now, conversely, suppose that the probability measure Pn converges mweakly to
some probability measure P on (R, B(R)) as n → ∞. Then
lim vln (t) = vl (t),
n→∞
t ∈ R,
l = 0, 1,
where vl (t), l = 0, 1, are the characteristic transforms of the measure P.
Proof of the lemma is given in [11].
Now let Pn and P be probability measures on (C, B(C)). The function
∫
v(t, k) =
|z|it eik arg z dP, t ∈ R, k ∈ Z,
C\{0}
is called the characteristic transforms of P . The measure P is uniquely determined
by its characteristic transform.
2. . Let Pn be a probability measure on (C, B(C)) with its characteristic
transform wn (t, k). Suppose that
Ëåììà
lim wn (t, k) = w(t, k),
n→∞
t ∈ R,
k ∈ Z,
where the function v(t, 0) is continuous at t=0. Then on (C, B(C)) there exists a
probability measure P such that Pn converges weakly in the sense of C to P as
n → ∞. In this case, v(t, k) is the characteristic transform of the measure P.
A. LAURINCIKAS
46
If Pn converges weakly in the sense of C to some probability measure P on
(C, B(C)) as n → ∞, then
t ∈ R,
lim vn (t, k) = v(t, k),
n→∞
k ∈ Z,
where v(t, k) is the characteristic transform of P.
Proof of the lemma is given in [8], see also [10].
Finally, let P be a probability measure on (X, B(X)). Then the functions
∫
|x|it sgnl xdPR ,
vl (t) =
t ∈ R, l = 0, 1,
R\{0}
∫
rit eikφ dPC ,
vl (t, k) =
t ∈ R,
k ∈ Z,
C\{0}
and
∫
|x|it1 sgnm xrit2 eikφ dP,
vm (t1 , t2 , k) =
t1 , t2 ∈ R,
k ∈ Z,
m = 0, 1,
X
where the last integrand is zero if x = 0 or r = 0, are called the characteristic
transforms of P .
3. . A probability measure P on (X, B(X)) is uniquely determined by
its characteristic transforms (vl (t), v(t, k), vm (t1 , t2 , k), l = 0, 1, m = 0, 1).
Ëåììà
Ëåììà 4. . Let Pn be a probability measure on (X, B(X)) with its characteristic
transforms (vln (t), vn (t, k), vmn (t1 , t2 , k), l = 0, 1, m = 0, 1), n ∈ N. Suppose that
lim vln (t) = vl (t),
n→∞
lim vn (t, k) = v(t, k),
n→∞
l = 0, 1,
t ∈ R,
t ∈ R,
k ∈ Z,
and
lim vmn (t1 , t2 , k) = vm (t1 , t2 , k),
n→∞
m = 0, 1,
t1 , t2 ∈ R,
k ∈ Z,
where the functions vl (t), l = 0, 1, v(t, 0), vm (0, t2 , 0) and vm (t1 , 0, 0), m = 0, 1, are
continuous at t = 0, t2 = 0 and t1 = 0, respectively. Then on (X, B(X)) there
exists a probability measure P such that Pn converges mweakly in the sense of X
to P as n → ∞. In this case, (vl (t), v(t, k), vm (t1 , t2 , k), l = 0, 1, m = 0, 1) are the
characteristic transforms of the measure P.
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
47
Ëåììà 5. . Let Pn and (vln (t), vn (t, k), vmn (t1 , t2 , k), l = 0, 1, m = 0, 1) be the
same as in Lemma 4. Suppose that Pn converges mweakly in the sense of X to some
probability measure P on (X, B(X)) as n → ∞. Then
lim vln (t) = vl (t),
l = 0, 1,
n→∞
t ∈ R,
lim vn (t, k) = v(t, k),
n→∞
t ∈ R,
k ∈ Z,
and
lim vmn (t1 , t2 , k) = vm (t1 , t2 , k),
m = 0, 1,
n→∞
t1 , t2 ∈ R,
k ∈ Z,
where (vl (t), v(t, k), vm (t1 , t2 , k), l = 0, 1, m = 0, 1) are the characteristic transforms
of the measure P.
Proofs of Lemmas 35 are given in [12].
Let g(m) be a multiplicative
function. We say that the function g(m) has the mean value M (g) if
Mean values of multiplicative functions.
1∑
g(m) = M (g).
x→∞ x
m≤x
lim
In this section, we will recall some known results on mean values of multiplicative
functions.
6. . In order that the mean value of the multiplicative function g(m),
|g(m)| ≤ 1, exist and be zero, it is necessary and sucient that one of the following
hypotheses should be satised:
10 For every u ∈ R,
∑ 1 − Reg(p)p−iu
= +∞.
p
p
Ëåììà
20 There exists a number u0 ∈ R such that the series
∑ 1 − Reg(p)p−iu0
p
p
= +∞
converges, and 2−riu0 g(2r ) = −1 for all r ∈ N.
The lemma is a corollary of the result of [4], see also [5].
Ëåììà 7. . Let g(m) = g(m; t1 , ..., tr ) be a multiplicative function, |g(m)| ≤ 1.
Suppose that there exists a function a(t1 , ..., tr ) such that the series
∑ 1 − Reg(p)e−ia(t1 ,...,tr )
p
p
A. LAURINCIKAS
48
converges uniformly in tj , |tj | ≤ T, j = 1, ..., r. Then, as x → ∞,
1∑
xia(t1 ,...,tr )
g(m) =
×
x m≤x
1 + ia(t1 , ..., tr )
)(
)
∞
∏(
∑
1
g(pα )
×
1−
1+
+ o(1)
α(1+ia(t1 ,...,tr ))
p
p
α=1
p≤x
uniformly in tj , |tj | ≤ T, j = 1, ..., r.
The assertion of the lemma is a particular case of a result from [13].
z
Proof of Theorem 6. Suciency. We suppose, for convenience, that 0 = 0
for all z ∈ C.
Denote by (vln (t), vn (t, k), vmn (t1 , t2 , k), l = 0, 1, m = 0, 1) the characteristic
transforms of the measure Pn in Theorem 6. Then we have that
n
1∑
vln (t) =
|g1 (m)|it sgnl g1 (m),
n m=1
t ∈ R, l = 0, 1,
n
1∑
vn (t, k) =
|g2 (m)|it eik arg g2 (m) ,
n m=1
t ∈ R,
k ∈ Z,
and
n
1∑
vrn (t1 , t2 , k) =
|g1 (m)|it1 sgnr g1 (m)|g2 (m)|it2 eik arg g2 (m) ,
n m=1
t1 , t2 ∈ R, k ∈ Z,
r = 0, 1.
Clearly, it suces to consider wrn (t1 , t2 , k) for k ∈ N0 .
In view of Theorem 3 we have that the probability measure
νn (g1 (m) ∈ A),
A ∈ B(R),
converges mweakly to a certain probability measure P on (R, B(R)), P ̸= Pa for
every a ∈ R, as n → ∞. Therefore, by the second part of Lemma 1, we obtain that
lim vln (t) = vl (t),
n→∞
t ∈ R,
l = 0, 1,
(3)
where vl (t), l = 0, 1, are the characteristic transforms of P . Since the characteristic
transforms are continuous functions, vl (t), l = 0, 1, are continuous at t = 0.
Similarly, in view of Theorem 4 we have that the probability measure
νn (g2 (m) ∈ A),
A ∈ B(C),
converges weakly to a certain probability measure P on (C, B(C)), P ({0}) ̸= 1, as
n → ∞. Therefore, by the second part of Lemma 2 we have that
lim vn (t, k) = v(t, k),
n→∞
t ∈ R,
k ∈ Z,
(4)
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
49
where w(t, k) is the characteristic transform of the limit measure P . Moreover, the
function v(t, 0) is continuous at t = 0.
It remains to consider the function vrn (t1 , t2 , k).
From the hypothesis of the theorem we have that
∑
g1 (p)g2 (p)=0
Let, for brevity,
∑′
1
< ∞.
p
(5)
means that the summation runs over those p for which
p
g1 (p)g2 (p) ̸= 0. Consider the series
def
Sr (t1 , t2 , k) =
∑ ′ 1 − Re sgnr g1 (p)|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p)
.
p
p
First we study the case r = 0. The hypotheses 10 and 20 show that the series
∑ ′ 1 − cos(tj log |gj (p)|)
∑ ′ sin2 ((tj /2) log |gj (p)|)
∑ ′ 1 − Re|gj (p)|itj
=
=2
p
p
p
p
p
p
(6)
converges uniformly in tj , |tj | ≤ t0 , for every xed t0 > 0, j = 1, 2. Therefore, in
virtue of the inequality
1 − Rez1 z2 ≤ 2(1 − Rez1 ) + 2(1 − Rez2 )
(7)
valid for |zj | ≤ 1, j = 1, 2, see [2], we have that the series S0 (t1 , t2 , 0) converges
uniformly in tj , |tj | ≤ t0 . This, (7) and Lemma 7 show that uniformly in tj , |tj | ≤ t0 ,
as n → ∞,
)(
)
∞
∏(
∑
1
|g1 (pα )|it1 |g2 (pα )|it2
v0n (t1 , t2 , 0) =
1−
1+
+ o(1).
(8)
α
p
p
α=1
p≤n
Now suppose that there exists k ∈ N such that the series
∑ 1 − ukg (p)
2
p
p
(9)
converges. Then using the hypothesis 20 and reasoning as in [3], p. 224230, we can
prove that there exists q ∈ N such that the series (9) converges if and only if q|k .
Then, for q|k , in view of (6), (7) and convergence of series (9) we obtain that the
series S0 (t1 , t2 , k) converges uniformly in tj , |tj | ≤ t0 . Thus, by Lemma 7, uniformly
in tj , |tj | ≤ t0 , as n → ∞,
)( ∑
α )
∞
∏(
|g1 (pα )|it1 |g2 (pα )|it2 eik arg g2 (p )
+o(1). (10)
v0n (t1 , t2 , k) =
1− ≥ 1p 1+
pα
α=1
p≤n
A. LAURINCIKAS
50
Now let g - k . Then the method of [3] allows to prove that, for all u ∈ R,
∑ ′ 1 − Reukg (p)p−iu
2
= +∞.
p
p
(11)
It is not dicult to see that in view of the identity
1 − z1 z2 z3 = 1 − z1 + z1 (1 − z2 ) + z1 z2 (1 − z3 )
we have that by (11) and (4)
∑ ′ 1 − Re|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p) p−iu
≥
p
p≤n
∑ ′ 1 − Re eik arg g2 (p) p−iu ∑ ′ 1 − Re|g1 (p)|it1 ∑ ′ 1 − Re|g2 (p)|it2
≥
−
−
−
p
p
p
p≤n
p≤n
p≤n
(∑
)1/2 ( ∑
)1/2
′ 1 − Re|g1 (p)|it1
′ 1 − Re|g2 (p)|it2
−2
−
p
p
p≤n
p≤n
(∑
)1/2 ( ∑
)1/2
′ 1 − Re|g1 (p)|it1
′ 1 − Reeik arg g2 (p) p−iu
−2
−
p
p
p≤n
p≤n
(∑
)1/2 ( ∑
)1/2
′ 1 − Re|g2 (p)|it2
′ 1 − Reeik arg g2 (p) p−iu
−2
→ +∞
p
p
p≤n
p≤n
as n → ∞ uniformly in tj , |tj | ≤ t0 , and all u ∈ R. Therefore, by (5)
∑ 1 − Re|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p) p−iu
p
p
= +∞
for all t1 , t2 , u ∈ R. Thus, in this case Lemma 6 shows that
lim v0n (t1 , t2 , k) = 0.
n→∞
(12)
If the series (9) does not converge for any k ∈ N, then we use the condition
∑ 1 − Reukg (p)p−iu
2
= +∞,
p
p
k ∈ N, u ∈ R, and, similarly to the case q - k , we obtain that, for all k ∈ N, t1 , t2 ∈ R,
lim v0n (t1 , t2 , k) = 0.
n→∞
Clearly,
(
)(
α )
∞
∑
1
|g1 (pα )|it1 |g2 (pα )|it2 eik arg g2 (p )
1−
=
1 +
α
p
p
α=1
(13)
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
( )
|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p)
1
=1−
+O 2
2
p
p
51
(14)
uniformly in tj , j = 1, 2. We clearly have seen that the series S0 (t1 , t2 , k), for q|k ,
converges uniformly in tj , |tj | ≤ t0 , j = 1, 2. So, in view of (12), for the convergence
of w0n (t1 , t2 , k), q|k , it remains to consider the series
def
S(t1 , t2 , k) =
Denote by
∑ ′′
∑ ′ Im|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p)
.
p
p
the summation in
∑′
over those primes p for which at least one of
∑ ′′′
inequalities | log |g1 (p)|| > 1 or | log |g2 (p)|| > 1 is satised, and let
mean the
p
∑′
summation in
over primes p satisfying | log |g1 (p)|| ≤ 1 and | log |g2 (p)|| ≤ 1.
p
p
p
Then we have that
S(t1 , t2 , k) =
∑ ′′ Im|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p)
+
p
p
∑ ′′′ Im|g1 (p)|it1 Re|g2 (p)|it2 Re eik arg g2 (p)
+
+
p
p
∑ ′′′ Re|g1 (p)|it1 Im|g2 (p)|it2 Im eik arg g2 (p)
+
+
p
p
∑ ′′′ Re|g1 (p)|it1 Re|g2 (p)|it2 Im eik arg g2 (p)
+
+
p
p
( ∑
)
′′′ Im|g1 (p)|it1 Im|g2 (p)|it2 Im eik arg g2 (p)
def
+ −
=
p
p
def
=
5 ∑
∑
j=1
j
.
(15)
The hypothesis
of theorem imply the uniform convergence in tj , j = 1, 2, for the
∑
series 1 . Furthemore, for q|k , the series
∑
2
=
∑ ′′′ (t1 log |g1 (p)| + O(t3 log2 |g1 (p)|))(1 + O(t2 log2 |g2 (p)|))
1
2
−
p
p
∑ ′′′ Im |g1 (p)|it1 Re |g2 (p)|it2 (1 − Re eik arg g2 (p) )
−
p
p
converges uniformly in tj , |tj | ≤ t0 , j = 1, 2. Similarly, we nd that the series
∑
3
=
∑ ′′′ (1 + O(t2 log2 |g1 (p)|))((t2 log |g2 (p)| + O(t3 log2 |g1 (p)|))
2
2
1
−
p
p
A. LAURINCIKAS
52
−
∑
4
=
∑
5
∑ ′′′ Re |g1 (p)|it1 Im |g2 (p)|it2 (1 − Re eik arg g2 (p) )
,
p
p
∑ ′′′ Im eik arg g2 (p) ∑ ′′′ (1 − Re |g1 (p)|it1 )Re |g2 (p)|it2 Im eik arg g2 (p)
−
−
p
p
p
p
∑ ′′′ (1 − Re |g2 (p)|it2 )Im eik arg g2 (p)
−
,
p
p
=
∑ ′′′ (1 − Re 2 |g1 (p)|it1 )1/2 (1 − Re 2 |g2 (p)|it2 )1/2 Im eik arg g2 (p)
≤
p
p
(∑
)1/2 ( ∑
)1/2
′′′ 1 − Re |g1 (p)|it1
′′′ 1 − Re |g2 (p)|it2
≤2
,
p
p
p
p
for q|k , converge uniformly in tj , |tj | ≤ t0 , j = 1, 2. Therefore, from this, (8), (10)
and (14) we deduce that, for q|k ,
lim v0n (t1 , t2 , k) = v0 (t1 , t2 , k),
n→∞
where
)(
α )
∞
∏(
∑
1
|g1 (pα )|it1 |g2 (pα )|it2 eik arg g2 (p )
v0 (t1 , t2 , k) =
1−
1+
.
p
pα
p
α=1
Hence, and from (12) and (13) we have that
lim v0n (t1 , t2 , k) = v0 (t1 , t2 , k),
n→∞
t1 , t2 ∈ R,
k ∈ Z,
and the functions v0 (0, t2 , 0) and v0 (t1 , 0, 0) are continuous at t2 = 0 and t1 = 0,
respectively.
It remains to study the characteristic transform w1n (t1 , t2 , k). We begin with
w1n (t1 , t2 , 0). First suppose that
∑ 1
< ∞.
p
(16)
g(p)<0
Then, similarly to the case r = 0, we obtain that the series S1 (t1 , t2 , 0) converges
uniformly in tj , |tj | ≤ t0 , j = 1, 2, and thus, by Lemma 7,
)
)(
∞
∏(
∑
1
|g1 (pα )|it1 sgng1 (pα )|g2 (pα )|it2
+ o(1)
v1n (t1 , t2 , k) =
1−
1+
p
pα
α=1
p≤n
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
53
uniformly in tj , |tj | ≤ t0 , j = 1, 2, as n → ∞. From this, using the above arguments,
we obtain that
lim v1n (t1 , t2 , 0) = v1 (t1 , t2 , 0),
(17)
n→∞
where
)(
)
∞
∏(
∑
1
|g1 (pα )|it1 sgng1 (pα )|g2 (pα )|it2
v1 (t1 , t2 , 0) =
1−
1+
,
α
p
p
p
α=1
and the functions v1 (0, t2 , 0) and v1 (t1 , 0, 0) are continuous at t2 = 0 and t1 = 0,
respectively.
Now let
∑ 1
= +∞
(18)
p
g1 (p)<0
This case is more complicated. We consider the series
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1 |g2 (p)|it2 p−iu
,
p
p
u ∈ R.
If u = 0, then in view of (6) and (18)
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1 |g2 (p)|it2
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1
≥
−
p
p
p≤n
p≤n
(∑
)1/2
∑ ′ 1 − Re |g2 (p)|it2
′ 1 − sgn g1 (p)Re |g1 (p)|it1
−
−2
×
p
p
p≤n
p≤n
(∑
)1/2
∑ ′ 2 − (1 − Re |g1 (p)|it1 )
′ 1 − Re |g2 (p)|it2
×
=
+ O(1)+
p
p
p≤n
p≤n
g1 (p)<0
( ∑
)1/2
′ 2 − (1 − Re |g1 (p)|it1 )
+O
→∞
p
p≤n
(19)
g2 (p)<0
as n → ∞.
Now suppose that u ̸= 0. Then, taking into account (6), we nd that, for σ > 1,
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1 |g2 (p)|it2 p−iu
=
pσ
p
(∑
)
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1 p−iu
′ 1 − Re |g2 (p)|it2
=
+O
+
σ
p
p
p
p
)1/2 ( ∑
)1/2 )
(( ∑
it
−iu
1
′ 1 − Re |g2 (p)|it2
′ 1 − sgng1 (p)Re |g1 (p)| p
=
+O
pσ
p
p
p
A. LAURINCIKAS
54
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1 p−iu
+
pσ
p
(( ∑
)1/2 )
′ 1 − sgn g1 (p)Re |g1 (p)|it1 p−iu
+O
+ O(1).
σ
p
p
=
(20)
It is not dicult to see that, for g1 (p) ̸= 0,
1 − sgn g1 (p)Re |g1 (p)|it1 p−iu =
= 1 − sgn g1 (p)Re p−iu + sgn g1 (p)Re (p−iu − |g1 (p)|it1 p−iu ) =
= 1 − sgn g1 (p) cos(u log p) + sgn g1 (p) cos(u log p)(1 − cos(t1 log |g1 (p)|))+
+sgn g1 (p) sin(u log p) cos(t1 log |g1 (p)|).
Therefore, in virtue of (6)
∑ ′ 1 − sgn g1 (p)Re |g1 (p)|it1 p−iu ∑ ′ 1 − sgn g1 (p) cos(u log p)
=
+
pσ
pσ
p
p
(∑
)
(( ∑
)1/2
′ 1 − cos(t1 log |g1 (p)|)
′ 1 − sgng1 (p) cos(u log p)
+O
+O
×
σ
p
p
p
p
(∑
)1/2 ) ∑
′ 1 − cos(t1 log |g1 (p)|)
′ 1 − sgn g1 (p) cos(u log p)
×
=
+
p
pσ
p
p
(( ∑
)1/2 )
′ 1 − sgn g1 (p) cos(u log p)
+O
+ O(1).
(21)
pσ
p
√
Now let ε > 0 is a small xed number, and a = arccos(1 − ε) < 2 ε. Then in view
of (5)
∑ ′ 1 − sgn g1 (p) cos(u log p)
∑ ′ 1 − cos(u log p)
=
+
pσ
pσ
p
g1 (p)>0
∑ ′ 1 + cos(u log p)
∑
∑
1
+
−
ε
−
ε
pσ
pσ
g1 (p)<0
g1 (p)>0
1−cos(u log p)<ε
∞
∑
∑ 1
∑
1
−
ε
≥
σ
σ
p
p
lπ−a
lπ+a
p
l=0
(
) exp{ u }<p≤exp{ u }
ε
1
≥
− cε3/2 log
→ +∞
2
σ−1
g1 (p)<0
1+cos(u log p)<ε
1
≥
pσ
≥ε
as σ → 1 + 0. Thus, (19)(22) and (5) show that, for all t1 , t2 , u ∈ R,
∑ 1 − sgn g1 (p)Re |g1 (p)|it1 |g2 (p)|it2 p−iu
→ +∞
pσ
p
(22)
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
55
as σ → 1+0. Moreover, the last sum monotonically increases as σ → 1+0. Therefore,
for all t1 , t2 , u ∈ R
∑ 1 − sgn g1 (p)Re g1 (p)Re |g1 (p)|it1 |g2 (p)|it2 p−iu
= +∞.
p
p
Therefore, by Lemma 6
lim v1 (t1 , t2 , 0) = 0.
n→∞
(23)
Now suppose that there exists at least one k ∈ N such that the series (9)
converges. If (16) is true, then similarly to the cases r = 0 and w1n (t1 , t2 , 0) we
nd that, for q|k , the series S1 (t1 , t2 , k) converges uniformly in tj , |tj | ≤ t0 , j = 1, 2,
and from this we deduce that
lim v1n (t1 , t2 , k) = v1 (t1 , t2 , k),
n→∞
(24)
where
)(
α )
∞
∏(
∑
1
|g1 (pα )|it1 sgng1 (pα )|g2 (pα )|it2 eik arg g2 (p )
1+
.
v1 (t1 , t2 , k) =
1−
α
p
p
α=1
p
Now suppose that (18) is valid. Then, the convergence of the series (9) shows
that, for q|k ,
∑ ′ 1 − sgn g1 (p)Re eik arg g2 (p)
∑ ′ 1 − Re eik arg g2 (p)
=
+
p
p
p
g1 (p)<0
∑ ′ 2 − (1 − Re eik arg g2 (p) )
+
= +∞.
p
(25)
g1 (p)>0
Now let u ̸= 0. Then, for q|k and σ > 1, by (22)
∑ ′ 1 − sgn g1 (p)Re eik arg g2 (pα ) p−iu ∑ ′ 1 − sgn g1 (p) cos(u log p)
=
+
σ
p
p
p
p
(∑
)
(( ∑
)1/2
′ 1 − cos(k arg g2 (p))
′ 1 − sgn g1 (p) cos(u log p)
+O
+O
×
p
pσ
p
p
(∑
)1/2 ) ∑
′ 1 − sgn g1 (p) cos(u log p)
′ 1 − cos(k arg g2 (p))
=
+
×
p
pσ
p
p
(∑
)1/2
′ 1 − sgn g1 (p) cos(u log p)
→ +∞
+O
σ
p
p
as σ → 1 + 0. Now this, (5), (25) and convergence of the series (6) allow to prove
that, for all t1 , t2 , u ∈ R and q|k ,
∑ 1 − sgn g1 (p)|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p) p−iu
→ +∞
pσ
p
A. LAURINCIKAS
56
as σ → 1 + 0. Hence, we have that, for all t1 , t2 , u ∈ R,
∑ 1 − sgn g1 (p)|g1 (p)|it1 |g2 (p)|it2 eik arg g2 (p) p−iu
= +∞,
p
p
therefore, Lemma 6 implies that, for q|k ,
lim v1 (t1 , t2 , k) = 0.
n→∞
(26)
If q - k , then, for all u ∈ R,
∑ ′ 1 − Re eik arg g2 (p) p−iu
= +∞.
p
p
(27)
Therefore, in the case of (16), for q - k and u ∈ R,
∑ ′ 1 − Re eik arg g2 (p) p−iu
= +∞.
p
g1 (p)>0
Hence, for q - k and u ∈ R,
∑ ′ 1 − sgn g1 (p)Reeik arg g2 (p) p−iu
∑ ′ 1 − Reeik arg g2 (p) p−iu
=
+
p
p
p
p
∑ ′ 1 + Reeik arg g2 (p) p−iu
+
= +∞.
p
p
Thus, the above arguments show that, for g - k ,
lim v1 (t1 , t2 , k) = 0.
n→∞
Now suppose that (18) and (27) take place. Then we have that
∑ 1
< ∞.
p
g1 (p)>0
Hence we have that, for all u ∈ R,
∑ ′ 1 − Reeik arg g2 (p) p−iu
= +∞.
p
g1 (p)<0
Therefore, for all u ∈ R,
∑ ′ 1 − sgn g1 (p)Reeik arg g2 (p) p−iu
∑ ′ 1 − Reeik arg g2 (p) p−iu
=
+
p
p
p
g1 (p)>0
(28)
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
57
∑ ′ 2 − (1 − Reeik arg g2 (pα ) p−iu )
+
= +∞.
p
g1 (p)<0
Hence we obtain that, for q - k ,
lim v1 (t1 , t2 , k) = 0.
n→∞
(29)
If, for all k ∈ N and u ∈ R,
∑ 1 − Reukg (p)p−iu
2
= +∞,
p
p
the reasoning similarly to the case q - k , we obtain that, for all k ∈ N, also
lim v1 (t1 , t2 , k) = 0.
n→∞
(30)
Now (3), (4), (15), (17), (23), (24), (26), (28)(30) and Lemma 4 complete the
proof of the suciency.
Necessity. Suppose that the measure Pn converges mweakly in the sense of
X to a certain probability measure P on (X, B(X)), PR ̸= Pa for every a ∈ R and
PC ({0}) ̸= 1, as n → ∞. Hence we nd that the probability measure
νn (g1 (m) ∈ A),
A ∈ B(R),
converges mweakly to a certain measure P on (R, B(R)), P ̸= Pa for every a ∈ R,
as n → ∞. Therefore, by Theorem 3 we obtain hypothesis 10 of the theorem.
Similarly, we have that the probability measure
νn (g2 (m) ∈ A),
A ∈ B(C),
converges weakly to a certain measure P on (C, B(C)), P ({0}) ̸= 1, as n → ∞.
Hence, by Theorem 4, we obtain hypotheses 20 and 30 of the theorem.
ÑÏÈÑÎÊ ÖÈÒÈÐÎÂÀÍÍÎÉ ËÈÒÅÐÀÒÓÐÛ
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A. LAURINCIKAS
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Department of Mathematics and Informatics
Vilnius University
Naugarduko 24
03225 Vilnius
Lithuania
E- mail: antanas.laurincikas@maf.vu.lt
Ïîëó÷åíî 13.05.2008