Mean behaviour of uniformly summable
Q-multiplicative functions
Dissertation
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
der Fakult¨
at f¨
ur Elektrotechnik, Informatik und Mathematik
der Universit¨
at Paderborn
vorgelegt von
Yi-Wei Lee-Steink¨
amper
Paderborn, den 7. September, 2005
Gutachter: Prof. Dr. Karl-Heinz Indlekofer
Prof. Dr. Imre K´
atai
Prof. Dr. Wolfgang Schwarz
Tag der m¨
undlichen Pr¨
ufung: 14. Dezember 2005
Dedicated to my family
4
Contents
Introduction 7
1 Additive functions 13
1.1 Definition and introduction . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 The Tur´
an-Kubilius inequality . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Finitely distributed functions . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Multiplicative functions of modulus ≤121
2.1 Definition.................................. 21
2.2 Mean-value theorem for multiplicative functions of modulus ≤1..... 22
3 Uniformly summable functions 27
3.1 Definition.................................. 27
3.2 Mean behaviour of uniformly summable multiplicative functions . . . . . 28
3.3 Mean behaviour of α-almost-periodic multiplicative functions . . . . . . 31
4Q-additive and Q-multiplicative functions 35
4.1 Definition.................................. 35
4.2 Generalized Tur´
an-Kubilius inequalities for Q-additive functions . . . . . 37
4.3 Limit distributions of Q-additive functions . . . . . . . . . . . . . . . . . 45
4.4 Mean-value theorem for Q-multiplicative functions . . . . . . . . . . . . 47
5
CONTENTS
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications 51
5.1 Mainresults................................. 51
5.2 Preliminaryresults ............................. 54
5.3 Proofofmainresults ............................ 66
5.4 Application to q-additivefunctions..................... 69
5.5 Characterization of almost-periodic q-multiplicative functions . . . . . . 71
6 Mean behaviour of uniformly summable Q-multiplicative functions 77
6.1 Mainresults................................. 77
6.2 Preliminaryresults ............................. 79
6.3 Proofofmainresults ............................ 85
Bibliography 91
6
Introduction
The theory of additive and multiplicative functions has made great progress during the
past years; in particular, we want to mention the mean-value theorems by H. Delange [9],
E. Wirsing [83], [84] and G. Hal´
asz [22], as well as, the first elementary proof of the
theorem of Hal´
asz by H. Daboussi and K.-H. Indlekofer [8].
H. Delange [9] characterized those multiplicative functions fwhich satisfy |f| ≤ 1and
for which a non-zero mean-value
M(f) = lim
x→∞
1
x
x
X
n=1
f(n)
exists. It turned out to be more difficult though to characterize those multiplicative func-
tions fwith |f| ≤ 1, for which a mean-value M(f)exists and is zero. It was done in
essentially two steps; for real-valued functions by E. Wirsing [83], [84], this included
the proof of an old conjecture, variously ascribed to Erd˝
os and Wintner, to the effect
that a mean-value M(f)always exists whenever fassumes only the values ±1; and for
complex-valued functions by G. Hal´
asz [22] using an analytic method. The first elemen-
tary proof of the theorem of Hal´
asz was given by H. Daboussi and K.-H. Indlekofer [8].
The Erd˝
os-Wintner conjecture includes the prime number theorem because the assertion
M(µ)=0, where µdenotes the M¨
obius function, is equivalent to the prime number
theorem
]{p|pprime, p ≤x}=: π(x)∼x
log xas x→ ∞ ,
7
Introduction
as was shown by E. Landau [56]. The elementary proofs by Wirsing and Daboussi-
Indlekofer provide, among other things, an elementary proof of the prime number the-
orem.
While these theorems use the hypothesis |f| ≤ 1, many results not based on this condi-
tion are also known now. In this context, let us mention the papers by H. Daboussi [6],
P.D.T.A. Elliott [13], K.-H. Indlekofer [25], J. Knopfmacher [54] and W. Schwarz [75].
All these results provide valuable methods for the investigation of additive and multi-
plicative functions, as well as, for prime number theory. However, the actual calculation
of values of additive and multiplicative functions requires knowledge of the prime factor
decomposition of a number, while usually its q-adic or Q-adic representation (“Cantor
expansion”) is given.
Let {qr}r≥1be a sequence of natural numbers with qr≥2, and let Q0= 1,Qr=qrQr−1
for r≥1. Each nonnegative integer nhas a unique Q-adic representation (“Cantor ex-
pansion”)
n=X
r≥0
εr(n)Qr
if the following condition is satisfied
0≤εr(n)< qr+1 , r ≥0.
In the case qr≡q≥2we will use standard notation of q-adic representation.
This motivates the investigation of functions that are additive or multiplicative with re-
spect to these representations. Such functions are called q-, Q-additive, or q-, Q-multiplicative,
respectively. Mean value theorems hold for this setting, too, and the methods and results
bear certain analogies to the classical case; however, there are also some peculiari-
ties. The case |f| ≤ 1for q-multiplicative functions has been treated by H. Delange [10]
to great extent, and its generalization to Q-adic representations for the case |f| ≤ 1by J.
8
Introduction
Coquet in his thesis [4]. The results for q-additive functions — which can be derived in
some cases from the theory of q-multiplicative functions — provide interesting statistical
tests for the randomness of data (see E. Manstaviˇ
cius [63]).
In this thesis, we prove, both for the q-adic case and general Q-adic representations, new
theorems about the average of multiplicative functions without the assumption |f| ≤ 1; it
turns out that the class of uniformly summable functions is the appropriate generalization.
In this context, we also investigate α-almost-periodic q-multiplicative functions.
To make the analogy to ”classical” additive and multiplicative functions apparent, it is
appropriate to summarize results related to these first.
We proceed as follows: Chapter 1 presents some well-known facts about additive func-
tions. G. H. Hardy and S. Ramanujan proved that ωand Ωhave the normal order log log n.
P. Tur´
an found a new proof of Hardy and Ramanujan’s result using an inequality which
is analogues to Tschebycheff’s inequality. This gave M. Kac the idea of thinking about
the role of independence in the application of probability theory to number theory. The
generalization of this inequality is the famous Tur´
an-Kubilius inequality. The important
theorems of P. Erd˝
os [14], M. Kac [15], [17] and A. Wintner [16] are introduced.
Since the main difficulties arise from the fact that the asymptotic density gives only a
finitely additive measure (or content or pseudo-measure) on the family of subsets of N,
where it is defined, one constructs a sequence of finite, purely probabilistic models which
approximate the number theoretical phenomena, and then use arithmetical arguments for
”taking the limit”. J. Kubilius [55] constructed such finite probability spaces on which in-
dependent random variables could be defined to mimic the behaviour of truncated additive
functions. K.-H. Indlekofer [46] presents an integration theory on Nusing the Stone-ˇ
Cech
compactification βNof Nwhich can be generalized to arbitrary sets.
In Chapter 2, we describe the mean behaviour of complex-valued multiplicative functions
fsuch that |f(n)| ≤ 1for every positive integer n. These functions fwhich satisfy
9
Introduction
|f(n)| ≤ 1for all n∈Nand for which a non-zero mean-value exists were characterized
by H. Delange [9] in 1961, but his method could not be modified to consider the case
M(f) = 0. In 1967, E. Wirsing [84] proved his celebrated mean-value theorem which
asserts, in particular, that any real-valued multiplicative functions fof modulus ≤1, has
a mean-value. This solved a famous conjecture of Erd˝
os and Wintner. His proof was
done by elementary methods (and thus, he gave another elementary proof of the prime
number theorem), but he could not handle the complex-valued case in its full generality.
Only by an analytic method, found by G. Hal´
asz [22] in 1968 the asymptotic behaviour
of X
n≤x
f(n)could be fully determined for all complex-valued multiplicative functions f
of modulus smaller than or equal to one.
The first elementary proof of the theorem of Hal´
asz was given by H. Daboussi and K.-
H. Indlekofer [8] in 1992. More general, K.-H. Indlekofer, I. K´
atai and R. Wagner [44]
in 2001, compare the asymptotic behavior of X
n≤x
f(n)and X
n≤x
g(n)for multiplicative
functions fand g, respectively, where |f| ≤ g. They obtain generalizations of Wirsing’s
result and extend the theorem of Hal´
asz.
In Chapter 3, we introduce the space L∗of uniformly summable functions; this notion
was introduced by K.-H. Indlekofer. Let α∈Rwith α≥1, and let
Lα:= {f|f:N→C,kfkα<∞}
be the vector-space of arithmetical functions fwith bounded semi-norm
kfkα:= Ãlim sup
N→∞
1
NX
n<N |f(n)|α!1
α
.
An arithmetical function f∈L1is said to be uniformly summable if
lim
K→∞ sup
N≥1
1
NX
n<N
|f(n)|≥K
|f(n)|= 0,
and the space of all uniformly summable functions is denoted by L∗.
Let β > α > 1, then
Lβ$Lα$L∗$L1.
10
Introduction
The idea of uniform summability turned out to provide the appropriate tools for describ-
ing the mean behaviour of a large class of multiplicative functions. As typical results, we
mention the theorem by K.-H. Indlekofer, which generalizes results of Daboussi, Delange,
Hal´
asz and Wirsing. In addition, the spaces Bα,Dαand Aαof α-even, α-limit-periodic
and α-almost-periodic arithmetical functions are considered. Finally, a complete char-
acterization of α-almost-periodic multiplicative functions given by K.-H. Indlekofer is
presented without proof.
The main topic of Chapter 4 is the investigation of q-additive, q-multiplicative functions,
and Q-additive, and Q-multiplicative functions, respectively. Observing that q-additive
functions are sums of “almost independent random variables”, we give a new proof of
the Tur´
an-Kubilius inequality for q-additive functions which is much shorter than the
proof given by M. Peter and J. Spilker [78] in 2001, and which extends this proof to
Q-additive functions. In the case of the q-adic scale, necessary and sufficient conditions
for the existence of an asymptotic distribution for a real-valued q-additive function and
the mean behaviour of q-multiplicative functions of modulus ≤1have been given by H.
Delange [10] in 1972. J. Coquet [4] considered in 1975 the same kind of problems in the
cases of Q-adic scales and obtained mainly sufficient conditions. Their main results are
formulated.
Chapter 5 and 6 contain our main results. The aim of Chapter 5 is to study the be-
haviour of the means 1
NX
n<N
f(n)and 1
NX
n<N |f(n)|αas N→ ∞,α > 0, where fis
uniformly summable and q-multiplicative, and we give a complete characterization of
these means. To our surprise, we find that for q-multiplicative functions the space Lα
for every α > 0coincides with the space L∗. Furthermore, applying our main results,
we investigate finitely distributed q-additive functions and find characterizations for q-
multiplicative functions belonging to the space D1of limit-periodic functions and the
11
Introduction
space A1of almost-periodic functions by their respective spectrum σ(f).
In Chapter 6, we extend the results of Chapter 5 to uniformly summable Q-multiplicative
functions. In the case of a bounded sequence {qr}r≥1we have similar theorems as in the
q-adic case. In the case of an unbounded sequence {qr}r≥1the situation is quite different.
Unavoidable for unbounded sequences {qr}r≥1is the existence of a so-called first digit
phenomenon.
We investigate the mean behaviour of uniformly summable Q-multiplicative functions
that belong to L2and for which the first digit condition
max
1≤j≤qr−1
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0 as r→ ∞
holds.
Acknowledgment
I like to express my gratitude to Professor Indlekofer for valuable discussions, helpful
suggestions and comments.
12
Chapter 1
Additive functions
In this chapter, we present some well-known facts about additive functions. Furthermore,
the Tur´
an-Kubilius inequality and Erd˝
os’ characterization of finitely distributed functions
is discussed.
1.1 Definition and introduction
We denote by N,N0,P,R, and Cthe sets of positive integers, non-negative integers,
prime, real, and complex numbers, respectively. An ”arithmetical function” is a map
f:N→C, defined on the set Nof natural number. The set CNof arithmetical functions
becomes a C-vector-space (CN,+,·)by defining addition and scalar multiplication as
follows:
(f+g) : n7→ f(n) + g(n), λ ·f:n7→ λ·f(n).
Definition 1.1.1. An arithmetical function gis additive if
g(m·n) = g(m) + g(n)(1.1)
whenever mand nare coprime. If (1.1) holds for all m, n, then fis called completely
additive. An additive function gis called strongly additive if the values of gat prime-
13
1 Additive functions
powers are restricted by the condition
g(pk) = g(p),if k= 1,2, . . . .
Because of the canonical representation
n=Y
p∈P
pαp(n)with pαp(n)kn
of the integers n∈Nwe have
gÃY
p∈P
pαp(n)!=X
p∈P
g¡pαp(n)¢.
G. H. Hardy and S. Ramanujan [23] considered the arithmetical functions ωand Ω, where
ω(n)and Ω(n)denote the number of different prime divisors and of all prime divisors -
i.e. counted with multiplicity - of an integer n, respectively. They proved that ωand Ω
have the normal order ”log log n”. Here we say, roughly, that an arithmetical function f
has the normal order F, if f(n)is approximately F(n)for almost all values of n.1More
precisely, this means that
(1 −ε)F(n)< f(n)<(1 + ε)F(n)
for every positive εand almost all values of n.
In 1934, P. Tur´
an [80] gave a simple proof of Hardy and Ramanujan’s result. It depended
upon the readily obtained estimation
X
n≤x
(ω(n)−log log x)2≤c·x·log log x.
for some constant c.
This inequality - reminding us of Tschebycheff’s inequality2- had a special effect, namely
1A property Eis said to hold for almost all nif lim
x→∞
x−1#{n≤x:Edoes not hold for n}= 0.
2At that time P. Tur´
an knew no probability (see chapter 12 of [12]). The first widely accepted axiom
system for probability theory, due to A. N. Kolmogorov, had only appeared in 1933.
14
1.1 Definition and introduction
gave M. Kac the idea of thinking about the role of independence in the application of
probability theory to number theory. Making essential use of the notation of independent
random variables, the central limit theorem, and sieve methods, M. Kac together with P.
Erd˝
os proved in 1939 [15], and 1940 [17] the following result:
Proposition 1.1.2 (Erd˝
os - Kac). For a real-valued strongly additive function f, let
A(x) := X
p≤x
f(p)
p(1.2)
and
B(x) := ÃX
p≤x
f2(p)
p!1/2
.(1.3)
Then, if |f(p)| ≤ 1, and if B(x)→ ∞as x→ ∞, the frequencies
Fx(z) := 1
x#{n≤x:f(n)−A(x)
B(x)≤z}
converge weakly to the limit law
G(z) := 1
√2πZz
−∞
e−w2/2dw
as x→ ∞(which we will denote by Fx(z)⇒G(z)).
Proof. see [12], Theorem 12.3.
Thus, for f(n) = ω(n), P. Erd˝
os and M. Kac obtained a much more general result than
G. H. Hardy and S. Ramanujan. In this case,
A(x) = X
p≤x
1
p= log log x+O(1),
and
B(x) = ÃX
p≤x
1
p!1/2
= (1 + o(1))(log log x)1/2,
so that
1
x#{n≤x:ω(n)−log log x
√log log x≤z} ⇒ 1
√2πZz
−∞
e−w2/2dw.
15
1 Additive functions
A second effect of the above mentioned paper of P. Tur´
an was that P. Erd˝
os, adopting
Tur´
an’s method of proof, showed in 1938 [14] that, whenever the three series
X
p
|f(p)|>1
1
p,X
p
|f(p)|≤1
f(p)
p,X
p
|f(p)|≤1
f2(p)
p
converge, then the real-valued strongly additive function fpossesses a limiting distribu-
tion F, i.e.
1
x#{n≤x:f(n)≤z} ⇒ F(z)
with some suitable distribution function F. It turned out that the convergence of these
three series was in fact necessary (see Erd˝
os and Wintner [16]).
All these results can be described as effects of the fusion of (intrinsic) ideas of probability
theory and asymptotic estimations. In this context, divisibility by a prime pis an event
Ap, and all the {Ap}are statistically independent of one another, where the underlying
”measure” is given by the asymptotic density
δ(Ap) := lim
x→∞
1
x#{n≤x:n∈Ap}
= lim
x→∞
1
xX
n≤x
p|n
1µ=1
p¶(1.4)
If the limit
M(f) := lim
x→∞
1
xX
n≤x
f(n)
exists, then we say that the function fpossesses an (arithmetical) mean-value M(f).
Then, for strongly additive functions f, we get
f=X
p
f(p)εp
where εpdenotes the characteristic function of Ap, and M(εp) := 1
p.
16
1.1 Definition and introduction
The main difficulties concerning the immediate application of probabilistic tools arise
from the fact that the arithmetical mean-value (1.4) defines only a finitely additive mea-
sure (or content or pseudo-measure) on the family of subsets of Nhaving an asymptotic
density; thus, one constructs a sequence of finite, purely probabilistic models, which ap-
proximate the number theoretical phenomena, and one then uses arithmetical arguments
for ”taking the limit”. This theory, starting with the above mentioned results of P. Erd˝
os,
M. Kac, and A. Wintner, was developed by J. Kubilius [55]. He constructed such finite
probability spaces on which independent random variables could be defined to mimic the
behaviour of truncated additive functions
X
p≤r
f(p)εp.
This approach is effective if the ratio log r
log xessentially tends to zero as xruns to infinity.
Then J. Kubilius was able to give necessary and sufficient conditions in order that the
frequencies
1
x#{n≤x:f(n)−A(x)≤zB(x)}
weakly converge as x→ ∞, assuming that fbelongs to a certain class of additive func-
tions. This opened the door for the investigation of the renormalisation of additive func-
tions, i.e. the question to determine when a given additive function fmay be renormalized
by functions α(x)and β(x), such that the frequencies
1
x#{n≤x:f(n)−α(x)
β(x)≤z}
possess a weak limit as x→ ∞(see Elliott [12], Kubilius [55], Levin and Timofeev [58]).
In [46], K.-H. Indlekofer presented an integration theory on N(which can be general-
ized to arbitrary sets) which is based on the following characterization of the Stone-ˇ
Cech
compactification βNof N:
If Ais an algebra in N, then
A:= {A⊂βN:A=clβNA, A ∈A}
17
1 Additive functions
is an algebra in βN.
If an algebra Ain N, and a finitely additive measure δon Aare given, then the function
¯
δon Adefined by δ(A) = δ(A),A∈A, is a premeasure on A. By a suitable closure of
the set of step functions he obtains spaces of number theoretical functions which contain
e.g. the M¨
obius function. Furthermore, the “construction” of these spaces yields new,
elementary proofs of the famous results of E. Wirsing, G. Hal´
asz and H. Delange for
multiplicative functions f,|f| ≤ 1. We will introduce this in Chapter 3.
1.2 The Tur´
an-Kubilius inequality
If gis strongly additive, then
1
xX
n≤x
g(n) = 1
xX
n≤xX
p|n
g(p) = 1
xX
p≤x
g(p)·[x/p],
and so g(n)is approximate to X
p≤x
g(p)
p. The so-called Tur´
an-Kubilius inequality gives an
estimation for the difference of the values of the function minus the “expectation”:
¯¯¯¯¯g(n)−X
p≤x
g(p)
p¯¯¯¯¯
in mean square.
Let gbe a complex-valued additive arithmetic function, then we set
g(n) = X
pkkn
g(pk).
For real number x > 0, we set
A(x) = X
pk≤x
g(pk)
pk,
E(x) = X
pk≤x
g(pk)
pk·µ1−1
p¶
18
1.3 Finitely distributed functions
and
D2(x) = X
pk≤x
|g(pk)|2
pk.
In its general form, the Tur´
an-Kubilius inequality appears as follows.
Proposition 1.2.1 (Tur´
an-Kubilius inequality). There exist constants c1,c2with the
property that for every x≥2, and for any additive function gthe inequalities
1
xX
n≤x|g(n)−A(x)|2≤c1·D2(x)
and
1
xX
n≤x|g(n)−E(x)|2≤c2·D2(x)
hold. In fact, it is possible to take c1= 30,c2= 20.
Proof. see [77], Theorem 4.1.
The Tur´
an-Kubilius inequality has often been applied to the study of additive and multi-
plicative functions. We formulate an analogous inequality in Chapter 4 for the q-additive
and Q-additive functions, and we use this inequality in Chapter 5 and 6.
1.3 Finitely distributed functions
In [18], P. Erd˝
os introduced the notion of finitely distributed functions on N:
A function gis said to be finitely distributed if there are positive constants c1and c2, and
an unbounded sequence of real numbers x1< x2< . . . such that for each xjat least k
positive integers a1< a2< . . . < ak≤xjmay be found, with k≥c1xj, and
|g(am)−g(an)| ≤ c21≤m≤n≤k.
For additive functions he proved the following characterization
19
1 Additive functions
Proposition 1.3.1 (Erd˝
os [18]). An additive function gis finitely distributed if and only
if there is a constant cand a function hsuch that
g(n) = clog n+h(n),
where the series X
|h(p)|>1
1
p,X
|h(p)|≤1
h2(p)
p
both converge.
It follows from Proposition (2.2.2) that
lim
x→∞
1
x¯¯¯¯¯X
n≤x
exp(itg(n))¯¯¯¯¯
always exists. It will become clear that gis finitely distributed if and only if there is a set
of real t-values of positive Lebesgue measure for which the value of this limit is not zero.
20
Chapter 2
Multiplicative functions of modulus ≤1
In this chapter, we describe the mean behaviour of complex-valued multiplicative func-
tions fsuch that |f(n)| ≤ 1for every positive integer n.
2.1 Definition
Definition 2.1.1. An arithmetical function fis multiplicative if f6= 0, and if for all pairs
m, n of positive integers the condition gcd(m, n) = 1 implies
f(m·n) = f(m)·f(n).(2.1)
If (2.1) holds for all m, n, then fis called completely multiplicative.
Every multiplicative function fsatisfies f(1) = 1, since f(n·1) = f(n)·f(1), and
an integer nmay be chosen for which f(n)6= 0. If f1and f2are multiplicative, then
the point-wise product f1·f2also is multiplicative, the same is true for the convolution-
product f1∗f2; if fis multiplicative and f(n)6= 0 for every n, then 1/f is multiplicative.
A multiplicative function fis determined by its values at the prime-powers:
fÃY
p∈P
pαp(n)!=Y
p∈P
f¡pαp(n)¢.
21
2 Multiplicative functions of modulus ≤1
In this formula, according to the fundamental theorem of arithmetic, an integer nis written
uniquely as
n=Y
p∈P
pαp(n)
as a product of prime powers where αp(n) = max{α:pα|n}.
An important multiplicative function is the M¨
obius function, defined by
µ(n) =
(−1)ω(n)if nis squarefree,
0 otherwise.
The Euler totient function given by
ϕ(n) = n·Y
p|nµ1−1
p¶
is another well-known multiplicative function which enumerates the number of coprime
residue classes (mod n).
2.2 Mean-value theorem for multiplicative functions of
modulus ≤1
The problem of establishing the existence of mean-values was considered by A. Wintner
in his book on Erathostenian averages [82], he asserted that if a multiplicative function f
may have only values ±1, then the mean-value M(f)always exists. But, the sketch of his
proof could not be substantiated, and the problem remained open as the Erd˝
os-Wintner
conjecture.
These functions fwhich satisfy |f(n)| ≤ 1for all n∈Nand for which a non-zero
mean-value exists were characterized by H. Delange [9] in 1961, he proved
Proposition 2.2.1 (Delange [9]). Let f:N→Cbe a multiplicative function satisfying
|f| ≤ 1. Then the following conditions are equivalent:
(A) The mean-value M(f) = lim
x→∞
1
x·X
n≤x
f(n)exists, and it is non-zero.
22
2.2 Mean-value theorem for multiplicative functions of modulus ≤1
(B) (i) The series S1(f) = X
p
f(p)−1
pis convergent,
(ii) X
0≤k<∞
f(pk)
pk6= 0 for all primes p.
The assumption |f| ≤ 1implies that ¯¯¯¯¯X
0≤k<∞
f(pk)
pk¯¯¯¯¯≥1
2for every prime p≥3. There-
fore, as did H. Delange, the validity of (B(ii)) has to be assumed only for p= 2, and it
may be substituted by Delange’s condition
f(2k)6=−1 for some k≥1.
But his method could not be modified to consider the case M(f) = 0.
To get an impression of the remaining case, we note that if f=µis the M¨
obius function,
then the validity of the assertion
1
xX
n≤x
µ(n)→0 (x→0)
essentially is (see E. Landau [56]) as difficult as to obtain as the proof of the prime number
theorem.
In his paper [84] of 1967, E. Wirsing proved his celebrated mean-value theorem which
asserts, in particular, that any real-valued multiplicative functions fof modulus ≤1has
a mean-value. This solved the afore-mentioned conjecture of Erd˝
os and Wintner.
In this paper, E. Wirsing adopts a more general formulation: he compares the behaviour
of X
n≤x
f(n)with that of X
n≤x
f∗(n), where f∗is a nonnegative multiplicative function
and |f| ≤ f∗. His proof was done by elementary methods (and thus, he gave another
elementary proof of the prime number theorem), but he could not handle the complex-
valued case in full generality. Only by an analytic method, found by G. Hal´
asz in 1968,
and published in his paper [22], the asymptotic behaviour of X
n≤x
f(n)could be fully
determined for all complex-valued multiplicative functions fof modulus smaller than or
equal to one. His main result is given by the following
23
2 Multiplicative functions of modulus ≤1
Proposition 2.2.2 (Hal´
asz [22]). Let fbe a complex-valued multiplicative arithmetical
function satisfying |f| ≤ 1.
(1) If there is a real number afor which the series
X
p
(1 −Re f(p)p−ia)
p(2.2)
is convergent, then the asymptotic relation
X
n≤x
f(n) = x1+ia
1 + ia Y
p≤xµ1−1
p¶Ã1 +
∞
X
m=1
p−m(1+ia)f(pm)!+o(x)
holds.
(2) If the series (2.2) is divergent for every real number a, then
lim
x→∞
1
xX
n≤x
f(n) = 0.
(3) In both cases, there are constants D,α, and a slowly oscillating function Lof
modulus |L|= 1 such that the asymptotic formula
X
n≤x
f(n) = Dx1+iαL(log x) + o(x)
holds.
The function L, and the constants α,Dmay explicitly be given (see for example Hal´
asz
[22]).
In 1986, A. Hildebrand [24] gave a new elementary proof of Wirsing’s theorem based on
a large sieve inequality which is simpler than Wirsing’s proof but does not work in the
complex-valued case.
In [8], H. Daboussi and K.-H. Indlekofer succeeded in finding an elementary proof of
Hal´
asz’s theorem, and thus, a new elementary proof of Wirsing’s result (see also In-
dlekofer [26] for a simplified and shorter proof).
24
2.2 Mean-value theorem for multiplicative functions of modulus ≤1
Remark. K.-H. Indlekofer, I. K´
atai and R. Wagner [44] compare the asymptotic behavior
of X
n≤x
f(n)and X
n≤x
g(n)for multiplicative functions fand g, respectively, where |f| ≤ g.
Their results extend relevant theorems by E. Wirsing and G. Hal´
asz. They established the
following theorem which generalizes Wirsing’s result and extends the theorem of Hal´
asz.
Proposition 2.2.3 (Indlekofer, K´
atai, Wagner [44]). Let gbe a multiplicative function
and X
p≤x
g(p)
p·log p∼τ·log x, x → ∞,
hold with a constant τ > 0. Furthermore, let g(p) = O(1) for all primes p, and let
X
pX
k≥2
g(pk)
pk<∞.
Besides this, if τ≤1, then let
X
pX
k≥2,pk≤x
g(pk) = Oµx
log x¶,
and let fbe a complex-valued function which satisfies |f(n)| ≤ g(n)for every positive
integer n. If there exists a real number a0such that the series
X
p
(g(p)−Re f(p)p−ia)
p(2.3)
converges for a=a0, then
X
n≤x
f(n) = xia0
1 + ia0Y
p≤xÃ1 +
∞
X
m=1
f(pm)
pm(1+ia0)!Ã1 +
∞
X
m=1
g(pm)
pm!−1X
n≤x
g(n)
+oÃX
n≤x
g(n)!.(2.4)
as x→ ∞. If the series (2.3) diverges for all a∈R, then
X
n≤x
f(n) = oÃX
n≤x
g(n)!(x→ ∞).(2.5)
25
2 Multiplicative functions of modulus ≤1
In both cases there are constants c, a0and a slowly oscillating function ˜
Lwith |˜
L(u)|= 1
such that
X
n≤x
f(n) = ³cxia0˜
L(log x) + o(1)´X
n≤x
g(n), as x → ∞.(2.6)
26
Chapter 3
Uniformly summable functions
In this chapter, the spaces of uniformly summable, α-even, α-limit-periodic and α-almost-
periodic arithmetical functions are considered. In addition, the mean behaviour of uni-
formly summable multiplicative functions and a complete characterization of α-almost-
periodic multiplicative functions given by K.-H Indlekofer are presented.
3.1 Definition
Let α∈Rwith α≥1, and let
Lα:= {f|f:N→C,kfkα<∞}
be the vector space of arithmetical functions fwith bounded semi-norm
kfkα:= Ãlim sup
N→∞
1
NX
n<N |f(n)|α!1
α
.
A characterization of multiplicative functions f∈Lα(α > 1) which possess a nonzero
mean-value M(f)was independently given by P.D.T.A. Elliott [13], and using a different
method, by H. Daboussi [6].
In 1980, K.-H. Indlekofer [25] introduced the following
27
3 Uniformly summable functions
Definition 3.1.1. An arithmetical function f∈L1is said to be uniformly summable if
lim
K→∞ sup
N≥1
1
NX
n<N
|f(n)|≥K
|f(n)|= 0,
and the space of all uniformly summable functions is denoted by L∗.
It is easy to show that, if β > α > 1,
Lβ$Lα$L∗$L1.
3.2 Mean behaviour of uniformly summable multiplica-
tive functions
The idea of uniform summability turned out to provide an appropriate tool for the descrip-
tion of the mean behaviour of a large class of multiplicative functions. As typical results,
we mention the theorem by K.-H. Indlekofer which generalizes results of H. Daboussi, H.
Delange, G. Hal´
asz, and E. Wirsing.
Proposition 3.2.1 (Indlekofer [25]). (A generalization of Delange’s result)
Let f:N→Cbe multiplicative, and let α≥1. Then, the following two assertions hold.
(i) If f∈L∗∩Lα, and if the mean-value
M(f) := lim
x→∞
1
xX
n≤x
f(n)
of fexists and is non-zero, then the series
X
p
f(p)−1
p,X
p
|f(p)|≤3
2
|f(p)−1|2
p,Xp
|f(p)−1|≥1
2
|f(p)|λ
p,X
pX
k≥2
|f(pk)|λ
pk(3.1)
converge for all λwith 1≤λ≤α, and, for each prime p,
1 +
∞
X
k=1
f(pk)
pk6= 0.(3.2)
28
3.2 Mean behaviour of uniformly summable multiplicative functions
(ii) If the series (3.1) converge, then f∈L∗∩Lα, and the mean-values M(f),M(|f|λ)
exist for all λwith 1≤λ≤α. If, in addition (3.2) holds, then M(f)6= 0.
Note that the membership of Lα∩L∗and the existence of a non-zero mean value together
are equivalent to a set of explicit conditions on prime powers. Further, observe that these
conditions imply the existence of the mean values M(|f|λ)for all 1≤λ≤α.
Proposition 3.2.2 (Indlekofer [28]). (A generalization of Wirsing’s result)
Let f∈L∗be a real-valued multiplicative function. Then, the existence of the mean value
M(|f|)implies the existence of M(f).
Note that Theorem 3.2.2 is an appropriate generalization of Wirsing’s result, for if fis
multiplicative and |f| ≤ 1, the mean value of M(|f|)always exists.
In this connection it is interesting to mention the following characterization of non-
negative multiplicative functions of L∗.
Proposition 3.2.3 (Indlekofer [29]). Let ε≥0, and let f∈L1+ε∩L∗be a non-negative
multiplicative function. If ||f||1>0, then f1+ε∈L?, and there exist positive constants
c1,c2such that, as x→ ∞,
M(f1+ε) = exp ÃX
p≤x
f1+ε(p)−1
p!(c1+o(1))
= exp ÃX
p≤x
f(p)−1
p!(c2+o(1))
from which we deduce that the existence of M(f1+ε)implies the existence of M(f).
A complete characterization of the asymptotic behaviour of the sums X
n≤x
f(n)as x→ ∞
for complex-valued multiplicative functions f∈L∗was given by K.-H. Indlekofer in
[30]. He proves the following
29
3 Uniformly summable functions
Proposition 3.2.4 (Indlekofer [30]). (A generalization of Hal´
asz’s result)
Let f∈L∗be multiplicative, and let kfk1>0. If we define
%(p) =
f(p)
|f(p)|if f(p)6= 0
1 otherwise
,
then the following two assertions hold.
(i) If there exists a constant a0∈Rsuch that the series
X
p
(1 −Re %(p)p−ia)
p(3.3)
converges for a=a0, then there exists a constant c0∈Csuch that, if x→ ∞,
1
xX
n≤x
f(n) = xia0exp ÃX
p≤x
f(p)p−ia0−1
p!(c0+o(1)),
where
c0=1
1 + ia0Y
pµ1−1
p¶Ã1 +
∞
X
k=1
f(pk)
pk(1+ia)!exp ½1−f(p)p−ia0
p¾.
If
A∗(x) := X
p≤x
Imf(p)p−ia0
p,
then
lim
x→∞ sup
x≤y≤x2|A∗(y)−A∗(x)|= 0.
(ii) If the series (3.3) diverges for all a∈R, then the mean-value M(f)of fexists and
equals zero.
This result generalizes the theorem of Hal´
asz [22] on multiplicative functions |f| ≤ 1.
We will extend above proposition to uniformly summable q-multiplicative functions in
Chapter 5.
30
3.3 Mean behaviour of α-almost-periodic multiplicative functions
3.3 Mean behaviour of α-almost-periodic multiplicative
functions
Definition 3.3.1. Let rbe a positive integer. An arithmetic function fis called
r-periodic, if f(n+r) = f(n)for every positive integer n,
r-even, if f(n) = f(gcd(n, r)) for every positive integer n.
fis termed periodic (resp. even) if there is an rfor which fis r-periodic (resp. r-even).
Obviously, an r-even function is r-periodic.
Standard examples of r-periodic functions are the exponential functions eβ, where β=a
r,
for a∈Z,r∈N, and where eβ(n) = exp(2πi ·β·n).
The Ramanujan sum cris a special exponential sum:
cr(n) = X
1≤a≤r
gcd(a,r)=1
exp ³2πi ·a
r·n´
=X
t|gcd(r,n)
t·µ³r
t´.
The vector space Brof r-even functions can be generated by the Ramanujan sums cd,
where d|r, i.e.,
Br= LinC[cd:d|r],
and each element of the vector space Drof r-periodic functions can be written as a linear
combination of exponential functions, i.e.,
Dr= LinC[ea/r : 1 ≤a≤r].
The vector space of all even and all periodic functions is denoted by B:=
∞
[
r=1
Brand
D:=
∞
[
r=1
Dr, respectively. Finally, we define the vector space
A= LinC[eβ:β∈R/Z]
31
3 Uniformly summable functions
of complex linear combinations of the functions eβ.
Using the semi-norm kfkα, the spaces
Bα=k.kα- closure of B(α-almost-even functions)
Dα=k.kα- closure of D(α-limit-periodic functions)
Aα=k.kα- closure of A(α-almost-periodic functions)
may be constructed.
The obvious inclusion relations B⊂D⊂Aimply
Bα⊂Dα⊂Aα,
where α≥1.
For γ < α, H¨
older’s inequality gives
X
n≤N|f(n)|γ≤(X
n≤N|f(n)|α)γ/α
·(X
n≤N
1)α/(α−γ)
,
therefore,
kfkγ≤ kfkαif γ≤α,
and so
Bα⊂Bγ,Dα⊂Dγ,and Aα⊂Aγ,if γ≤α.
Furthermore, we have the inclusions
B1$D1$A1$L∗.
For every function f∈A1, the mean-value M(f), and for every β∈R, the Fourier
coefficient
ˆ
f(β) := lim
N→∞
1
NX
0≤n<N
f(n)e−β(n)
exist (see, for example, W. Schwarz and J. Spilker [77] Chap. IV and VI).
32
3.3 Mean behaviour of α-almost-periodic multiplicative functions
For f∈L1, the Fourier-Bohr spectrum σ(f)is defined by
σ(f) = (β∈R/Z: lim sup
N→∞ ¯¯¯¯¯
1
NX
n≤N
f(n)e−β(n)¯¯¯¯¯>0).
If f∈A1, then β∈σ(f)if and only if ˆ
f(β)6= 0.
In his paper [27], K.-H. Indlekofer gave a complete characterization of α-almost-periodic
multiplicative functions. He proved the following results.
Proposition 3.3.2 (Indlekofer [27]). Let f∈A1be multiplicative. Then, M(|f|)=0if
and only if σ(f) = ∅.
Proposition 3.3.3 (Indlekofer [27]). Let f∈Aαbe multiplicative. Then, fis α-limit-
periodic.
Proposition 3.3.4 (Indlekofer [27]). Let f:N→Cbe multiplicative. Then, the follow-
ing assertions are equivalent.
(i) f∈Aα, and kfk1>0.
(ii) f∈Aα, and the spectrum σ(f)of fis non-empty.
(iii) f∈Lα∩L∗, and there exists a Dirichlet-character χsuch that the mean-value
M(fχ)of fχ exists and is different from zero.
(iv) There exists a Dirichlet-character χsuch that the series
X
p
f(p)χ(p)−1
p,X
p
|f(p)|≤3/2
|f(p)χ(p)−1|2
p,(3.4)
and X
p
||f(p)|−1|>1/2
|f(p)|λ
p,X
pX
k≥2
|f(pk)|λ
pk(3.5)
converge for all λwith 1≤λ≤α.
33
3 Uniformly summable functions
Remark. The equivalence of (ii) and (iv) was proved by H. Daboussi [7]. The equivalence
of (ii), (iii) and (iv) was shown by K.-H. Indlekofer in [30], Corollary 7.
In Chapter 5, we give a complete characterization of α-almost-periodic q-multiplicative
functions.
34
Chapter 4
Q-additive and Q-multiplicative
functions
In this chapter, we start our investigation of q-additive and q-multiplicative functions and
Q-additive and Q-multiplicative functions, respectively. We give a new proof of the Tur´
an-
Kubilius inequality for q-additive functions, and we extend this proof to Q-additive func-
tions.
4.1 Definition
Let {qr}r≥1with qr≥2be a sequence of natural numbers, and let Q0= 1,Qr=qrQr−1
when r≥1. For each nonnegative integer nhas a unique representation
n=X
r≥0
εr(n)Qr(4.1)
if the following condition is satisfied
0≤εr(n)< qr+1 , r ≥0.
If εk(n)6= 0 and εk+j(n)=0for all j≥1, then εk(n)and kwill be called the first digit
and the order of n, respectively.
35
4Q-additive and Q-multiplicative functions
The function g:N0→C, satisfying the relation
g(n) = X
r≥0
g(εr(n)Qr)
for each n∈Nhaving the form (4.1) with g(0) = 0, will be called Q-additive function.
Similarly, we say that f:N0→Cis a Q-multiplicative function if f(0) = 1 and
f(n) = Y
r≥0
f(εr(n)Qr)
for each n∈Nhaving the form (4.1).
In the case qr≡q≥2, we will use standard notation of q-adic representation.
We consider the so-called q-additive functions g:N0→Cwhich are defined by
g(n) = X
r≥0
g(εr(n)qr) and f(0) = 0,
and the q-multiplicative functions f:N0→Cwhich are defined by
f(n) = Y
r≥0
f(εr(n)qr) and f(0) = 1.
These functions were first introduced by A. O. Gelfond [21]. The sum of digits X
r≥0
εr(n)
of nis a typical and mostly investigated example of q-additive functions (see for example
Delange [11]; Coquet [5]). Exponentiating a q-additive function gives a q-multiplicative
function.
We recall that a real-valued function g(n)has an asymptotic distribution if there is a dis-
tribution function Gsuch that for all continuity points yof G, the probability measures
defined by Nx(y) := x−1]{n≤x;g(n)≤y}tend to G(y)as xtend to infinity.
36
4.2 Generalized Tur´
an-Kubilius inequalities for Q-additive functions
4.2 GeneralizedTur´
an-Kubilius inequalitiesforQ-additive
functions
Observing that q-additive functions are sums of “almost independent random variables”,
we prove the following inequality which is interesting in itself.
Theorem 4.2.1. Let Fbe an arbitrary nonnegative-valued increasing function satisfying
the inequality
F(2x)≤ρF(x)
for some constant ρ > 0. Let g:N0→Cbe q-additive and let cqR−1≤N < (c+1)qR−1
with R∈Nand for c∈Nwith 0< c < q.
We set
ER(g) =
R−2
X
r=0
1
q
q−1
X
a=0
g(aqr),
and
ER,c(g) = ER(g) + 1
c
c
X
a=1
g(aqR−1).
Then, the following assertions hold.
(i) For some constant M > 0, we have
1
NX
n<N
F(|g(n)−ER,c(g)|)
≤M·
F
ÃR−2
X
r=0
1
q
q−1
X
a=0 |g(aqr)|2+1
c
c
X
a=1 |g(aqR−1)|2!1/2
+
R−2
X
r=0
1
q
q−1
X
a=0
F(|g(aqr)|) + 1
c
c
X
a=1
F(|g(aqR−1)|)).(4.2)
(ii) If, in addition, Ffulfills
F(x+y)ÀF(x) + F(y),
37
4Q-additive and Q-multiplicative functions
then, for some constant M0>0, we have
1
NX
n<N
F(|g(n)−ER,c(g)|)
≤M0·F
ÃR−2
X
r=0
1
q
q−1
X
a=0 |g(aqr)|2+1
c
c
X
a=1 |g(aqR−1)|2!1/2
.(4.3)
To prove Theorem 4.2.1, we use the Burkholder’s inequality (see Burkholder [3], Ruzsa
[68] and Indlekofer [35])
EFﯯ¯¯
n
X
j=1
ξj¯¯¯¯¯!¿Fµ³XE(|ξj|2)´1/2¶+XEF(|ξj|),(4.4)
where the ξj’s are independent variables of zero mean, E denotes an expectation, and F
is an arbitrary nonnegative-valued increasing function satisfying the inequality
F(2x)≤ρF(x)
for some constant ρ; the value of the implied constant depends on this ρ.
Proof of Theorem 4.2.1.
(i) Each nonnegative integer n < N has a unique representation
n=
R−1
X
r=0
εr(n)qr,
which 0≤εr(n)< q. We obtain
g(n) =
R−1
X
r=0
g(εr(n)qr).
Let ηr(n) = g(εr(n)qr), then η0, . . . , ηR−1are independent random variables in the
Laplace space {0,1,··· ,(c+ 1)qR−1}, and g=
R−1
X
r=0
ηr.
We define the function g∗
g∗(aqr) =
g(aqr) for r < R −1,0≤a < q ,
or r=R−1,0≤a≤c;
0 for r > R −1,0≤a < q ,
or r=R−1, c < a < q .
38
4.2 Generalized Tur´
an-Kubilius inequalities for Q-additive functions
Then, ξr=ηr−E(ηr)are independent random variables of zero mean, and
g∗−E(g∗) =
R−1
X
r=0
ξr,
where E(g∗) = ER,c(g).
Applying the Burkholder’s inequality (4.4), we obtain
1
(c+ 1)qR−1X
n<(c+1)qR−1
F(|g(n)−ER,c(g)|)
¿F
R−2
X
r=0
1
q
q−1
X
a=0 ¯¯¯¯¯g(aqr)−1
q
q−1
X
b=0
g(bqr)¯¯¯¯¯
2
+1
c
c
X
a=1 ¯¯¯¯¯g(aqR−1)−1
c
c
X
b=1
g(bqR−1)¯¯¯¯¯
2
1/2
| {z }
I1
+
R−2
X
r=0
1
q
q−1
X
a=0
Fﯯ¯¯g(aqr)−1
q
q−1
X
b=0
g(bqr)¯¯¯¯¯!+1
c
c
X
a=1
Fﯯ¯¯g(aqR−1)−1
c
c
X
b=1
g(bqR−1)¯¯¯¯¯!.
| {z }
I2
For the estimation of the first term (I1), we observe
1
q
q−1
X
a=0 ¯¯¯¯¯g(aqr)−1
q
q−1
X
b=0
g(bqr)¯¯¯¯¯
2
≤1
q
4
q−1
X
a=0 |g(aqr)|2+4
q¯¯¯¯¯
q−1
X
b=0
g(bqr)¯¯¯¯¯
2
≤1
qÃ4
q−1
X
a=0 |g(aqr)|2+ 4
q−1
X
b=0 |g(bqr)|2!
= 8 Ã1
q
q−1
X
a=0 |g(aqr)|2!.
In the same way, we get
1
c
c
X
a=1 ¯¯¯¯¯g(aqR−1)−1
c
c
X
b=1
g(bqR−1)¯¯¯¯¯
2
≤8Ã1
c
c
X
a=1 ¯¯g(aqR−1)¯¯2!.
39
4Q-additive and Q-multiplicative functions
For the second term (I2), we have
q−1
X
a=0
Fﯯ¯¯g(aqr)−1
q
q−1
X
b=0
g(bqr)¯¯¯¯¯!≤
q−1
X
a=0
Fµ2 max
0≤a<q |g(aqr)|¶
≤ρ
q−1
X
a=0
Fµmax
0≤a<q |g(aqr)|¶
≤qρ
q−1
X
a=0
F(|g(aqr)|).
In the same way, we get
c
X
a=1
Fﯯ¯¯g(aqR−1)−1
c
c
X
b=1
g(bqR−1)¯¯¯¯¯!≤cρ
c
X
a=1
F¡¯¯g(aqR−1)¯¯¢.
Since
1
NX
n<N
F(|g(n)−ER,c(g)|)
≤1
NX
n<(c+1)qR−1
F(|g∗(n)−ER,c(g)|)
≤c+ 1
c·1
(c+ 1)qR−1X
n<(c+1)qR−1
F(|g∗(n)−ER,c(g)|)
the inequality (4.2) follows.
(ii) Since
q−1
X
a=0
F(|g(aqr)|)¿FÃq−1
X
a=0 |g(aqr)|!
≤F
√qÃq−1
X
a=0 |g(aqr)|2!1/2
¿F
Ãq−1
X
a=0 |g(aqr)|2!1/2
,
40
4.2 Generalized Tur´
an-Kubilius inequalities for Q-additive functions
we then have
R−2
X
r=0
1
q
q−1
X
a=0
F(|g(aqr)|) + 1
c
c
X
a=1
F(|g(aqR−1)|)
¿
R−2
X
r=0
F
Ã1
q
q−1
X
a=0 |g(aqr)|2!1/2
+F
Ã1
c
c
X
a=1 |g(aqR−1)|2!1/2
¿F
ÃR−2
X
r=0
1
q
q−1
X
a=0 |g(aqr)|2+1
c
c
X
a=1 |g(aqR−1)|2!1/2
.
The inequality (4.3) follows.
For F(x) = xpwith p≥1, we obtain a recent result by M. Peter and J. Spilker [78]
1
NX
n<N |g(n)−ER,c(g)|p
≤M00 ·ÃR−2
X
r=0
1
q
q−1
X
a=0 |g(aqr)|2+1
c
c
X
a=1 |g(aqR−1)|2!p/2
for some constant M00 >0.
If p= 2, we obtain an analog of the Tur´
an-Kubilius inequality from Theorem 4.2.1.
Corollary 4.2.2. Let g:N0→Cbe q-additive, cqR−1≤N < (c+ 1)qR−1with R∈N
and some c∈Nwith 0< c < q.
We set
ER(g) =
R−2
X
r=0
1
q
q−1
X
a=0
g(aqr),
and
ER,c(g) = ER(g) + 1
c
c
X
a=1
g(aqR−1).
Then,
1
NX
n<N |g(n)−ER,c(g)|2≤2ÃR−2
X
r=0
1
q
q−1
X
a=0 |g(aqr)|2+1
c
c
X
a=1 |g(aqR−1)|2!.(4.5)
41
4Q-additive and Q-multiplicative functions
Analogously, we get
Theorem 4.2.3. Let Fbe an arbitrary nonnegative-valued increasing function satisfying
the inequality
F(2x)≤ρF(x)
with some constant ρ > 0. Let g:N0→Cbe Q-additive, let cQR−1≤N < (c+ 1)QR−1
with R∈Nand some c∈Nwith 0< c < qR.
We set
ER(g) =
R−1
X
r=1
1
qr
qr−1
X
a=0
g(aQr−1),
and
ER,c(g) = ER(g) + 1
c
c
X
a=1
g(aQR−1).
Then, for some constant M1>0,
1
NX
n<N
F(|g(n)−ER,c(g)|)
≤M1·
F
ÃR−1
X
r=1
1
qr
qr−1
X
a=0 |g(aQr−1)|2+1
c
c
X
a=1 |g(aQR−1)|2!1/2
+
R−1
X
r=1
1
qr
qr−1
X
a=0
Fﯯ¯¯g(aQr−1)−1
qr
qr−1
X
b=0
g(bQr−1)¯¯¯¯¯!
+1
c
c
X
a=1
Fﯯ¯¯g(aQR−1)−1
c
c
X
b=1
g(bQR−1)¯¯¯¯¯!).(4.6)
Proof. Each nonnegative integer n < N has a unique representation
n=
R−1
X
r=0
εr(n)Qr,
which 0≤εr(n)< qr+1. We obtain
g(n) =
R−1
X
r=0
g(εr(n)Qr).
42
4.2 Generalized Tur´
an-Kubilius inequalities for Q-additive functions
Let ηr(n) = g(εr(n)Qr), then η0, . . . , ηR−1are independent random variables in the
Laplace space {0,1,··· ,(c+ 1)QR−1}, and g=
R−1
X
r=0
ηr.
We define the function g∗
g∗(aQr) =
g(aQr) for r < R −1,0≤a < qr+1 ,
or r=R−1,0≤a≤c;
0 for r > R −1,0≤a < qr+1 ,
or r=R−1, c < a < qR.
Then ξr=ηr−E(ηr)are independent random variables of zero mean, and
g∗−E(g∗) =
R−1
X
r=0
ξr,
where E(g∗) = ER,c(g).
Applying the Burkholder’s inequality (4.4), we obtain
1
(c+ 1)QR−1X
n<(c+1)QR−1
F(|g(n)−ER,c(g)|)
¿F
R−1
X
r=1
1
qr
qr−1
X
a=0 ¯¯¯¯¯g(aQr−1)−1
qr
qr−1
X
b=0
g(bQr−1)¯¯¯¯¯
2
+1
c
c
X
a=1 ¯¯¯¯¯g(aQR−1)−1
c
c
X
b=1
g(bQR−1)¯¯¯¯¯
2
1/2
+
R−1
X
r=1
1
qr
qr−1
X
a=0
Fﯯ¯¯g(aQr−1)−1
qr
qr−1
X
b=0
g(bQr−1)¯¯¯¯¯!
+1
c
c
X
a=1
Fﯯ¯¯g(aQR−1)−1
c
c
X
b=1
g(bQR−1)¯¯¯¯¯!.
43
4Q-additive and Q-multiplicative functions
We estimate the first summand as in Theorem 4.2.1.
1
qr
qr−1
X
a=0 ¯¯¯¯¯g(aQr−1)−1
qr
qr−1
X
b=0
g(bQr−1)¯¯¯¯¯
2
≤1
qr
4
qr−1
X
a=0 |g(aQr−1)|2+4
qr¯¯¯¯¯
qr−1
X
b=0
g(bQr−1)¯¯¯¯¯
2
≤1
qrÃ4
qr−1
X
a=0 |g(aQr−1)|2+ 4
qr−1
X
b=0 |g(bQr−1)|2!
= 8 Ã1
qr
qr−1
X
a=0 |g(aQr−1)|2!.
In the same way, we get
1
c
c
X
a=1 ¯¯¯¯¯g(aQR−1)−1
c
c
X
b=1
g(bQR−1)¯¯¯¯¯
2
≤8Ã1
c
c
X
a=1 |g(aQR−1)|2!.
Since 1
NX
n<N
F(|g(n)−ER,c(g)|)
≤1
NX
n<(c+1)QR−1
F(|g∗(n)−ER,c(g)|)
≤c+ 1
c·1
(c+ 1)QR−1X
n<(c+1)QR−1
F(|g∗(n)−ER,c(g)|),
the inequality (4.6) follows.
Corollary 4.2.4. Let g:N0→Cbe Q-additive, cQR−1≤N < (c+ 1)QR−1with R∈N
and some c∈Nwith 0< c < qR.
We set
ER(g) =
R−1
X
r=1
1
qr
qr−1
X
a=0
g(aQr−1),
and
ER,c(g) = ER(g) + 1
c
c
X
a=1
g(aQR−1).
44
4.3 Limit distributions of Q-additive functions
Then,
1
NX
n<N |g(n)−ER,c(g)|2≤2ÃR−1
X
r=1
1
qr
qr−1
X
a=0 |g(aQr−1)|2+1
c
c
X
a=1 |g(aQR−1)|2!.(4.7)
4.3 Limit distributions of Q-additive functions
In the case of the q-adic scale, necessary and sufficient conditions for the existence of
an asymptotic distribution for a real-valued q-additive function have been given by H.
Delange [10] in 1972, he proved the following theorem
Proposition 4.3.1 (Delange [10]). Let gbe a real-valued q-additive function. Then ghas
a limit distribution if and only if the series
∞
X
r=0
q−1
X
a=0
g(aqr),
and ∞
X
r=0
q−1
X
a=0
g2(aqr)
converges.
The limit distribution has as characteristic function the infinite product
∞
Y
r=0
1
qÃ1 +
q−1
X
a=1
exp(itg(aqr))!,
which converges for all real t.
This is similar to the theorem of Erd˝
os - Wintner [16] for the ordinary additive functions.
J. Coquet [4] considered in 1975 the same kind of problems in the cases of the Q-adic
scales and obtained mainly sufficient conditions, he proved the following theorems
45
4Q-additive and Q-multiplicative functions
•If {qr}r≥1is bounded:
Proposition 4.3.2 (Coquet [4]). Let gbe a real-valued Q-additive function.
Then, ghas a limit distribution if and only if the series
∞
X
r=1
1
qrÃqr−1
X
a=0
g(aQr−1)!,
and ∞
X
r=1
1
qrÃqr−1
X
a=0
g2(aQr−1)!
converge.
The limit distribution has as characteristic function the infinite product
∞
Y
r=1
1
qrÃ1 +
qr−1
X
a=1
exp(itg(aQr−1))!
which converges for all real t.
•If {qr}r≥1is unbounded:
Proposition 4.3.3 (Coquet [4]). Let gbe a real-valued Q-additive function.
We set
g∗(aQr−1) =
g(aQr−1) if |g(aQr−1)| ≤ 1,
1 if |g(aQr−1)|>1,
and
β∗
r= sup
1≤j≤qr−1Ã1
j+ 1
j
X
a=0
g∗(aQr−1)!2
.
If β∗
r→0, and the series
∞
X
r=1
1
qrÃqr−1
X
a=0
g∗(aQr−1)!,
46
4.4 Mean-value theorem for Q-multiplicative functions
and ∞
X
r=1
1
qrÃqr−1
X
a=0
g∗(aQr−1)2!
are convergent, then ghas a limit distribution, its characteristic function is
∞
Y
r=1
1
qrÃ1 +
qr−1
X
a=1
exp(itg(aQr−1))!.
4.4 Mean-value theorem for Q-multiplicative functions
In the same paper, H. Delange [10] asserts that for every q-multiplicative function fwith
|f| ≤ 1, where Nx=blog x
log qc,
m(x) := 1
xX
n<x
f(n) =
Nx−1
Y
r=0
1
qÃq−1
X
a=0
f(aqr)!+o(1),
as x→ ∞.
From this, he deduced that lim
x→∞ |m(x)|always exists and equals
∞
Y
r=0 ¯¯¯¯¯
1
q
q−1
X
a=0
f(aqr)¯¯¯¯¯
which is nonzero if and only if
∞
X
r=0
q−1
X
a=0
Re (1 −f(aqr)) (4.8)
converges, and
q−1
X
a=0
f(aqr)6= 0 (for all r∈N0).(4.9)
Furthermore, he proved that lim
x→∞ m(x)exists, and is nonzero if and only if (4.9) holds
and the series
∞
X
r=0
q−1
X
a=0
(1 −f(aqr)) (4.10)
47
4Q-additive and Q-multiplicative functions
is convergent.
As an analogue of the Delange result, J. Coquet [4] proved the following mean-value
theorems for Q-multiplicative functions modulus ≤1.
•If {qr}r≥1is bounded:
Proposition 4.4.1 (Coquet [4]). Let fbe a Q-multiplicative function with |f| ≤ 1.
(i) If the mean-value of fexists, and is nonzero, then the series
∞
X
r=1
1
qrÃqr−1
X
a=0
(1 −f(aQr−1))!(4.11)
converges, and
1 +
qr−1
X
a=1
f(aQr−1)6= 0
for all r∈N.
(ii) If the series (4.11) converges, then the mean-value of fis equal to
∞
Y
r=1 (1
qrÃ1 +
qr−1
X
a=1
f(aQr−1)!),
which converges.
•If {qr}r≥1is unbounded:
Proposition 4.4.2 (Coquet [4]). Let fbe a Q-multiplicative function with |f| ≤ 1.
(i) If max
1≤j≤qr−1(1
j+ 1
j
X
a=0
(1 −Re f(aQr−1)))→0, as r→ ∞, and the series
∞
X
r=1
1
qrÃqr−1
X
a=0
(1 −f(aQr−1))!(4.12)
48
4.4 Mean-value theorem for Q-multiplicative functions
converges, then the mean-value of fis equal to
∞
Y
r=1 (1
qrÃ1 +
qr−1
X
a=1
f(aQr−1)!),
which converges.
(ii) If the mean-value of fexists, and it is nonzero and
∞
X
r=1
1
qr
qr−1
X
a=0
(1 −Re f(aQr−1)) <∞,
then the series (4.12) converges and
1 +
qr−1
X
a=1
f(aQr−1)6= 0
for all r∈N.
It appears to be essential to have information on the difference
1
xX
0≤n<x
f(n)−Y
0<r≤r(x)
1
qrX
0≤a<qr
f(aQr−1),
where f(·)is any Q-multiplicative function of modulus ≤1, and more precisely, to get a
characterization of
lim
x→∞
1
xX
0≤n<x
f(n)−Y
0<r≤r(x)
1
qrX
0≤a<qr
f(aQr−1)
= 0.(4.13)
In fact, if the sequence {qr}r≥1is bounded, the relation 4.13 is true always. But if {qr}r≥1
is unbounded, the situation is quite different. For example, in [1], G. Barat constructed a
Q-multiplicative function hwith values 1 or -1 such that
lim
x→∞ Y
0<r≤r(x)
1
qrX
0≤a<qr
h(aQr−1)
49
4Q-additive and Q-multiplicative functions
exists, and it is a positive number while
lim inf
x→∞
1
xX
n<x
h(n)
is less than or equal to zero.
This difference is due to the existence of a first digit phenomenon which is unavoidable
for unbounded sequences {qr}r≥1(see E. Manstaviˇ
cius [63]).
50
Chapter 5
Mean behaviour of uniformly
summable q-multiplicative functions
and its applications
Theaimofthischapteris to study thebehaviourofthe means 1
NX
n<N
f(n)and 1
NX
n<N |f(n)|α
as N→ ∞, for α > 0, where fis uniformly summable and q-multiplicative. To our
surprise, we find that for q-multiplicative functions the space Lαfor every α > 0coin-
cides with the space L∗. Furthermore, applying our main results, we investigate finitely
distributed q-additive functions and find characterizations for q-multiplicative functions
belonging to the space D1of limit-periodic functions and the space A1of almost-periodic
functions by their respective spectrum σ(f).
5.1 Main results
Here we recall that an arithmetical function f∈L1is said to be uniformly summable if
lim
K→∞ sup
N≥1
1
NX
n<N
|f(n)|≥K
|f(n)|= 0,
51
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
and the space of all uniformly summable functions is denoted by L∗.fis q-multiplicative
if f(0) = 1 and
f(aqr+b) = f(aqr)·f(b)
for every pair of integer (a, b)satisfying
0≤a < q and 0 ≤b < qr.
Definition 5.1.1. Let fbe q-multiplicative function, we define
g
ΠR,α := Y
r<R
(1 + gur,α),
and
ΠR:= Y
r<R
(1 + ur)
with gur,α := 1
q
q−1
X
a=1
(|f(aqr)|α−1) and ur:= 1
q
q−1
X
a=1
(f(aqr)−1).
The following theorem describes a complete characterization of q-multiplicative uni-
formly summable functions.
Theorem 5.1.2. Let fbe a q-multiplicative function. Then, the following assertions are
equivalent.
(i) f∈L∗, and kfk1>0.
(ii) Let α > 0. The series
∞
X
r=0
1
q
q−1
X
a=0
(|f(aqr)|α−1)2(5.1)
is convergent, and for some constants c1(α), c2(α)∈R, for all Rand for some
sequence {Ri},Ri→ ∞, the inequalities
X
r<R
1
q
q−1
X
a=0
(|f(aqr)|α−1) ≤c1(α)<∞,(5.2)
52
5.1 Main results
and
X
r<Ri
1
q
q−1
X
a=0
(|f(aqr)|α−1) ≥c2(α)>−∞ (5.3)
hold.
(iii) f∈Lα, and kfkα>0for all α > 0.
The mean behaviour of such functions is given in
Theorem 5.1.3. Let f∈L∗be a q-multiplicative function, and let kfk1>0. Further, let
qR−1≤N < qRwith R∈N. Then, as N→ ∞,
1
NX
n<N
f(n) = ΠR+o(1)
and, for every α > 0,
1
NX
n<N |f(n)|α=g
ΠR,α +o(1).
An immediate consequence is the following
Corollary 5.1.4. Let fbe q-multiplicative. Then, the following assertions hold.
(i) Let f∈L∗. If the mean-value M(f)of fexists, and if it is different from zero, then
the series
∞
X
r=0
q−1
X
a=0
(f(aqr)−1),(5.4)
and
∞
X
r=0
q−1
X
a=0 |f(aqr)−1|2(5.5)
converge, and
q−1
X
a=0
f(aqr)6= 0 for each r ∈N0.
53
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
(ii) If the series (5.4) and (5.5) converge, then f∈L∗, and the mean-value M(f)of f
exists,
M(f) =
∞
Y
r=0 Ã1
q
q−1
X
a=0
f(aqr)!,
and kf−fRk1→0as R→ ∞, where
fR(n) = Y
r≤R
f(εr(n)qr) 0 ≤εr(n)< q.
(iii) Let f∈L∗. If the mean-value M(f)of fexists, and if it is different from zero, then
the mean-value M(|f|α)of |f|αexists for each α > 0(and is different from zero).
The case of mean-value zero is contained in
Corollary 5.1.5. Let f∈L∗be q-multiplicative. Then, the mean-value M(f)of fis zero
if and only if ΠR=o(1) as R→ ∞.
5.2 Preliminary results
To prove our main theorem, we need to show the following lemmata
Lemma 5.2.1. Let f∈L∗be q-multiplicative and let kfk1>0. Then,
∞
X
r=0
1
q
q−1
X
a=0
(|f(aqr)|α−1)2<∞
for all α > 0.
Proof. Because of kfk1>0, we can find a sequence {xi}such that
X
n<xi
ε<|f(n)|α<K
1>> xi,
as i→ ∞for some suitable ε, K > 0.
54
5.2 Preliminary results
We define an q-additive function gby
g(aqr) =
log(|f(aqr)|α)if f(aqr)6= 0
1if f(aqr) = 0.
Then, X
n<xi
−c1<g(n)<c2
1³xi
with c1= log 1/ε and c2= log K.
For real numbers t, we define the functions
H(x, t) = X
n<x
exp(itg(n)),
for any x > 0.
Delange [10] proved that the limit l(t) := lim
x→∞
1
x|H(x, t)|always exists, and l(t)6= 0
holds if and only if
∞
X
r=0
1
q
q−1
X
a=0
(1 −cos(tg(aqr)))
converges.
Further, we define the function Dby
D(ν) =
¡sin πν
πν ¢2if ν6= 0,
1 if ν= 0.
Then, for each real number y, we have
Z∞
−∞
e2πiνyD(ν)dν =
1−|y|if |y| ≤ 1,
0 otherwise.
Interchanging summation and integration shows that for positive λ
Z∞
−∞
λ|H(x, t)|2D(λt)dt =X
n1,n2≤x
|g(n1)−g(n2)|≤λµ1−1
λ|g(n1)−g(n2)|¶.
We divide by xi, let xi→ ∞, and apply Lebesgue’s theorem for dominated convergence.
If λis sufficiently large, then
Z∞
−∞
λl(t)2D(λt)dt > 0.
55
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
More exactly, if g(n)satisfies the condition given in the definition of finitely distributed
functions, and if λ≥2c2, then the value of this integral is at least as large as c2
1/2.
It follows that there is a set E, of positive Lebesgue measure, on which l(t)>0.
Now, ∞
X
r=0
(1 −cos(tg(aqr))) <∞
for every 0≤a≤q−1and for all t∈E. It means
∞
X
r=0
(1 −cos(tg(aqr))) ≤c
for all t∈E∗, where E∗is some subset of Eand m(E∗)>0. This is equivalent to
∞
X
r=0
sin2µt
2g(aqr)¶≤c < ∞
for all t∈E∗.
In view of the inequality
sin2(x±y)≤2 sin2x+ 2 sin2y
and applying Steinhaus’s lemma1we can find a T > 0such that for all 0≤a≤q−1,
and for |t| ≤ T, we get
∞
X
r=0
(1 −cos(tg(aqr))) ≤4c < ∞(5.6)
Integrating (5.6) from 0 to Tand multiplying with 1/T, we have
∞
X
r=0
h(Tg(aqr)) ≤4c < ∞(5.7)
where h(u) = 1 −sin u
ufor u6= 0 and h(0) = 0.
Since h(u)≥0for all real numbers u, and h(u)≥1/2for u≥2, we conclude that
|g(aqr)| ≥ 2/T for only finitely many r.
1(see [12], Lemma (1.1) The differences generated by a set of real numbers of positive measure, cover
an open interval about the origin.)
56
5.2 Preliminary results
Thus, there is an Ma>0such that |g(aqr)| ≤ Mafor all r≥0, and there is ma>0such
that h(u)≥mau2for |u| ≤ TMa.
Hence, ∞
X
r=0
(g(aqr))2≤2qlog 2
maT2,
and the series
∞
X
r=0
1
q
q−1
X
a=0
(g(aqr))2converges.
Since (log |x|)2³(|x|−1)2if ||x|−1| ≤ 1/2, the proof of Lemma 5.2.1 is finished.
Lemma 5.2.2. Let fbe q-multiplicative and R∈N. Then,
qR−1
X
n=0 |f(n)|α=qRg
ΠR,α
for every α > 0, and
qR−1
X
n=0
f(n) = qRΠR.
Proof. Induction over Ryields the following formulas
qR+1−1
X
n=0 |f(n)|α=
q−1
X
a=0
qR−1
X
l=0 |f(aqR+l)|α
,
and
qR+1−1
X
n=0
f(n) =
q−1
X
a=0
qR−1
X
l=0
f(aqR+l)
,
which prove Lemma 5.2.2.
Lemma 5.2.3. Let f∈L∗be q-multiplicative and kfk1>0. Then,
g
ΠR,α = (c(α, |f|) + o(1)) exp ÃX
r<R gur,α!
for all α > 0with some constant c(α, |f|)∈R.
57
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
Proof. It is easy to see that, because of the convergence of the series in Lemma 5.2.1, we
have
g
ΠR,α =Y
r<R
(1 + gur,α)
= exp ÃX
r<R
log(1 + gur,α)!
= exp ÃX
r<R gur,α +OÃX
r<R
(gur,α)2!!
= (c(α, |f|) + o(1)) exp ÃX
r<R gur,α!
for all α > 0, and some constant c(α, |f|)∈R.
Lemma 5.2.4. Let f∈L∗be q-multiplicative with kfk1>0, and let α > 0. Then, there
are some constants c1(α), c2(α)∈Rsuch that
X
r<R
1
q
q−1
X
a=0
(|f(aqr)|α−1) ≤c1(α)<∞(5.8)
for all Rand
X
r<Ri
1
q
q−1
X
a=0
(|f(aqr)|α−1) ≥c2(α)>−∞ (5.9)
for some sequence {Ri},Ri→ ∞.
Proof. Since f∈L1and kfk1>0, by Lemma 5.2.3, we get the inequalities (5.8) and
(5.9) for α= 1. Now, let α > 0, and let
||f(aqr)|−1| ≤ 1
2,
58
5.2 Preliminary results
then
|f(aqr)|α−1 = (|f(aqr)|−1 + 1)α−1
=α(|f(aqr)|−1) + O((|f(aqr)|−1)2),
which implies the inequalities (5.8) and (5.9) for all α > 0.
Remark. Let f∈L∗be q-multiplicative with kfk1>0, and let α > 0.
If
X
n<x |f(n)|α³x
then
X
r<R
1
q
q−1
X
a=0
(|f(aqr)|α−1) = O(1)
as R→ ∞.
The next lemma shows a general method for getting upper estimations.
Lemma 5.2.5. Let fbe q-multiplicative and let qR−1≤N < qRwith R∈N. Then, for
every h∈N, we have
¯¯¯¯¯X
n<N
f(n)¯¯¯¯¯≤
h
X
r=1 ¯¯¯¯¯¯qR−rΠR−r
r−1
Y
t=1
f(εR−t(N)qR−t)
εR−r(N)−1
X
a=0
f(aqR−r)¯¯¯¯¯¯
+ÃR−1
Y
r=R−h|f(εr(N)qr)|!·O(qR−h),
where the O-constant depends only on f.
Proof. Let N=cqR−1+b, where 1≤c < q, and b=X
r<R−1
εr(N)qr< qR−1, where
59
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
0≤εr(N)≤q−1. Then,
X
n<N
f(n)
=
c−1
X
a=0
qR−1−1
X
l=0
f(aqR−1+l)
+
b−1
X
l=0
f(cqR−1+l)
=
c−1
X
a=0
f(aqR−1)
qR−1−1
X
l=0
f(l) + f(cqR−1)
b−1
X
l=0
f(l)
=qR−1ΠR−1
c−1
X
a=0
f(aqR−1)
+qR−2ΠR−2f(cqR−1)
εR−2(N)−1
X
a=0
f(aqR−2)
+qR−3ΠR−3f(cqR−1)f(εR−2(N)qR−2)
εR−3(N)−1
X
a=0
f(aqR−3)
+
.
.
.
+qR−hΠR−hf(cqR−1)f(εR−2(N)qR−2)···f(εR−h+1(N)qR−h+1)
εR−h(N)−1
X
a=0
f(aqR−h)
+f(cqR−1)f(εR−2(N)qR−2)···f(εR−h+1(N)qR−h+1)f(εR−h(N)qR−h)
bh−1
X
l=0
f(l),
where bh< qR−h, and ¯¯¯¯¯
bh−1
X
l=0
f(l)¯¯¯¯¯≤
qR−h−1
X
l=0 |f(l)|=O(qR−h).
In the following lemmata 5.2.6, 5.2.7 and 5.2.8, we collect some more properties of q-
multiplicative functions f∈L∗with kfk1>0.
60
5.2 Preliminary results
Lemma 5.2.6. Let f∈L∗be q-multiplicative and kfk1>0. Then, the series
∞
X
r=0
1
q
q−1
X
a=0 |f(aqr)−1|2
is convergent if and only if
X
r<Ri
1
q
q−1
X
a=0
(Re f(aqr)−1) ≥c3>−∞
for some constant c3∈Rand some sequence {Ri}with Ri↑ ∞.
Proof. We have
X
r<R
1
q
q−1
X
a=0 |f(aqr)−1|2=X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1)2
+2X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1)
−2X
r<R
1
q
q−1
X
a=0
(Re f(aqr)−1)
=: P1+2 P2−2P3.
By Lemma 5.2.1, P1is convergent and, by Lemma 5.2.4, P2is bounded from above for
some sequence {Ri}for Ri→ ∞. Thus, Lemma 5.2.6 holds.
Lemma 5.2.7. Let f∈L∗be q-multiplicative, and let kfk1>0. If
X
r<Ri
1
q
q−1
X
a=0
(Re f(aqr)−1) ≥c3>−∞
for some constant c3, and for some sequence {Ri}with Ri↑ ∞, then
ΠR:= Y
r<R
(1 + ur)
= (c(f) + o(1)) exp ÃX
r<R
ur!
for some constant c(f)6= 0.
61
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
Proof. If X
r<Ri
1
q
q−1
X
a=0
(Re f(aqr)−1) ≥c3>−∞ for some constant c3, and for some
sequence {Ri}with Ri↑ ∞, then by Lemma 5.2.6, we have
∞
X
r=0 |ur|2≤
∞
X
r=0
1
q
q−1
X
a=0 |f(aqr)−1|2<∞,
and we obtain
ΠR:= Y
r<R
(1 + ur)
= exp ÃX
r<R
ur+OÃX
r<R |ur|2!!
= (c(f) + o(1)) exp ÃX
r<R
ur!
with some constant c(f)6= 0.
Lemma 5.2.8. Let f∈L∗be q-multiplicative, and let kfk1>0. If
lim
R→∞ X
r<R
1
q
q−1
X
a=0
(Re f(aqr)−1) = −∞,
then ΠR→0as R→ ∞.
Proof. Obviously,
|ΠR|= exp ÃX
r<R
log |1 + ur|!,
and
log |1 + ur|=1
2log((1 + Re ur)2+ (Imur)2)
=1
2log(1 + 2Re ur+|ur|2)
≤Re ur+1
2|ur|2.
62
5.2 Preliminary results
Since
|ur|2≤q−1
q·1
q
q−1
X
a=0 |f(aqr)−1|2
=q−1
q(1
q
q−1
X
a=0
(|f(aqr)|−1)2+2
q
q−1
X
a=0
(|f(aqr)|−1) −2Re ur),
we observe
Re ur+1
2µq−1
q·(−2Re ur)¶=1
qRe ur
which implies
|ΠR| ¿ exp ÃX
r<R
1
q·1
q
q−1
X
a=0
(Re f(aqr)−1)!,
and the assertion of Lemma 5.2.8 follows.
Remark. Let f∈L∗be q-multiplicative and kfk1>0. Then, by Lemma 5.2.7 and
Lemma 5.2.8, ΠR=o(1) if and only if
X
r<R
1
q
q−1
X
a=0
(Re f(aqr)−1) → −∞
as R→ ∞.
Using the Tur´
an-Kubilius inequality (Corollary 4.2.2), we prove
Lemma 5.2.9. Let f∈L∗be q-multiplicative, kfk1>0and qR−1≤N < qRwhere
R∈N. Further, let ∞
X
r=0
1
q
q−1
X
a=0 |f(aqr)−1|2<∞.
Then, for any h∈N,¯¯¯¯¯
1
NX
n<N
f(n)−ΠR¯¯¯¯¯≤˜cq−h+o(1)
as N→ ∞, with some constant ˜c∈Rwhich only depends on f.
Proof. We set
fR(n) =
R
Y
r=0
f(er(n)qr).
63
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
Then, for any h∈N, we have
¯¯¯¯¯
1
NX
n<N
f(n)−ΠR¯¯¯¯¯≤1
NX
n<N |f(n)−fR−h(n)|
+1
N¯¯¯¯¯X
n<N
fR−h(n)−NΠR−h+1¯¯¯¯¯+|ΠR−h+1 −ΠR|
=: P1+P2+4
Ad P1:
We choose r0∈Nsuch that |f(aqr)−1| ≤ 1
10 for all r, a ∈Nwith r > r0,0≤a < q,
and we define the function gR
gR(n) :=
X
r>R
log f(er(n)qr) for R≥r0,
0 for R < r0.
Then, the functions gRare q-additive.
Now,
1
NX
n<N |f(n)−fR−h(n)|
=1
NX
n<N |fR−h(n)||exp(gR−h(n)) −1|
≤1
NX
n<N |gR−h(n)|(|f(n)|+|fR−h(n)|)
≤Ã1
NX
n<N |gR−h(n)|2!1/2
Ã2
NX
n<N |f(n)|2!1/2
+Ã2
NX
n<N |fR−h(n)|2!1/2
.
Applying the Tur´
an-Kubilius inequality for q-additive functions (Corollary 4.2.2), we ob-
64
5.2 Preliminary results
tain
1
NX
n<N |gR−h(n)|2
≤2
NX
n<N ¯¯¯¯¯gR−h(n)−X
R−h<r<R
1
q
q−1
X
a=0
gR−h(aqr)¯¯¯¯¯
2
+2
NX
n<N ¯¯¯¯¯X
R−h<r<R
1
q
q−1
X
a=0
gR−h(aqr)¯¯¯¯¯
2
≤4ÃX
R−h<r<R−1
1
q
q−1
X
a=0 |log f(aqr)|2+1
c
c
X
a=1 |log f(aqR−1)|2!+
+2 ¯¯¯¯¯X
R−h<r<R
1
q
q−1
X
a=0
log f(aqr)¯¯¯¯¯
2
,
where cqR−1≤N < (c+ 1)qR−1with some integer cwith 0< c < q.
Now, since his fixed, and log f(aqr)→0for r→ ∞, such that
lim
R→∞ lim
N→∞
1
NX
n<N |gR−h(n)|2= 0.
Using Lemmata 5.2.1, 5.2.3 and 5.2.4 for α= 2 show f,fR−h∈L2, and thus,
1
NX
n<N |f(n)−fR−h(n)|=o(1).
Ad P2:
For all 0≤a < q with 0≤n < qR−h+1, we have
fR−h(aqR−h+1 +n) = f(n),
and for all l∈N, we get
lqR−h+1−1
X
n=0
fR−h(n) = l
qR−h+1−1
X
n=0
f(n) = lqR−h+1ΠR−h+1 .
Further, for N=lqR−h+1, we obtain
1
NX
n<N
fR−h(n)−ΠR−h+1 = 0
65
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
and for lqR−h+1 < N < (l+ 1)qR−h+1 and l≥1, we conclude
¯¯¯¯¯X
n<N
fR−h(n)−NΠR−h+1¯¯¯¯¯
=¯¯¯¯¯¯−(N−lqR−h+1)ΠR−h+1 +
N−1
X
n=lqR−h+1
fR−h(n)¯¯¯¯¯¯
=¯¯¯¯¯¯−(N−lqR−h+1)ΠR−h+1 +fR−h(lqR−h+1)
N−lqR−h+1−1
X
n=0
f(n)¯¯¯¯¯¯
≤c(N−lqR−h+1)
< cqR−h+1
with some constant cdepending only on f.
Ad 4: Obviously (cf. proof of Lemma 5.2.8)
|ΠR−ΠR−h+1|=|ΠR−h+1|¯¯¯¯¯ÃR−1
Y
r=R−h+1
1
q
q−1
X
a=0
f(aqr)!−1¯¯¯¯¯
≤c
R−1
X
r=R−h+1 ¯¯¯¯¯
1
q
q−1
X
a=0
(f(aqr)−1)¯¯¯¯¯.
Since his fixed, and f(aqr)tends to 1 as rruns to infinity, we have |ΠR−ΠR−h|=o(1)
as R→ ∞.
5.3 Proof of main results
Proof of Theorem 5.1.2. The implication (i) ⇒(ii) is proved as follows.
If f∈L∗and kfk1>0, we conclude, by Lemma 5.2.1, that the series (5.1) is convergent.
Lemma 5.2.4 shows the inequalities (5.2) and (5.3) for all α > 0.
66
5.3 Proof of main results
Proof of (ii) ⇒(iii).
By Lemma 5.2.2 and the convergence of (5.1), we show, as in the proof of Lemma 5.2.3,
1
qR
qR−1
X
n=0 |f(n)|α=g
ΠR,α = (c(α, |f|) + o(1)) exp(X
r<R gur,α)
for all α > 0with some constant c(α, |f|)∈R. Observing, if qR−1≤N < qR,
1
NX
n<N |f(n)|α¿1
qRX
n<qR|f(n)|α=g
ΠR,α,
and the inequality (5.2) gives f∈Lα, and (5.3) implies kfkα>0.
The implication (iii) ⇒(i) is obvious.
Proof of Theorem 5.1.3. First, we assume that ΠR=o(1). Then, by Lemma 5.2.5
1
NX
n<N
f(n) = o(1). Now, let ΠR6=o(1). Then, by Lemma 5.2.6 and Lemma 5.2.9, we
have 1
NX
n<N
f(n) = ΠR+o(1).
Furthermore g
ΠR,α 6=o(1), because of 0<kfk1≤ kfkαfor all α > 0. Then, by Lemma
5.2.1 and Lemma 5.2.9 the second assertion of Theorem 5.1.3 follows.
Proof of Corollary 5.1.4. (i) Let f∈L∗be q-multiplicative. If the mean-value M(f)of
fexists and is nonzero, then obviously kfk1>0.
We know that (see the proof of Lemma 5.2.8)
|ΠR| ¿ exp ÃX
r<R
1
q2
q−1
X
a=0
(Re f(aqr)−1)!.
Further,
∞
X
r=0
1
q
q−1
X
a=0
(Re f(aqr)−1) > c3>−∞ for some constant c3∈R, since the
mean-value M(f)of fexists, and it is different from zero.
67
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
By Lemma 5.2.6, the series (5.5) converges, and Lemma 5.2.7 gives
ΠR:= Y
r<R
1
q
q−1
X
a=0
f(aqr)
= (c(f) + o(1)) exp ÃX
r<R
1
q
q−1
X
a=0
(f(aqr)−1)!,
with some constant c(f)6= 0.
Since the mean-value M(f)of fexists, and it is nonzero, the series (5.4) converges, and
q−1
X
a=0
f(aqr)6= 0 for each r∈N0.
(ii) If the series (5.4) and (5.5) converge then the infinite product lim
R→∞ ΠRexists, and it is
zero if and only if a factor equals zero. Thus, 0<g
ΠR,1for all Rand
X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1) > c4>−∞
for some constant c4∈R.
Now,
X
r<R
1
q
q−1
X
a=0 |f(aqr)−1|2=X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1)2
+2X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1)
−2X
r<R
1
q
q−1
X
a=0
(Re f(aqr)−1),
holds, and the convergence of the series (5.4) and (5.5) shows that the series
X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1)2,
and X
r<R
1
q
q−1
X
a=0
(|f(aqr)|−1)
converge. Then, by Theorem 5.1.2, we have f∈Lαand kfkα>0.
Furthermore, by Lemma 5.2.6 and Lemma 5.2.9, we know that the mean-value M(f)of
68
5.4 Application to q-additive functions
fexists, and M(f) =
∞
Y
r=0 Ã1
q
q−1
X
a=0
f(aqr)!.
A small modification of the proof for the estimation of P1in Lemma 5.2.9 yields, because
of the convergence of the series (5.4) and (5.5), that kf−fRk1→0as R→ ∞.
(iii) Using Theorem 5.1.2 and the same arguments as above, we conclude that the series
∞
X
r=0
q−1
X
a=0
(|f(aqr)|α−1),
and ∞
X
r=0
q−1
X
a=0
(|f(aqr)|α−1)2
converge, and thus the mean-value M(|f|α)of |f|αexists for each α > 0(and it is
different from zero).
Proof of Corollary 5.1.5. Obvious.
5.4 Application to q-additive functions
Let us now turn to q-additive functions. Here we recall that a function f:N→Cis
q-additive if f(0) = 0 and
f(aqr+b) = f(aqr) + f(b)
for every pair of integer (a, b)satisfying
0≤a < q and 0 ≤b < qr.
The main results are as follows.
Theorem 5.4.1. Let gbe q-additive. Then, the following assertions hold.
(i) If gis finitely distributed, then the series
∞
X
r=0
q−1
X
a=0
(g(aqr))2converges.
69
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
(ii) If, for some α(x),
1
x]{n≤x:g(n)−α(x)≤y} ⇒ G(y)
where Gis a distribution function, then gis finitely distributed.
(iii) Let
∞
X
r=0
q−1
X
a=0
(g(aqr))2converge, and let α(x) := X
r<Nx
1
q
q−1
X
a=0
g(aqr),Nx:= blog x
log qc.
Then,
1
x]{n≤x:g(n)−α(x)≤y} ⇒ G(y),
where Gis some distribution function.
Assertion (iii) of Theorem 5.4.1 has already been proved by J. Coquet (see [4], Theorem
II. 4).
Proof.
(i) The assertion is an immediate consequence of the proof of Lemma 5.2.1.
(ii) We choose the number γsufficiently large, and such that ±γare continuity points of
the limiting distribution of g(n)−α(x). Then,
S:= 1
x]{n≤x:g(n)−α(x)≤γ}>1
2.
Moreover, let mand nbe any two elements of S, then
|g(m)−g(n)| ≤ |g(m)−α(x)|+|α(x)−g(n)| ≤ 2γ,
from which it is clear that g(n)is finitely distributed.
(iii) Let
ϕx(t) := 1
xX
n<x
eitg(n).
Then, we shall prove that
ϕx(t)e−itα(x)→ϕ(t) (x→ ∞)
70
5.5 Characterization of almost-periodic q-multiplicative functions
for all t∈R, where ϕ(t)is continuous at t= 0.
By Theorem 5.1.3, we have 1
xX
n<x
eitg(n)=Y
r<NxÃ1 + 1
q
q−1
X
a=0 ¡eitg(aqr)−1¢!+o(1).
Let ur(t) = 1
q
q−1
X
a=0 ¡eitg(aqr)−1¢and vr(t) = it
q
q−1
X
a=0
g(aqr).
For |t| ≤ T, we obtain
|ur(t)| ≤ T
q
q−1
X
a=0 |g(aqr)|,
|ur(t)|2≤T2(q−1)
q2
q−1
X
a=0
(g(aqr))2
and
|ur(t)−vr(t)| ≤ T2
2q
q−1
X
a=0
(g(aqr))2.
Hence, the infinite product
∞
Y
r=0
(1 + ur(t))e−vr(t)is uniformly convergent for t∈[−T, T]
and defines the characteristic function of a distribution function G.
5.5 Characterization of almost-periodic q-multiplicative
functions
The aim of this section is to find corresponding characterizations for q-multiplicative
functions belonging to D1and A1, respectively. Here, we recall that fis called α-almost-
periodic, if for every ε > 0, there is a linear combination hof exponential functions2
eβ,β∈R, such that kf−hkα≤ε. The linear space of α-almost-periodic functions is
denoted by Aα. If hcan always be chosen to be periodic then fis called α-limit-periodic.
The linear space of α-limit-periodic functions is denoted by Dα. We have the inclusions
D1$A1$L∗.
2eβ:N→Cwith eβ(n) := exp(2πiβn)is a q-multiplicative function.
71
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
A first step in this direction was done by J. Spilker [79] who proved the following
Proposition 5.5.1 (Spilker [79], Theorem 4). Let fbe q-multiplicative and the following
two series ∞
X
r=0
q−1
X
a=0
(f(aqr)−1) (5.10)
and ∞
X
r=0
q−1
X
a=0 |f(aqr)−1|2(5.11)
converge. Then,
(i) f∈Dα, α ≥1.
(ii) M(f) =
∞
Y
r=0 Ã1
q
q−1
X
a=0
f(aqr)!.
(iii) ˆ
f(β) =
∞
Y
r=0 Ã1
q
q−1
X
a=0
f(aqr)e−c
b(aqr)!if β=c
b,
0 if βirrational.
Remark. Assertion (iii) of Proposition 5.5.1 is not correct as it stands. Choose, for
example, f= 1 and β=1
pwhere pis a prime which does not divide q. Then, ˆ
f(β)=0,
and for all r∈N0,
q−1
X
a=0
f(aqr)e−1
p(aqr) = 1−e(qr+1/p)
1−e(qr/p)6= 0,
i.e. the infinite product
∞
Y
r=0 Ã1
q
q−1
X
a=0
f(aqr)e−1
p(aqr)!does not converge in this case.
We shall characterize the q-multiplicative functions f∈D1and f∈A1\D1by their
respective spectrum σ(f). First we show that the spectrum is empty only in the trivial
case. We prove
Theorem5.5.2. Let f∈A1be q-multiplicative. Then M(|f|) = 0 if and only if σ(f) = ∅.
72
5.5 Characterization of almost-periodic q-multiplicative functions
In the special case that the mean value exists and is different from zero, using Corollary
5.1.4, we obtain
Theorem 5.5.3. For every q-multiplicative function f, the following assertions are equiv-
alent:
(a) f∈D1and the mean-value M(f)is nonzero.
(b) The series (5.10) and (5.11) are both convergent and
q−1
X
a=1
f(aqr)6= 0 for each
r∈N0.
(c) f∈L∗and the mean-value M(f)exists and is nonzero.
(d) f∈Dαfor all α≥1and the mean-value M(f)is nonzero.
(e) f∈A1and the mean-value M(f)is nonzero.
(f) f∈Aαfor all α≥1and the mean-value M(f)is nonzero.
(g) f∈Lαfor all α≥1and the mean-value M(f)exists and is nonzero.
We use the following well-known result to prove Theorem 5.5.2 and Theorem 5.5.3.
Lemma 5.5.4. (see [77] Chap. VI.8. Proposition 8.2 ) For α≥1and every arithmetical
function f,f∈Aαif and only if f∈A1and |f| ∈ Aα.
ProofofTheorem5.5.3. The implications “(a)⇒(e)⇒(c)”are obviousand “(c)⇒(b)⇒(a)”
hold by Corollary 5.1.4, (i) and (ii). Using Lemma 5.5.4 together with Corollary 5.1.4 for
|f|α,α≥1, gives “(c)⇒(d)”, whereas the implications “(d)⇒(f)⇒(g)⇒(c)” are again
obvious. This proves Theorem 5.5.3.
Proof of Theorem 5.5.2. If M(|f|)=0then obviously σ(f) = ∅. Assume that
M(|f|)6= 0. Then, by Theorem 5.5.3, |f| ∈ A2and M(|f|2)6= 0, and Lemma 5.5.4
implies f∈A2. By Parseval’s equality M(|f|2) = X
β∈σ(f)|M(f·e−β)|2, and σ(f) = ∅
73
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
implies M(|f|) = M(|f|2) = 0. This contradiction proves Theorem 5.5.2.
Concerning the description of the spectrum σ(f)for q-multiplicative functions f∈D1or
f∈A1\D1we establish
Theorem 5.5.5. Let f∈D1be q-multiplicative with non-empty spectrum σ(f).
(a) If M(f)6= 0 then
σ(f)⊂ {β|β=c
bmod 1,c
b∈Q;p prime, p|b⇒p|q;
q−1
X
a=0
f(aqr)e−β(aqr)6= 0 for all r ∈N0}.
(b) If M(f) = 0 then there exists some β0∈Q/Zsuch that
σ(f)⊂ {β|β=β0+c
bmod 1,c
b∈Q;p prime, p|b⇒p|q;
q−1
X
a=0
f(aqr)e−β(aqr)6= 0 for all r ∈N0}.
Corollary 5.5.6. Let f∈A1\D1be q-multiplicative with non-empty spectrum σ(f).
Then there exists some β0∈(R\Q)/Zsuch that
σ(f)⊂ {β|β=β0+c
bmod 1,c
b∈Q;p prime, p|b⇒p|q;
q−1
X
a=0
f(aqr)e−β(aqr)6= 0 for all r ∈N0}.
Proof of Theorem 5.5.5 and Corollary 5.5.6. Let f∈D1be q-multiplicative and let the
mean-value M(f)be nonzero. Then the series (5.10) and (5.11) both converge for f. Let
β∈σ(f). Then β∈R/Zand the mean-value M(f·e−β)is nonzero. Putting g=f·e−β
implies that
∞
X
r=0
q−1
X
a=0 |g(aqr)−1|2(5.12)
74
5.5 Characterization of almost-periodic q-multiplicative functions
is convergent. We show that this happens if and only if β=c
bis a rational number and
each prime divisor of bdivides q. We consider three cases.
•Case 1: Let βbe irrational. The function e−βis q-multiplicative and its absolute
value is equal to 1. By Delange’s result [10] for q-multiplicative functions fof
absolute value less or equal to 1 whose mean-value M(f)exists, the series
∞
X
r=0
1
q
q−1
X
a=0 |e−β(aqr)−1|2(5.13)
converges if and only if the representation M(e−β) =
∞
Y
r=0 Ã1
q
q−1
X
a=0
(e−β(aqr))!
holds. Since M(eβ)=0and 1
q
q−1
X
a=0
(e−β(aqr)) 6= 0 for all r∈N0the series (5.13)
diverges.
•Case 2: Let β=c
bbe rational and assume there is a prime pwhich divides b, but
does not divide q. Then for all rthe numbers c
bqrare not integers. This implies:
¯¯¯exp ³−c
bqr´−1¯¯¯≥¯¯¯¯1−exp µ−1
b¶¯¯¯¯,
and the series ∞
X
r=0
1
q
q−1
X
a=0 |e−c
b(aqr)−1|2(5.14)
diverges.
•Case 3: Let β=c
bbe rational, and assume that for each prime divisor of bdivides
q, too. Then for all a= 0,1,··· , q −1and all r≥r0, we have e−β(aqr) = 1.
Now
|1−e−β(aqr)|2¿ |1−g(aqr)|2+|1−f(aqr)|2.
Since the series (5.11) and (5.12) converge, cases 1 and 2 can not occur. Therefore, the
mean-value M(f·e−β)is zero for the cases 1 and 2. In case 3 the series
∞
X
r=0
1
q
q−1
X
a=0
(g(aqr)−1)2
75
5 Mean behaviour of uniformly summable q-multiplicative functions and its
applications
and ∞
X
r=0
1
q
q−1
X
a=0
(g(aqr)−1)
converge. Then
M(g) =
∞
Y
r=0
1
q
q−1
X
a=0
g(aqr)(5.15)
and the mean-value M(g)is nonzero if and only if each factor of (5.15) is nonzero. This
proves (a).
For the proof of (b) and Corollary 5.5.6, let the mean-value of fbe zero, and let β0∈R/Z
such that the mean-value of f·e−β0is nonzero . Then f·e−β0∈D1. Since f∈A1\D1
if and only if β0is irrational, (a) yields (b) and Corollary 5.5.6.
Example. Let f=eβwhere β∈(R\Q)/Z. Then, obviously, the mean-value M(f)
equals zero and σ(f) = {β}.
76
Chapter 6
Mean behaviour of uniformly
summable Q-multiplicative functions
In this chapter, we extend the results of Chapter 5 to uniformly summable Q-multiplicative
functions. In the case of a bounded sequence {qr}r≥1, we have similar theorems as in
the q-adic case. In the case of an unbounded sequence {qr}r≥1, the situation is quite
different. Unavoidable for unbounded sequences {qr}r≥1is the existence of a so-called
first digit phenomenon. We investigate the mean behaviour of uniformly summable Q-
multiplicative functions that belong to L2and for which the first digit condition
max
1≤j≤qr−1
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0 as r→ ∞
holds.
6.1 Main results
Let {qr}r≥1with qr≥2, be a sequence of natural numbers, and let Q0= 1,Qr=qrQr−1
when r≥1. Here, we recall that fis Q-multiplicative if f(0) = 1, and
f(aQr+b) = f(aQr)·f(b)
77
6 Mean behaviour of uniformly summable Q-multiplicative functions
for every pair of integer (a, b)satisfying
0≤a < qr+1 and 0 ≤b < Qr.
Definition 6.1.1. Let fbe Q-multiplicative function, we define
f
ΠR:= Y
r<R
(1 + eur),
and
ΠR:= Y
r<R
(1 + ur)
where eur:= 1
qr
qr−1
X
a=0
(|f(aQr−1)|−1) and ur:= 1
qr
qr−1
X
a=0
(f(aQr−1)−1).
Theorem 6.1.2. Let f∈L2be Q-multiplicative function. If
max
1≤j≤qr−1
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0,(6.1)
then, for QR−2≤N < QR−1,N→ ∞
(a) 1
NX
n<N
f(n) = ΠR+o(1),
(b) 1
NX
n<N |f(n)|=e
ΠR+o(1).
Theorem 6.1.3. Let f∈L2be Q-multiplicative function. If the conditions
(i) max
1≤j<qr
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0,
(ii)
∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞,
(iii)
∞
X
r=1
1
qr
qr−1
X
a=0
(f(aQr−1)−1) converges,
78
6.2 Preliminary results
(iv) 1 +
qr−1
X
a=1
f(aQr−1)6= 0.
hold, then, the mean-value M(f)of fexists and is different from zero.
Theorem 6.1.4. Let f∈L2be Q-multiplicative function. Suppose the mean-value M(f)
of fexists and is different from zero,
max
1≤j<qr
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0,
and ∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞.
Then,
∞
X
r=1
1
qr
qr−1
X
a=0
(f(aQr−1)−1)
converges, and
1 +
qr−1
X
a=1
f(aQr−1)6= 0.
6.2 Preliminary results
To prove our main theorem, we need to show the following lemmata
Lemma 6.2.1. Let z1, . . . , zk∈Care complex numbers, then
|z1···zk−1| ≤
k
Y
j=1
max(|zj|,1)
k
X
j=1 |zj−1|.
Proof.
|z1···zk−1| ≤ |zk||z1···zk−1−1|+|zk−1|
≤max(|zk|,1)(|z1···zk−1−1|+|zk−1|)
≤
k
Y
j=1
max(|zj|,1)
k
X
j+1 |zj−1|.
79
6 Mean behaviour of uniformly summable Q-multiplicative functions
Lemma 6.2.2. Let fbe Q-multiplicative and R∈N. Then,
QR−1−1
X
n=0
f(n) = QR−1ΠR,
and
QR−1−1
X
n=0 |f(n)|=QR−1f
ΠR.
Proof. Induction over Ryields the following formulas
QR−1
X
n=0
f(n) =
qR−1
X
a=0 ÃQR−1−1
X
l=0
f(aQR−1+l)!,
and
QR−1
X
n=0 |f(n)|=
qR−1
X
a=0 ÃQR−1−1
X
l=0 |f(aQR−1+l)|!
for all R≥0, which proves Lemma 6.2.2.
Lemma 6.2.3. Let fbe Q-multiplicative function and
∞
X
r=1
1
qr
qr−1
X
a=0
(|f(aQr−1)|−1)2<∞,
then
f
ΠR= (c1(f) + o(1)) exp ÃX
r≤Reur!
for some constant c1(f)∈R.
Proof. Since
∞
X
r=1
(eur)2=
∞
X
r=1
1
q2
rÃqr−1
X
a=0
(|f(aQr−1)|−1)!2
≤
∞
X
r=1
1
qr
qr−1
X
a=0
(|f(aQr−1)|−1)2<∞,
80
6.2 Preliminary results
is easy to see that
f
ΠR=Y
r<R
(1 + eur)
= exp ÃX
r<R
log(1 + eur)!
= exp ÃX
r<R eur+OÃX
r<R
(eur)2!!
= (c(|f|) + o(1)) exp ÃX
r<R eur!.
As a consequence we get
Corollary 6.2.4. Let fbe Q-multiplicative function and
∞
X
r=1
1
qr
qr−1
X
a=0
(|f(aQr−1)|−1)2<∞,
then, f∈L1, and kfk16= 0 if and only if for some constants c1, c2∈R
X
r<R eur≤c1<∞,
as R→ ∞, and X
r<Rieur≥c2>−∞
for some sequence {Ri},Ri→ ∞.
Lemma 6.2.5. Let fbe Q-multiplicative function and
∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞,
then, |ΠR| ³ 1if and only if
X
r≤R
(2Re ur+|ur|2) = O(1),
as R→ ∞.
81
6 Mean behaviour of uniformly summable Q-multiplicative functions
Proof. We get
|ΠR|=Y
r<R |1 + ur|
= exp Ãlog Y
r<R |1 + ur|!
= exp ÃX
r<R
log |1 + ur|!.
Since
log |1 + ur|=1
2log(1 + 2Re ur+|ur|2)
³2Re ur+|ur|2,
we obtain
|ΠR|= exp ÃX
r<R
log |1 + ur|!
³exp ÃX
r<R
(2Re ur+|ur|2)!
Lemma 6.2.5 yields the following
Corollary 6.2.6. Let fbe Q-multiplicative function and
∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞,
then,
(i) |ΠR| → c6= 0 if and only if the series X
r≤R
(2Re ur+|ur|2)converges,
(ii) |ΠR| → 0if and only if the series X
r≤R
(2Re ur+|ur|2)diverges,
as R→ ∞.
82
6.2 Preliminary results
Example 6.2.7. Let
ur=−1
r+i√2
√r,
where ur=1
r+ 1
r
X
a=1
f(ar!) −1with f(ar!) = 1 −r+ 1
r2+i(r+ 1)√2
r√r.
It is easy to see that, the series
∞
X
r=1
Re urdiverges, but the series
∞
X
r=1
(2Re ur+|ur|2) =
∞
X
r=1
1
r2
converges.
Lemma 6.2.8. Let fbe Q-multiplicative function. If
max
1≤j<qr
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0,
and
∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞,
then,
|ΠR|= (c2(f) + o(1)) exp ÃX
r≤R
2Re ur!
for some constant c2(f)∈R.
Proof. Since
∞
X
r=1 |ur|2=
∞
X
r=1 ¯¯¯¯¯
1
qr
qr−1
X
a=0
(f(aQr−1)−1)¯¯¯¯¯
2
≤
∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞,
83
6 Mean behaviour of uniformly summable Q-multiplicative functions
and applying the proof of Lemma 6.2.5, we obtain
|ΠR|= exp ÃX
r<R
log |1 + ur|!
³exp ÃX
r<R
(2Re ur+|ur|2)!
= (c2(f) + o(1)) exp ÃX
r≤R
2Re ur!.
Lemma 6.2.9. Let fbe Q-multiplicative function. If
max
1≤j<qr
1
j+ 1
j
X
a=0 |f(aQr−1)−1|2→0,
and
∞
X
r=1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2<∞,
then,
ΠR= (c3(f) + o(1)) exp ÃX
r≤R
ur!
for some constant c3(f)∈R.
Proof. By the proof of Lemma 6.2.8, we know that,
∞
X
r=1 |ur|2<∞.
Hence,
ΠR:= Y
r<R
(1 + ur)
= exp ÃX
r<R
ur+OÃX
r<R |ur|2!!
= (c3(f) + o(1)) exp ÃX
r<R
ur!.
84
6.3 Proof of main results
6.3 Proof of main results
Proof of Theorem 6.1.2. We set
fT(n) =
T
Y
r=1
f(εr(n)Qr−1)
and let M > QT−1, then
1
MX
n<M |fT(n)|2=1
MÃX
m<QT|f(m)|2·](n¯¯¯¯¯
T
X
r=1
εr(n)Qr−1=m∧n < M )!.
Put cQR1−1≤M < (c+ 1)QR1−1with 0< c < qR1, and to estimate the above term, we
obtain
1
MX
n<M |fT(n)|2≤KQT·(c+ 1)qR1−1qR1−2···qT−1
cQR1−1≤2K,
where Kis constant. Thus fTis also a L2function.
Then, for each h∈N,
¯¯¯¯¯
1
NX
n<N
f(n)−ΠR¯¯¯¯¯≤1
NX
n<N |f(n)−fR−h(n)|
+1
N¯¯¯¯¯X
n<N
fR−h(n)−NΠR−h¯¯¯¯¯+|ΠR−h−ΠR|
=P1+P2+∆ .
Ad P1:
We define the Q-additive function g
g(n) =
∞
X
r=1
g(εr(n)Qr−1)
=X
R−h<r<R
g(εr(n)Qr−1)
=X
R−h<r<R |f(εr(n)Qr−1)−1|
85
6 Mean behaviour of uniformly summable Q-multiplicative functions
where
g(εr(n)Qr−1) :=
|f(εr(n)Qr−1)−1|) if R−h < r ≤R ,
0 otherwise .
Applying Lemma 6.2.1, we obtain
1
NX
n<N |f(n)−fR−h(n)|
=1
NX
n<N |fR−h(n)|¯¯¯¯¯Y
R−h<r≤R
f(εr(n)Qr−1)−1¯¯¯¯¯
≤1
NX
n<N |fR−h(n)|Y
R−h<r≤R
max(|f(εr(n)Qr−1)|,1) X
R−h<r≤R|f(εr(n)Qr−1)−1|
≤1
NX
n<N |fR−h(n)|Y
R−h<r≤R
max(|f(εr(n)Qr−1)|,1)g(n)
≤Ã1
NX
n<N
(g(n))2!1/2
1
NX
n<N Ã|fR−h(n)|Y
R−h<r≤R
max(|f(εr(n)Qr−1)|,1)!2
1/2
.
Using the Tur´
an-Kubilius inequality for Q-additive functions (Corollary 4.2.4), we have
1
NX
n<N
(g(n))2
≤2
NX
n<N Ãg(n)−X
R−h<r<R
1
qr
qr−1
X
a=0
g(aQr−1)!2
+2
NX
n<N ÃX
R−h<r<R
1
qr
qr−1
X
a=0
g(aQr−1)!2
≤4ÃX
R−h<r<R−1
1
qr
qr−1
X
a=0 |f(aQr−1)−1|2+1
c
c
X
a=1 |f(aQR−2−1)|2!
+2 ÃX
R−h<r<R
1
qr
qr−1
X
a=0 |f(aQr−1)−1|!2
,
where cQR−2≤N < (c+ 1)QR−2,0< c < qR−1.
86
6.3 Proof of main results
Applying the Cauchy-Schwarz inequality, we conclude
2ÃX
R−h<r<R
1
qr
qr−1
X
a=0 |f(aQr−1)−1|!2
≤2hX
R−h<r<R
1
(qr)2Ãqr−1
X
a=0 |f(aQr−1)−1|!2
≤2hX
R−h<r<R
r
(qr)2
qr−1
X
a=0 |f(aQr−1)−1|2.
Since his fixed, and concerning condition (6.1), we obtain
lim
N→∞
1
NX
n<N
(g(n))2= 0.
Now, we define the Q-multiplicative function ˜
fwith
˜
f(aQr−1) :=
|f(aQr−1)|if 1 ≤r≤R−h
max(|f(aQr−1)|,1) if R−h < r ≤R
,
then,
|fR−h(n)|Y
R−h<r≤R
max(|f(εr(n)Qr−1)|,1) = ˜
f(n),
and
1
NX
n<N
(˜
f(n))2≤1
cQR−2X
n<(c+1)QR−2
(˜
f(n))2
=1
cQR−2Y
1≤r≤R−h
(qr)Ã1 + 1
qr
qr−1
X
a=0 |f(aQr−1)|2−1!
·Y
R−h<r≤R−2
(qr)Ã1 + 1
qr
qr−1
X
a=0 |˜
f(aQr−1)|2−1!
·(c+ 1) Ã1 + 1
c+ 1
c
X
a=1 |˜
f(aQR−2)|2−1!
=Q1·Q2·Q3,
where cQR−2≤N < (c+ 1)QR−2with 0< c < qR−1.
87
6 Mean behaviour of uniformly summable Q-multiplicative functions
The first product Q1equals
Q1=1
cQR−2Y
1≤r≤R−h
(qr)Ã1 + 1
qr
qr−1
X
a=0 |f(aQr−1)|2−1!
=1
cQR−2X
n<(c+1)QR−2|fR−h(n)|2.
Since fR−h∈L2, such that the product Q1is bounded.
The equivalent of condition (6.1) is
max
1≤j<qr
1
j+ 1
j
X
a=0
|f(aQr−1)|>1
|f(aQr−1)|2−1 = O(1),
for r→ ∞, therefor the products Q2and Q3are bounded.
Thus, 1
NX
n<N
(˜
f(n))2≤4K˜ch−1is bounded, where Kis constant only depends on fR−h,
and ˜cis constant only depends on ˜
f.
Ad P2:
For all a≥0,0≤n < QR−h,
fR−h(aQR−h+n) = f(n)
holds, and for 1≤l≤qR−h, applying Lemma 6.2.2, we have
lQR−h−1
X
n=0
fR−h(n) =
l−1
X
a=0
QR−h−1
X
n0=0
fR−h(aQR−h+n0)
=
l−1
X
a=0
QR−h−1
X
n0=0
f(n0)
=l
QR−h−1
X
n0=0
f(n0)
=lQR−hΠR−h.
Further, for N=lQR−h, we obtain
1
NX
n<N
fR−h(n)−ΠR−h= 0,
88
6.3 Proof of main results
and for lQR−h< N < (l+ 1)QR−h, we conclude
¯¯¯¯¯X
n<N
fR−h(n)−NΠR−h¯¯¯¯¯
=¯¯¯¯¯¯−(N−lQR−h)ΠR−h+
N−1
X
n=lQR−h
fR−h(n)¯¯¯¯¯¯
=¯¯¯¯¯¯−(N−lQR−h)ΠR−h+fR−h(lQR−h+1)
N−lQR−h−1
X
n=0
f(n)¯¯¯¯¯¯
≤c(N−lQR−h)
< cQR−h,
with some constant conly depends on f.
Ad ∆:
We get
|ΠR−ΠR−h|=|ΠR−h|¯¯¯¯¯ÃR−1
Y
r=R−h
1
qr
qr−1
X
a=0
f(aQr−1)!−1¯¯¯¯¯
≤c
R−1
X
r=R−h¯¯¯¯¯
1
qr
qr−1
X
a=0
(f(aQr−1)−1)¯¯¯¯¯
≤c
R−1
X
r=R−h
√qr−1
qrÃqr−1
X
a=0 |(f(aQr−1)−1)|2!1/2
≤c√hÃR−1
X
r=R−h
qr−1
q2
r
qr−1
X
a=0 |(f(aQr−1)−1)|2!1/2
.
Since condition (6.1) holds, and his fixed, we have |ΠR−ΠR−h|=o(1), as R→ ∞.
Altogether, we obtain
¯¯¯¯¯
1
NX
n<N
f(n)−ΠR¯¯¯¯¯≤˜c1
qR−1qR−2···qR−h+1
+o(1).
89
6 Mean behaviour of uniformly summable Q-multiplicative functions
The proof of assertion (b) is analogous.
Proof of Theorem 6.1.3. By Lemma 6.2.9 and Theorem 6.1.2, the assertion of Theorem
6.1.3 follows.
Proof of Theorem 6.1.4. Follows by Lemma 6.2.9.
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