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ЧЕБЫШЕВСКИЙ СБОРНИК Том 10 Выпуск 1 (2009)
УДК 519.14
THE JOINT DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS
A. Laurincikas
Аннотация
In the paper, the existence of a joint limit distribution for a real multiplicative and a complex-valued multiplicative functions is considered. For one class of multiplicative functions, the necessary and sufficient 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, non-negative integers, integers, real and complex numbers, respectively. A function g : N ^ C is called multiplicative if g ф 0,m G N, and g(mn) = g(m)g(n) for all m,n G 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 G N, (m, n) = 1. This definition implies f (1) = 0. The main problem of the probabilistic number theory are asymptotic distribution laws for additive and multiplicative functions. We recall
nGN
//„(...) = -#{1 < m < n : ...}, n
where in place of dots a condition satisfied bv m is to be written. As usual, bv B(S) denote the class of Borel sets of the space S, and by p a prime number. The first 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-weaklv to P as n ^ ж 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
I u if |u| < 1, w = \
II if |u| > 1.
Теорема 1. ([6]). Let g(m) > 0 be a'multiplicative function. Then the probability measure
Vn(g(m) G A), A G B(R), (1)
converges m-weakly to a certain probability measure P on (R, B(R)), P({0}) = 1, as n ^ ж, if and only if the series
II9ip) — 111 II9ip) — HI2
p p p p
converge.
The first result on real multiplicative functions of an arbitrary sign belongs to A, Bakstvs [1], We recall that a probability measure P on (R, B(R)) is called symmetric if, for some a G R,
Teopema 2. ([1]). Let g(m) be a real multiplicative function. Then the probability measure (1) converges m-weakly to a certain non-symmetric probability measure on (R, $(R)), as n ^ to, if and only if the series (2) and
converge, and there exists a G N such, that g(2a) = —1.
The problem of the existence of limit distribution for real multiplicative functions was completely solved in [14], Let, for a G R and A G B(R),
Теорема 3. ([Ц]). Let g(m) be a real 'multiplicative function. Then the probabi-
(R, B(R))
P = Pa for every a G R, as n ^ ж, if and only if the series
Let Pra and P be probability measures on (C, B(C)). We say that Pn converges weakly in the sense of C to P as n ^ to if Pn converges weaklv to P as n ^ to, and, additionally, limn^^ Pn({0}) = P({0}).
Teopema 4. ([4]). Let g(m) be a complex-valued 'multiplicative function. Then the probability measure
P(-ж, a) = 1 — P(-ж, a].
| log |g(p)||<1
converge.
Now let g(m) be a complex-valued multiplicative function. Define
and
bg Ig(p)1 if e 1 < |g(p)| < e,
1 if|g(P)| < e-1 or |g(p)| > e.
vn(g(m) G A), A G B(C),
converges weakly to a certain probability measure P on (C, B(C)),P({0}) = 1, as n ^ to, if and only if the following two hypotheses hold:
10 The series
y^ Vgip) y^ vg(P)
p ’ ^ p
p p
converge;
20 Either, for all m G N and aU t G R,
1 — Re Mm(p)p-it
> ------------------= +oo,
p
p
or there exists at least one m G N such that the series
1 — um(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 difficult to
Pn P
measures on (R2, B(R2)), We say that Pn converges m-weakly to P as n ^ ж, if Pn P
lim Pn(R x {0}) = P(R x {0}),
n—
lim Pn({0} x R) = P({0} x R),
n—^
lim Pn({0} x {0}) = P({0} x {0}).
n—^
Теорема 5. ([7]). Let g1(m) and g2(m) be real 'multiplicative functions. Then the probability measure
Vn((g1(m), g2(m)) G A), A G B(R2),
converges 'm-weakly to a certain probability measure P on (R2, B(R2)),P(R x A) = Pa(A), P(A x R) = Pb(A), A G B(R2), for every a, b G R, as n ^ ж, if and only if the series
y^ l°g \gjip)\ j = i 2 1 bg2 \gj[jp)\ y^ y^ 1
9№> P ’ ’ ’ идШ0Р1+1о^\9М\’ UgfcuP
I log 1gj(p)ll<1 converge.
A twodimensional limit theorem for complex-valued multiplicative function was proved in [9],
The aim of this paper is to prove a twodimensional limit theorem for a real and a complex-valued multiplicative functions. Let, for brevity, X = R x C. Let Pn and P be probability measures on (X, B(X)). We say that Pn converges m-weakly in the sense of X to P as n ^ ж if Pn converges weaklv to P as n ^ ж, and
lim Pn(R x {0}) = P(R x {0}),
n—<^
lim Pn({0} x C) = P({0} x C),
n—<^
lim Pn({0} x {0}) = P({0} x {0}).
n—^
Define Pr(A) = P(A x C), A G B(R), and Pc(A) = P(R x A), A G B(C).
Теорема 6. . Let g1(m) be a real andg2(m) be a complex-valued multiplicative functions such that the series
- and Y^ -pp gl(p)>0 gl(p)<0
do not diverge simultaneously. Then the probability measure
Pn(A) = Vn((g1(m),g2(m)) G A), A G B(X),
X
(X, B(X)), PR = Pa for every a G R and PC({0}) = 1, as n ^ ж, if and only if the following hypotheses are satisfied:
10
£
gl(p)=0 I log |gi(p)||<1
converge;
20 The series
bg|gi(p)|
V
£
gi(p)=o
E
p
i log2 \giip) I
p 1 +log2 \gi{p)[
Vg2 ip) ^ p
E
gi(p)=o
p
converge;
30 Either, for all k G N and a 11 u G
+ж,
or there exists at least one k G N such that the series
1 P)
p
converges.
1
Characteristic transforms. For the proof of Theorem 6 we will apply the method of eharaeteristie transforms. Therefore, we will recall definitions of characteristic transforms of probability measures on (R, 0(R)), (C, 0(C)) and (X, 0(X)) as well as the correspondence between probability measures and their characteristic transforms.
Let P be a probability measure on (R, 0(R)), The functions
vj(t) = J Ix^sgn^xdP, t G R, l = 0,1,
R\{0}
PP
by its characteristic transforms,
JIemma 1. . Let Pn be a probability meas ure on (R, 0(R)) with its characteristic transforms v1n(t), l = 0,1. Suppose that
lim vto(t) = vi(t), t G R,1 = 0,1,
n—<^
where the functions vj(t), l = 0,1, are continuous at t = 0. Then on (R, 0(R)) there exists a probability measure P such, that Pn converges m-weakly to P as n ^ to. In this case, vj(t),/ = 0,1, are the characteristic transforms of the measure P.
Now, conversely, suppose that the probability measure Pn converges m-weakly to some probability measure P on (R, B(R)) as n ^ to. Then
lim vi„(t) = vi(t), t G R, l = 0,1,
n—
where vj(t), / = 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, 0(C)), The function v(t,k)= J |z|VfcargzdP, t G R, k G Z,
C\{0}
PP
by its characteristic transform,
JIemma 2. . Let Pn be a probability measu,re on (C, 0(C)) with, its characteristic transform wn(t, k). Suppose that
lim wn(t, k) = w(t, k), t G R, k G Z,
n—^
where the function v(t, 0) is continuous at t=0. Then on (C, 0(C)) ttere a
Pn C
n ^ to. In this case, v(t, k) is tte characteristic transform of the measure P.
If Pn converges weakly in the sense of C to some probability measure P on (C, B(C)) as n ^ to, then
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
where the last integrand is zero if x = 0 or r = 0, are called the characteristic P
Лемма 3. . A probability measure P on (X, B(X)) is uniquely determined by its characteristic transforms (^(t), v(t, k), vm(ti, t2, k), l = 0,1,m = 0,1).
Лемма 4. . Let Pn be a probability measu,re on (X, B(X)) with its characteristic transforms (vln(t), vn(t, k), vmn(t1; t2, k),l = 0,1, m =0,1), n G N. Suppose that
where the functions v(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)) ttere ercisfc a probability measure P such, that Pn converges m-weakly in the sense of X to P as n —> to. In this case, (v^(t), v(t, k), vm(t1, t2, k), l = 0,1,m = 0,1) are tte characteristic transforms of the measure P.
lim vn(t, k) = v(t, k), t G R, k G Z,
and
vm(ti,t2,k) =
x|it^nmxrit2eifc^dP, t1,t2 G R, k G Z, m = 0,1
X
lim vln(t) = v^(t), l = 0,1, t G R,
n—
lim vn(t, k) = v(t, k), t G R, k G Z
n—^
and
lim vmn(ti,t2,k) = vm(ti,t2,k), m = 0,1, ti,t2 G R, k G Z,
n—^
JIemma 5. . Let Pn and (vln(t), vn(t, k), vmn(ti, t2, k), l = 0,1,m = 0,1) be the
Pn X
probability measure P on (X, B(X)) as n ^ to. Then
lim vln(t) = vj(t), l = 0,1, t G R,
n—^
lim vn(t, k) = v(t, k), t G R, k G Z,
n—^
and
lim vmn(ti,t2,k) = vm(ti,t2,k), m = 0,1, ti,t2 G R, k G Z,
n—^
where (vj(t), v(t, k), vm(ti, t2, k), l = 0,1, m = 0,1) are Йе characteristic transforms of the measure P.
Proofs of Lemmas 3-5 are given in [12].
Mean values of multiplicative functions. Let g(m) be a multiplicative function. We say that the function g(m) has the mean value M(g) if
lim — g(m) = M(g). x—^ x 1'
m<x
In this section, we will recall some known results on mean values of multiplicative functions.
Лемма 6. . In order that the mean value of the 'multiplicative function g(m), |g(m)| < 1, exist and be zero, it is necessary and sufficient that one of the following hypotheses should be satisfied:
10 For every u G R,
1 — R eg(p)p-
,-ги
1 - ney\JJ)p _
^ p 00’
p
20 There exists a number u0 G R such that the series
1 — Reg(p)p~vao ^ p °°
converges, and 2 rm°g(2r) = — 1 for a 11 r G N.
The lemma is a corollary of the result of [4], see also [5]
JIemma 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
y-v 1 — Reg{jp)e т(*ь-,ir)
p
converges uniformly in t,, | < T, j = 1,r. Then, as x ^ to,
1 __^ xia(t1 V^r)
- _ 1 + m(t1,...,tr)x
m<x
p<x V V a=1 ^ 7
uniformly in t,, |t,| < T, j = 1,..., r.
The assertion of the lemma is a particular case of a result from [13],
Proof of Theorem 6. Sufficiency. We suppose, for convenience, that 0z = 0 for all z e 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
1 n
vin(t) = - V] |gi(m)|rtsgn^i(m), i G R, I = 0,1,
m=1 1 n
vn(t,k) = -V |g2(m)|Vfcargfl2(m), t Gl, fcGZ, n
m=1
and
Vm(tl,t2,k) = ~ V tl,t2 G R’ fc f Z’
n ' r = 0,1.
m=1
Clearly, it suffices to consider wrn(t1; t2, k) for k e N0.
In view of Theorem 3 we have that the probability measure
Vn(g1(m) e A), A e B(R),
converges m-weakly to a certain probability measure P on (R, B(R)),P = Pa for every a e R, as n ^ to. Therefore, bv the second part of Lemma 1, we obtain that
lim vin(t) = vz(t), t e R, l = 0,1, (3)
n—
where vz (t), l = 0,1, are the characteristic transforms of P. Since the characteristic transforms are continuous functions, vz(t), l = 0,1, are continuous at t = 0, Similarly, in view of Theorem 4 we have that the probability measure
Vn(g2(m) e A), A e B(C),
converges weakly to a certain probability measure P on (C, B(C)),P({0}) = 1, as n ^ to. Therefore, by the second part of Lemma 2 we have that
lim vn(t,k) = v(t,k), t e R, k e Z, (4)
n
n
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(ti, t2, k).
From the hypothesis of the theorem we have that
5] I < °°- (5)
gl(p)g2(p)=0
Let, for brevity, means that the summation runs over those p for which
p
g1(p)g2(p) = 0. Consider the series
I^*11 ^ (P) I**2 argg2 (p)
P
O ^ * n */ V-'l-Re Sgn’-flTiO?) IflT! (p) |iAl |^2(p) |it2ei Sr{ti,t2,k) = 2_^ -----------------------------------------------------------------
p
First we study the case r = 0. The hypotheses 10 and 20 show that the series y^/ 1 - Re\gj(p)\itj = y^/ 1 - cos (tj log | (p) |) = sm2{{tj/2)log\gj{p)\)
p p p
p p p
(6)
converges uniformly in tj, |tj| < t0, for every fixed t0 > 0, j = 1, 2, Therefore, in virtue of the inequality
1 — Rez1z2 < 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 —— to,
0,0) = n K) 0+E lgl(p°)'^(p°)l‘fa) +*)• 0)
p<n ' p/ V a=1 P 7
Now suppose that there exists k G N such that the series
(9)
pp
20
prove that there exists q G 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 uniformlv in tj, |tj | < t0. Thus, bv Lemma 7, uniformly in tj, |tj| < t0, as n — to,
/ \/ ~ |g1(pa) 1**1 |o2(Pa) 1**2eikargg2(pa)\
Von(ti,t2, k) = ( 1- > 1 Pj ( 1+^-------------------—------------J +0(1). (10)
p<n ' ' ' a=1 P '
Now let g \ k, Then the method of [3] allows to prove that, for all u G R,
(p)p-‘- = +co (11)
pp
It is not difficult to see that in view of the identity
1 — Z1Z2Z3 = 1 — Z1 + ^1(1 — Z2) + ^1^2(1 — Z3)
we have that by (11) and (4)
. ^ 1 - Re |(p) ||(p) Ieifc argS2
p ~
p<n
Y^/l-R eeifcargfl2(p)p-iM 1 - Rel^iCp)!^1 1 - Re|^2(p)|it2
— z_^ p z_^ p z_^ p
p<n p<n p<n
1 — Re|g1(p)|itl \1/2/^ / 1 — Re|g2(p)|it2 \ 1/2
pp p<n ' x p<n
1 - Re|fl,i(p)litiy/2/^/ 1 - Reeifcargfl2^p_i“^1/2
pp p<n 7 v p<n
/ ^2' 1 - Re\g2{p)\it2\1/2 1 - Reeifcargfl2(-P')p_i“\ 1/2 +oo
pp p<n ' x p<n 7
as n — to uniformly in tj, |tj | < t0, and all u G R. Therefore, bv (5)
X ^ 1 - Re I (/1 (p) |Ul IQ2 ip) |it2 eik arg fl2 _
p
t1, t2, u G R
lim V0ra(t1,t2,k) = 0. (12)
n—<^
kGN ^ 1 — Reu! (p)p-iu
> -----------------------= +°°,
p
p
k G N, u G R, and, similarly to the case q { k, we obtain that, for all k G N, t1, t2 G R,
lim V0„(t1,t2,k) = 0. (13)
n—
Clearly,
1 ^f1 + £ |^i(pa)|itl|^2(pa)|it2eifcargfl2(p“)
_1 k(p)r%2(p)r^fcargfl2(p) |
P2 VP2/
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
CV, , M def ^'Im\gi(p)\^\g2(p)f2e^92^
s(ti,t2,k) = 2^------------------------------------•
p
Denote bv " the summation in ; over those primes p for which at least one of pp inequalities | log |g1(p)|| > 1 or | log |g2(p)|| > 1 is satisfied, and 1 et J2'" mean the
p
summation in over primes p satisfying | log |g1(p)|| < 1 and | log |g2(p)|| < 1.
p
Then we have that
Sihhk) = Y” lm\9i(p)\Ul\92(p)\lt2elkaTg92{p) |
’ ’ ^ p
p
y. - /// Im | gi (p) |ltl Re | g2 (p) |%t2 Re e%k arg 92 ^
^ p
p
x - /// Re | gi (p) |%tl Im | g2 (p) |%t2 Im elk arg 92 ^
^ p
p
x - in Re | gi (p) |ltl Re | g2 (p) |%t2 lme%k arg 92 ^
^ p
p\ y-v iii Im|gi(p)|rtlIm|g2(p)|rt2Imetfcargs'2(^)^ def
pp p
5
def
j.
j=1 1
The hypothesis of theorem imply the uniform convergence in tj ,j = 1, 2, for the 1 q| k
2=
p
— £
(t 1 log1 g1 (p) 1 + O(t3 log21 g1 (p) 1 ))(1 + °(t2 fog21 g2 (p) 1))
p
m to |g1(p) |**^Re |g2(p)|it2 (1 — Re eik arg g2(p))
p
p
converges uniformly in tj, |tj| < t0, j = 1, 2, Similarly, we find that the series ./// (1 + O(t2log2 |g1(p)|))((t2log |g2(p)I + O(t3log2 |g1(p)|))
3=
p
Re \giijp)\%tllm |<72(_p)I**2(1 — Re etfcargfl2^) p '
p
x - Im etfcargfl2(p) (1 — Re | <7i (p) |rtl )Re |g2(p)|rt2Im etfcargfl2(p)
Z^4 p p
pp
,„ (1 _ Re Iflr2(p)|it2)lm eifcargfl2(p)
pp p
y-v y~v»> (1 — Re 2|,i(p)|-)1/2(l — Re 2|,2(p)|-)1/2lm
<
p
< 2 / ^ /// 1-Re 1/2 / ^ /// 1-Re |g2(p)|if2\ 1/2
p p p p for q|k, converge uniformly in tj, |tj| < t0, j = 1, 2, Therefore, from this, (8), (10)
q| k
lim V0n(t1,t2, k) = V0(t1,t2, k),
n—<^
where
]\ / |g1(p“)|i*l |g2(p“)|**2e*kargg2(p“)
V0l
(ii, t2, k) — J^[ ( 1---------------) ( 1 + ^
Hence, and from (12) and (13) we have that
lim v0n(t1,t2,k) = V0(t1,t2,k), t1,t2 G R, k G Z,
n—<^
and the functions v0(0,t2, 0^d 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
Y, I<oc- (16)
s(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, bv Lemma 7,
m*,*, *>=n(i--)(i+f: lgi(p“)|i,lsgngiip°)lg2(p°)r‘2)+o(D
uniformly in tj, |tj| < t0, j = 1, 2, as n — to. From this, using the above arguments, we obtain that
lim V1n(t1,t2, 0) = V1(t1,t2, 0), (17)
n—^
where
1 \ ^ I g1 (pa) Irtl sgng1 (pa) | g2 (pa) | **2
V1(
p/\ — pa
p v v a=1
and the functions v1(0,t2, 0^d v1(t1, 0, 0) are continuous at t2 = 0 and t1 = 0, respectively.
Now let
I= +oc (18)
gl(p)<0
This case is more complicated. We consider the series
/ 1 - sgn gi(p)Re \gi(p)\Ul\g2(p)\it2p~iu u(_R
p
u=0
y^/ 1 - sgn ffi(p)Re \gi(p)\Ul\g2(p)\it2 > y^' 1 - sgn ffi(p)Re \gi(p)\Ul
1 - Re \92(p)\it2 J ^2' 1 - sgn gi(p)Re \gi(p)\Ul\1/2
pp
p< n p< n
x
x| y^'1-R.c te(p)l‘ay/2_ y*' 2 — (1 - Re |gl(p)|*) : e(1)
p< n p p< n p
gl(\)<0
(19)
p< n p
g2(p)<0
as n — to.
Now suppose that u = 0, Then, taking into account (6), we find that, for a > 1,
/ 1 — sgn #i(p)Re \gi(p)\Ul\g2(p)\it2p iu
pf
p
^ / 1 - sgn ffi(p)Re \gi(p)\Ulp iu , Qf ^2> 1 - Re \g2(p)\it2
pCT \ *■—' p
+
+0/ / J2' 1 ~s&n9i(p)Re \9i(p)\UiP iu\1/2 / 1 - Re \g2(p)\it2^ 1/2
£'
1 — sgn g^p)Re |g1(p)|itlp pa
+Q ^ ' 1 ~Sgn gi^Re \9i(p)\itlrP iu^j 1/2^j + (20)
It is not difficult to see that, for g1(p) = 0,
1 — sgng1(p)Re |g1(p)|itlp-iu =
= 1 — sgn g1(p)Re p-iu + sgn g1(p)Re (p-iu — Ig^p)^p-iu) =
= 1 — sgn g1(p) cos(ulogp) + sgn g1(p) cos(ulogp)(1 — cos(t1 log |g1(p)|))+ +sgn g1(p) sin(ulogp) cos(t1 log |g1 (p)|).
Therefore, in virtue of (6)
1 — sgn g^p)Re |g1(p)|itlp-iu v-^' 1 — sgn g1(p) cos(u logp)
y-v> 1 — sgn gi(p)Ke \gi(p)rLp
pCT pff
1 /2
+0i ^2'1 - cos(^iIogMp)IA + o(Yj^ /1 _sgn£i(-P)cos(Ml°g-PA
p / V V po /
pp
1 /2
x 1 - cos(tj log \gi(p)|) \ 7 \ = y^1 1 — sgn 01 (p) cos(mlogp)
Z^ p y y Z^ per
pp +°((£' 1 ~ Sg“ 9l(^COS(Ml°gp))‘/2) +0(1). (21)
Now let e > 0 is a small fixed number, and a = areeos(l — e) < 2i/s. Then in view of (5)
' 1 — sgn gi(p) cos[u logp) ' 1 _ cos(m logp)
Z^ p(T Z_/ per
p gl(p)>0
^—"\ ' 1 + cos(u log p) ^—T 1 ^—T 1
+ >------------------£ > ----------£ > — >
po- pff pCT
gl(p)<0 gl(p)>0 gl(p)<0
1—cos(u log p)<£ 1+cos(u log p)<£
^_ 1 ^ ______________ 1
Sf£^-f£ £
P 1=0 eXp{^}<p<exp{^}
- _ C£3/2) log -*■ +°° (22) as a — 1 + 0 Thus, (19)-(22) and (5) show that, for all t1, t2, u G R,
1 — sgn g1(p)Re |g1(p)|itl |g2 (p) |it2p—iu
>-----------------------------------> + OO
po
as a — 1+0. Moreover, the last sum monotonically increases as a — 1+0. Therefore,
t1, t2, u G R
1 — sgn g1(p)Re g1(p)Re Ig^p)^ |g2(p)|it2p—
> = +oo.
p
p
Therefore, by Lemma 6
lim v1(t1,t2, 0) = 0. (23)
n—<^
Now suppose that there exists at least one k G N such that the series (9) converges. If (16) is true, then similarly to the cases r = 0 and w1n(t1,t2, 0) we find that, for qIk the series S1(t1,t2, k) converges uniformly in tj, ItjI < t0, j = 1, 2, and from this we deduce that
lim V1n(t1,t2, k) = V1(t1,t2, k), (24)
n—^
where
1 ^ Ig1 (pa) Irtlsgng1 (pa) Ig2(pa) Iit2eikargg2(pa)'
V1(
(ti,t2,k) = n 1 - - 1 + Y1
p/\ --------- p
p N 7 N a=1
Now suppose that (18) is valid. Then, the convergence of the series (9) shows qI k
1 — sgn g^p)Re eikargg2(p) ^' 1 — Re eikargg2(p)
p T7<0 p
f gl(p)<0
v^' 2 — (1 — Re eikargg2(p)) . ,
+ £ ----------------------------1 = +°°- (25)
gl(p)>0
Now let u = 0. Then, for qIk and a > 1, bv (22)
1 — sgn g^p)Re eikargg2(pa)p—^' 1 — sgn g1(p) cos(u logp)
y-y I — sgn gi{p)ixe >p /
+ ^ / 1 - cos(fc argg2(p))\ +o(Yj^/ 1 -sgn ^i(p)cos(m1°SP)\1/2
p / V V po /
' p / \ \ p /
' 1 — cos(fc a,Tgg2(p)) \ ' 1 — sSn S'i(p) cos(M logp)
x| p J J = <L ^ +
pp
X
+o( £'
po
p
+
as a — 1+0. Now this, (5), (25) and convergence of the series (6) allow to prove t1, t2, u G R qIk
1 — sgn g1(p)Ig1(p)Iitl Ig2(p) Iit2eikargg2(p)p—
>---------------------------------------------------------> +oo
po
as a — 1 + 0, Hence, we have that, for all t1, t2, u G R,
1 — sgn g1(p)Ig1(p)Iitl Ig2(p)Iit2 eik arg g2(p)p—
/ --------------------------------------------------------= +°°)
p
p
qI k
lim v1(t1, t2, k) = 0. (26)
n—<^
If q f k, then, for all u G R,
^^' 1 — Re eik arg g2(p)p—
p
+to. (27)
Therefore, in the ease of (16), for q f u G R,
1 — Re eik arg g2(p)p—
'
+.
p
gl(p)>0
Hence, for q f ^d u G R,
1 - sgn g1{p)E£eikarg92(-p')p-iu _ 1 - Reeikaig32^p-iu
pp
pp
v^' 1 + Reeifc arg g2(p)p—
> -------------------------= +oo.
p p
gfk
lim v1(t1, t2, k) = 0. (28)
n—^
Now suppose that (18) and (27) take place. Then we have that
y i<oo.
p
gl(p)>0
u G R
V^' 1 — Reeifc arg g2(p)p—
> ------------------------= +00.
p
gl(p)<0
u G R
1 - sgn g1{p)E£eikarg92(-p')p-iu _ 1 - Reeifcarg92(p}p~iu
P , P
p gl(p)>0
v^' 2 — (1 — Reeifc argg2(p )p—iu)
+ > ----------- ------------------- —- = +00.
p
gl(p)<0
qfk
lim v1(t1, t2, k) = 0. (29)
n—<^
If, for all k G N and u G R,
^ 1 — Reu! (p)p—
/ ---------------------= +°°)
p
p
q f k k G N
lim v1(t1, t2, k) = 0. (30)
n—<^
Now (3), (4), (15), (17), (23), (24), (26), (28)-(30) and Lemma 4 complete the proof of the sufficiency.
Necessity. Suppose that the measure Pn converges m-weaklv in the sense of
X to a certain probability measure P on (X, B(X)), PR = Pa for everv a G R and
PC({0}) = 1, as n — to. Hence we find that the probability measure
Vn(g1(m) G A), A G B(R),
converges m-weaklv to a certain measure P on (R, B(R)), P = Pa for eve rv a G R,
as n — to. Therefore, by Theorem 3 we obtain hypothesis 10 of the theorem. Similarly, we have that the probability measure
Vn(g2(m) G A), A G B(C),
converges weakly to a certain measure P on (C,B(C)),P({0}) = 1, as n — to. Hence, bv Theorem 4, we obtain hypotheses 20 and 30 of the theorem.
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[9] Laurincikas A. On the distribution of complex-valued multiplicative functions // J. Theorie Nombres Bordeaux. 1996. No 8. P. 183-203.
[10] Laurincikas A. Limit Theorems for the Riemann Zeta - Function, Kluwer, Dordrecht. 1996.
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Department of Mathematics and Informatics
Vilnius University
Naugarduko 24
03225 Vilnius
Lithuania
E- mail: [email protected]
Получено 13.05.2008
