Uniformly summable multiplicative
functions on additive arithmetical
semigroups
Dissertation
zur Erlangung des Doktorgrades
der Fakult¨at f¨ur Elektrotechnik, Informatik und Mathematik
der Universit¨at Paderborn
vorgelegt von
Anna Bar´at
Betreuer
Prof. Dr. Dr. h.c. mult. Karl-Heinz Indlekofer
2011
i
Abstract
We present the additive arithmetical semigroups and summarize the improvements
on prime number theorems and mean-value theorems on additive arithmetical semi-
groups. We start with definitions and examples, then compare the approaches,
which have been used to prove prime number theorems. Thereafter, we give a short
outline of the convolution theory and generating functions.
Then we proceed with complex-valued multiplicative functions on additive arith-
metical semigroups. First we summarize some results for multiplicative functions of
modulus ≤1, and more generally for uniformly summable multiplicative functions.
Afterwards, we prove new mean-value theorems for uniformly summable multiplica-
tive functions on additive arithmetical semigroups. These theorems are more general
than the previous results because our conditions on the additive arithmetical semi-
groups are weaker and we can prove our mean-value theorems for a larger class of
functions. In the proof we use some tauberian theorems by Indlekofer, and some
ideas of the proof of mean-value theorems for multiplicative functions in the classical
number theory.
Finally, we give an application of our results by proving a characterization of finitely
distributed functions on additive arithmetical functions and the Three-series theo-
rem on additive arithmetical semigroups.
ii
iii
Zusammenfassung
Wir pr¨asentieren die additiven arithmetischen Halbgruppen und fassen die wichtig-
sten Ergebnisse ¨uber Primzahl- und Mittelwerts¨atze auf additiven arithmetischen
Halbgruppen zusammen. Wir beginnen mit Definitionen und Beispielen, danach
vergleichen wir die Ans¨atze, die verwendet wurden, um Primzahls¨atze zu beweisen.
Anschließend geben wir einen kurzen ¨
Uberblick ¨uber Faltung und erzeugende Funk-
tionen.
Dann betrachten wir komplexwertige multiplikative Funktionen auf additiven arith-
metischen Halbgruppen. Zuerst fassen wir einige Ergebnisse f¨ur multiplikative Funk-
tionen vom Betrag ≤1 zusammen, dann allgemeiner f¨ur gleichgradig summierbare
multiplikative Funktionen. Danach beweisen wir neue Mittelwerts¨atze f¨ur gleich-
gradig summierbare multiplikative Funktionen auf additiven arithmetischen Halb-
gruppen. Diese S¨atze sind allgemeiner als die bisherigen, weil unsere Bedingungen an
die additiven arithmetischen Halbgruppen schw¨acher sind und weil wir eine gr¨oßere
Klasse von Funktionen behandeln.
In dem Beweis benutzen wir eine Methode ¨uber Taubers¨atze von Indlekofer, und
einige Ideen des Beweises der Mittelwerts¨atze f¨ur multiplikative Funktionen in der
klassischen Zahlentheorie.
Schließlich geben wir als eine Anwendung unserer Ergebnisse eine Charakterisierung
von endlich verteilten additiven Funktionen auf additiven arithmetischen Halbgrup-
pen und einen Beweis f¨ur den Drei-Reihen-Satz auf additiven arithmetischen Halb-
gruppen.
iv
v
Acknowledgements
I would like to start by thanking my advisor Prof. Dr. Karl-Heinz Indlekofer
for many things: for choosing the interesting subject, for the guidance, for the
continuous support and advice in many aspects.
I would like to thank my colleagues at the University of Paderborn for the friendly
and inspiring environment. Prof. Indlekofer’s group at the University of Paderborn
has always provided me with a very good research environment.
I am also very grateful to Prof. Dr. Karl-Heinz Indlekofer, Prof. Dr. Imre K´atai,
Prof. Dr. Oleg Klesov, Dr. Richard Wagner, Dr. L´aszl´o Germ´an for stimulating
discussions and exciting work done jointly. For fruitful discussions and comments I
also thank Erdener Kaya.
I also want to thank Prof. Dr. Toni Machi’ for his interest in my work and for a
pleasant stay in Rome.
For financial support I am very grateful to the University of Paderborn and the
Deutsche Forschungsgemeinschaft (DFG).
I am most indebted to my parents Dr. ´
Agnes Hajdu Bar´atn´e and J´anos Bar´at who
have supported me not only during my studies but throughout my whole life in every
imaginable way. I want to thank my brother Dr. J´anos Bar´at, who has always been
a great advisor and an inspiration to me, since my childhood.
Furthermore, I would like to thank Gilles Gnokam for being an on-going source of
support and motivation during the time of this research.
This thesis is dedicated to my children.
i
Contents
Notations iii
Introduction 1
1 Basic definitions and facts about additive arithmetical semigroups 4
1.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Special conditions and their consequences . . . . . . . . . . . . . . . . 6
1.3 Convolution................................ 11
1.4 Generatingfunctions ........................... 13
2 Investigation of arithmetical functions 16
2.1 Mean-value theorems for multiplicative functions of modulus ≤1 . . . 17
2.2 New mean-value theorems for uniformly
summable multiplicative functions . . . . . . . . . . . . . . . . . . . . 21
2.3 Characterization of uniformly summable functions . . . . . . . . . . . 24
2.4 Indlekofer’smethod............................ 27
2.5 Lemmata ................................. 29
2.6 Proof of the new mean-value theorems . . . . . . . . . . . . . . . . . 36
3 Applications 56
3.1 Finitely distributed additive functions . . . . . . . . . . . . . . . . . . 56
3.2 Three-series theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Bibliography 60
ii
Notations
We use the following conventions: Nis the set of the natural numbers; and Ndoes
not contain zero, while N∪ {0}will be denoted by N0. The real- and imaginary
parts of a complex number zare written Re z and Im z. We use the the well-known
Landau symbols little-oand big-Oin this thesis, as well as the following notations to
describe the limiting behaviour of a function: f(n)g(n) (f(n)g(n)) as n→ ∞
means, that there exists a positive constant Cand a natural number n0such that
|f(n)| ≤ Cg(n) (C|f(n)| ≥ g(n)) for all n≥n0. Summarizing the two assertions
f(n)g(n) and f(n)g(n) as n→ ∞, we write f(n)g(n) as n→ ∞.
The places where frequently used symbols are defined are indicated in the following
table. Some symbols denote objects that depend on n,z, or a, where nrepresents
an integer, zand yrepresent complex numbers, and arepresents an element of
an additive arithmetical semigroup. To emphasize the difference between functions
defined on complex numbers or on additive arithmetical semigroups we shall in
general use the extra ” ˜ ” sign over an alphabetic character (for example ˜
f) for
functions defined on an additive arithmetical semigroup.
We use the notation pk||afor pkdivides exactly awhen pk|abut pk+1 6 |a.
Numbers in brackets like (1.1) indicate equation numbers, while numbers without
brackets like 1.1 indicate numbers of definitions.
Symbol Definition page
∂(a) 1.1 4
G, G(n) 1.1 4
P1.1 4
P(n) (1.1) 4
ˆ
ZG(z),ˆ
Z(z) (1.2) 5
q(1.3) 5
A6
ˆ
H(z) (1.8) 6
r(n) (1.12) 8
˜
f(a) 11
∗(1.24) 11
1B(a) (1.26) 12
˜µ(a) (1.27) 12
˜ω(a) (1.29) 12
Symbol Definition page
˜
Λ(a) (1.30) 13
ˆ
F(z) (1.32) 13
ˆ
P(z) (1.35) 14
Λ(n) (1.37) 15
Λf(n) (2.4) 17
M(n, ˜
f), M(˜
f) 2.1 17
Lα2.7 21
L∗2.8 22
γ(n) 2.21 27
λ(n) (2.21) 27
F(y) 2.23 27
Z(y) (2.22) 27
iii
Introduction
Results from several different areas of mathematics have found a common gener-
alization in the theory of arithmetical semigroups. An arithmetical semigroup is
a free abelian semigroup generated by a countable set of prime elements with an
associated mapping: a multiplicative norm, that leads to multiplicative arithmetical
semigroups or an additive norm, that defines an additive arithmetical semigroup. In
the case of multiplicative arithmetical semigroups one has to investigate the behav-
ior of Dirichlet series, and for additive arithmetical semigroups we deal with power
series.
Some of the research has focused on specific questions: e.g. polynomials over finite
fields as composed of irreducibles or graphs as composed of connected components.
This research example was the first instance in which many theorems were proven,
and it also had a strong influence on the notation used in arithmetical semigroups.
Others, such as Knopfmacher in his first books [30] and [31], began with given
classical theorems and tried to prove them under the weakest possible conditions
on the number of elements below a given size. It is this approach we follow in this
thesis in the case of additive arithmetical semigroups.
There has been a considerable amount of research on the prime element theorem and
therefore, a need to summarize the latest developments in this field. We focus on
the other important question in the field, the mean-value theorems for multiplicative
functions. The results in this thesis are improvements of the fundus presented by
Knopfmacher and Zhang [32], the only recent book on the subject.
The idea of developing an arithmetical theory based mainly on the foundation of
the axiom referred to here as Axiom A#seems to have first been pointed out in
papers by Fogels [14] and [15]. However, in these papers, Fogels dealt only with
some very special consequences of Axiom A#, and referred only to polynomial rings
and algebraic function fields over finite fields in order to motivate the axiom. These
papers were followed by books written by J. Knopfmacher [30] and [31].
The zeta function of an additive arithmetical semigroup is the associated power
series, where the nth coefficient gives the total number of elements in the additive
arithmetical semigroup with degree n, which is by definition always finite. In the first
investigations on additive arithmetical semigroups, the zeta function was regarded
as having a pole on the boundary of its circle of convergence. Such an assumption is
1
INTRODUCTION
Axiom A#(see [30]), which is a requirement on the coefficients of the zeta function.
Knopfmacher stated that the zeta function has no zeros on its circle of convergence
(see [31]), but Indlekofer, Manstaviˇcius and Warlimont ([26]) have given examples of
additive arithmetical semigroups satisfying Axiom A#with a zero on the boundary
of the circle of convergence. This is an improvement of the prime number theorem
on additive arithmetical semigroups. A major difference from the classical number
theory is that the zeta function in the classical number theory has no zeros on the
boundary; in the case of an additive arithmetical semigroup there can be one zero.
Further, if there is a zero on the boundary then there are no further zeros in the
circle of convergence. The third difference is that in contrast to the natural numbers
on additive arithmetical semigroups, the Chebyshev lower estimate does not hold in
general.
The argument has been developed in two different ways: some mathematicians, such
as Knopfmacher and Zhang, have used requirements on the coefficients of the zeta
function, where others, such as Indlekofer, have used the boundary behaviour of the
zeta function and Axiom ¯
A#introduced by Indlekofer [20]. If Axiom A#or Axiom
¯
A#holds, then the zeta function is meromorphic on its circle of convergence with a
simple pole.
Following this, the investigations have continued with the restriction that the circle
of convergence is a natural boundary for the zeta function. Several mathemati-
cians have made improvements in this case, particularly Indlekofer, Knopfmacher,
Manstaviˇcius, Warlimont and Zhang.
Another important area of investigation is complex-valued multiplicative functions
on an additive arithmetical semigroup and mean-value theorems.
First, mean-value theorems for multiplicative functions on additive arithmetical
semigroups were considered with functions of modulus ≤1. Several mathemati-
cians, such as Bar´at, Indlekofer, Manstaviˇcius, Warlimont, Wehmeier and Zhang,
have contributed to this field. In this thesis we consider a larger class of functions,
uniformly summable multiplicative functions, instead of the restriction on the mod-
ulus of the function. On the other hand, our condition on the additive arithmetical
semigroup is more general.
The class of uniformly summable functions has been defined by Indlekofer (see [17])
for functions defined on Nand correspond to integrable functions with respect to
the asymptotic density.
In this thesis, we proceed as follows: Chapter 1 presents, just for reference, some
well-known facts about the subject. We summarize the basic definitions and the
most important conditions on additive arithmetical semigroups. The remaining
part deals with multiplicative functions.
Chapter 2 contains the latest mean-value theorems for multiplicative functions on
additive arithmetical semigroups for multiplicative functions of modulus ≤1. Then,
2
we formulate our new mean-value theorems for unifomly summable multiplicative
functions with nonzero mean-value. The necessary assumptions on the semigroup
are weaker than the conditions of the previous results. We proceed to give a char-
acterization of uniformly summable functions on additive arithmetical semigroups.
For the proof of our theorems we introduce the new method developed by Indlekofer
that compares the coefficients of power series. Later, we prove some lemmas and
give the proof of our theorems.
In Chapter 3 we give an application of our results by proving a characterization of
finitely distributed additive functions defined on an additive arithmetical semigroup.
Finally, we apply our results to prove the well-known Three-series theorem for ad-
ditive arithmetical semigroups, and give an outline of our work with Indlekofer and
Kaya on the more general Two-series theorem (see [3]) which was motivated among
others by our previous work with Indlekofer and Wagner on Stone-Cech compacti-
fications (see [4]).
3
Chapter 1
Basic definitions and facts about
additive arithmetical semigroups
In this chapter, we introduce the additive arithmetical semigroups, and some func-
tions related to them, such as the zeta function; and collate the most used conditions
on additive arithmetical semigroups with the historical results on prime number the-
orems. We briefly expound the convolution on complex-valued functions on additive
arithmetical semigroups and define the generating functions.
1.1 Definition and examples
Definition 1.1. We call (G, ∂) an additive arithmetical semigroup, if Gis a com-
mutative semigroup with identity element 1G, generated by a countable set Pof
primes and ∂is an integer valued degree mapping ∂:G→N0, which satisfies
(i) ∂(ab) = ∂(a) + ∂(b) for all a, b ∈G,
(ii) the total number G(n) of elements of degree nin Gis finite for each n≥0.
Therefore, ∂(1G) = 0, ∂(p)>0 for all p∈P,G(0) = 1 and Gis countable.
In this work (unless otherwise stated), Gdenotes an additive arithmetical semigroup
related to an integer valued mapping ∂.
We write
(1.1) P(n) := #{p∈P:∂(p) = n}.
In using these notations, we shall be particularly concerned with arithmetical con-
sequences of assumptions on the total number G(n) of elements of degree nin G, or
on the total number P(n) of primes of degree nin G.
J. Knopfmacher gave numerous natural examples (see [31]). We give here the most
natural one.
4
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
Example 1.2. Galois polynomial rings. Let Fq[X] denote a polynomial ring
in an indeterminate Xover the finite Galois field Fqwith qelements (qa prime
power). The subset Gq=G(q, X) consisting of all monic polynomials in Fq[X]
forms a semigroup under multiplication. In particular, Gqtogether with the usual
degree mapping on polynomials forms an additive arithmetical semigroup such that
Gq(n) = qn(n= 0,1,2, ...).
Example 1.3. Let (G, ∂) be an additive arithmetical semigroup such that there
exists exactly one prime element of degree nfor each n∈N, i.e. P(n) = 1 for all
n∈N. In this case G(n) is the number of ways to partition the number ninto a
sum of positive integers, so G(1) = 1, G(2) = 2, G(3) = 3, G(4) = 5, etc.
To investigate additive arithmetical semigroups, it is essential to work with the
(generating) power series
ˆ
ZG(z) := ˆ
Z(z) =
∞
X
n=0
G(n)zn
(1.2)
=
∞
Y
n=1
(1 −zn)−P(n)
= exp
∞
X
n=1
1
nX
d|n
dP(d)zn
which we call the zeta function ˆ
ZGof G. In the case of the Galois polynomial rings
Gq=G(q, X) (qa prime power), we obtain
ˆ
ZGq(z) =
∞
X
n=0
Gq(n)zn=
∞
X
n=0
qnzn=1
1−qz
that is convergent for |z|< q−1. For the additive arithmetical semigroup described
in Example 1.3, we obtain the fundamental identity (|z|<1)
ˆ
Z(z) =
∞
X
n=0
G(n)zn=
∞
Y
n=1
(1 −zn)−1
the famous identity studied by Hardy and Ramanujan. Using their circle method in
complex analysis, they found an asymptotic expression for G(n), namely
G(n)∼eπ√2
3n
4n√3,
as well as efficient methods to actually compute G(n).
Here we restrict ourselves to additive arithmetical semigroups satisfying
(1.3) G(n)qnn%
with some q > 1 and %∈Rand generating zeta functions the circle of convergence
of which is equal to |z|< q−1with radius q−1.
5
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
1.2 Special conditions and their consequences
To get more precise results on additive arithmetical semigroups we need an assump-
tion stronger than (1.3). Therefore Knopfmacher introduced the following condition.
Axiom A#(see J. Knopfmacher [30]). There exist constants A > 0, q > 1 and ν
with 0 ≤ν < 1 (all depending on G), such that
(1.4) G(n) = Aqn+O(qνn)as n→ ∞.
If Axiom A#holds, we get
(1.5) ˆ
Z(z) = A
1−qz +ˆ
H1(z),
where
(1.6) ˆ
H1(z) =
∞
X
n=0
cnzn
with
(1.7) cn=G(n)−Aqn=O(qνn).
Thus ˆ
H1is holomorphic for |z|< q−ν.Put
(1.8) ˆ
H(z) := A+ (1 −qz)ˆ
H1(z).
Then the zeta function can be written as
(1.9) ˆ
Z(z) = ˆ
H(z)
1−qz .
Obviously ˆ
H(0) = 1 and ˆ
H(q−1) = A, furthermore ˆ
His holomorphic for |z|< q−ν.
Whereas Axiom A#gives conditions for the power series coefficients cnof ˆ
H1,
Indlekofer formulated in [20] an assumption on the boundary behaviour of ˆ
H. This
reads as
Axiom ¯
A#(see Indlekofer [20]). There exist constants q > 1 and νwith 0 ≤ν < 1
(all depending on G) such that
(i) the function ˆ
Hdefined by (1.8) is holomorphic in the open disc |z|< q−ν, and
ˆ
H(q−1)>0,
6
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
(ii) the function ¯
Hdefined by ¯
H(y) := ˆ
H(q−νy) is an element of the Nevanlinna
class N.
Remark 1.4. If Axiom A#or Axiom ¯
A#holds, then the zeta function ˆ
Z(z) is
meromorphic for |z|< q−νwith a simple pole at z=q−1with residue A. Knopf-
macher stated that ˆ
Z(z)6= 0 for |z| ≤ q−1(see [31]), but the proof was not
correct for z=−q−1, since Indlekofer, Manstaviˇcius and Warlimont ([26]) have
given examples of additive arithmetical semigroups Gsatisfying Axiom A#with
ν= 1/2 and ˆ
Z(−q−1) = 0. On the other hand, these authors have also shown in
[26] that, if Gsatisfies Axiom A#with ν < 1/2, then ˆ
Z(−q−1)6= 0. Indlekofer,
Manstaviˇcius and Warlimont also proved that if ˆ
Z(−q−1) = 0, then ˆ
Z(z)6= 0 for
all |z|< q−ν, z 6=−q−1. As a result the authors showed the following prime number
theorem:
Let 0 < ε < 1−ν, if ˆ
Z(−q−1)6= 0, then there exist l=l(ε)∈N0,0< ε0< ε and
complex numbers βi(i= 1, . . . , l) -the zeros of ˆ
Z(z) in the disc |z| ≤ q−ν−ε- such
that:
ˆ
Λ(z) = z
q−1−z−z
l
X
i=1
1
βi−z+zR(z)
where q−1<min
i=1,...,l |βi| ≤ max
i=1,...,l |βi| ≤ q−ν−εand R(z) is holomorphic for |z| ≤ q−ν−ε0,
furthermore
(1.10) P(n) = qn
n−1
n
l
X
i=1
β−n
i+Oεqn(ν+ε0)
n.
as n→ ∞. If ˆ
Z(−q−1) = 0, then
(1.11) P(n) = 1+(−1)n+1qn
n+Oεqn(ν+ε)
n
for all 0 <ε<1−νas n→ ∞.
Axiom ¯
A#provides a better remainder term in the prime number theorem (see [20],
Theorem 2, Theorem 5 and Corollary 3). For instance, if ˆ
Z(−q−1) = 0, then Axiom
A#yields (1.11) for some (ε > 0), whereas Axiom ¯
A#implies
P(n) = (1 −(−1)n)qn
n+ 2anqnν
with
an=Zπ
−π
e−intdm(t),
where mis the (real) measure occuring in the factorization of the Nevanlinna func-
tion ¯
H(cf. [20], Proposition). Furthermore, Axiom ¯
A#follows from this asymptotic
formula for P(n) (cf. [20], Corollary 3).
7
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
Subsequently, the investigations continued with the restriction that |z|=q−1is a
natural boundary for the zeta function. Zhang required assumptions on the coeffi-
cients of the zeta function and introduced
Axiom A(see Zhang [32]). There exist constants A > 0, 1 < q < ∞and a
real-valued function rsuch that
(1.12) G(n) = (A+r(n))qn
with ∞
X
n=0
sup
m≥n|r(m)|<∞.
Zhang and Warlimont proved the following prime number theorem (see [32], The-
orem 5.4.1): if
(1.13) G(n) = Aqn+Oqnn−γ
with A > 0, q > 1,and γ > 2, then either
(1.14) P(n) = qn
n+Oqnn−γ
or
(1.15) P(n) = (1 −(−1)n)qn
n+O(qnn−γ+1).
Indlekofer assumed a boundary behaviour on ˆ
Husing the notation of (1.8) (see
[20]). This reads as
Axiom A1(see Indlekofer [21]). There exists a constant q > 1 (depending on G),
such that
(i) the function ˆ
His holomorphic in the disc |z|< q−1, and continuous on |z| ≤
q−1with A:= ˆ
H(q−1)>0,
(ii) the derivative ˆ
H0of ˆ
His bounded on |z|< q−1.
Indlekofer showed if Axiom A1holds and if ˆ
Z(−q−1)6= 0, then
(1.16) P(n) = qn
n+oqn
n
and if ˆ
Z(−q−1) = 0, then
(1.17) nP(n)
qn+(n−1)P(n−1)
qn−1= 2 + o(1) .
8
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
Furthermore, the asymptotic formula
(1.18) X
m≤n
mP(m)q−m=n+o(n1/2), n → ∞
holds for the mean of the number of prime elements P(n).
A small change of Axiom A1leads to the abstract prime number theorem and to the
asymptotic formula (1.18) with remainder term o(1). This modification is contained
in
Axiom A2(see Indlekofer [21]). The conditions of Axiom A1hold, and in addition,
the power series of ˆ
H0converges absolutely for |z| ≤ q−1.
Axiom A2yields the following prime number theorem (see [22], Theorem 1):
(1.19) P(n) = qn
n+Oqnmax
n
4≤m≤n|h(m)|q−mas n→ ∞
if ˆ
Z(−q−1)6= 0, and
(1.20) P(n) = (1 −(−1)n)qn
n+O
qnX
n
8≤m|h(m)|q−m
as n→ ∞
if ˆ
Z(−q−1) = 0.
It may be observed, that if
∞
X
n=1
nk|h(n)|q−n<∞
then
max
n
4≤m≤n|h(m)|q−m=o(n−k)
and X
n≤m|h(m)|q−m=o(n−k).
Remark 1.5. If the condition (1.13) holds with γ > 2, then (1.19) and (1.20) imply
(1.14) and (1.15), respectively.
For the investigation of the mean-value of a multiplicative function it is sufficient to
assume conditions weaker than the above assumptions, which were used to prove a
prime number theorem on additive arithmetical semigroups.
9
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
There are several approaches to weaken the assumption on the additive arithmetical
semigroup. Instead of (ii) in Axiom A1Bar´at and Indlekofer assumed
(1.21) X
n≤N
X
d|n
dP(d)qn
2
=O(N),as N→ ∞.
This estimate is an assumption about the mean behaviour of the prime coefficients
P(n), which does not yield the Chebyshev upper estimate
(1.22) P(n) = Oqn
n.
It is natural to ask for the Chebyshev lower estimate:
qn
nP(n).
The asymptotic (1.11) or (1.20) show that the prime number theorem does not yield
the Chebyshev lower estimate. This indicates a major divergence of the theory of
additive arithmetical semigroups from the classical number theory. The question
arises: what can we say about an additive arithmetical semigroup if the Chebyshev
lower estimate is satisfied? Indlekofer investigated this case and considered additive
arithmetical semigroups with the following condition (see [23], Example 4)
(1.23) 0 < c1≤nP(n)q−n≤X
d|n
dP(d)q−n≤c2<∞(n∈N).
Beside (1.23) there are no further assumptions about the additive arithmetical semi-
group; nevertheless, we can prove an estimate for G(n). The zeta function can be
written as (see (1.2))
ˆ
Z(z) = exp
∞
X
n=1
1
n(X
d|n
dP(d))zn
.
Therefore
|ˆ
Z(z)|=ˆ
Z(|z|) exp
∞
X
n=1
1
n(X
d|n
dP(d))qn|z|n(cos(nt)−1)
≤ˆ
Z(|z|) exp c1
∞
X
n=1
|z|n
nqn(cos(nt)−1)!
=ˆ
Z(|z|)
1−|qz|
1−qz
c1
.
10
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
In addition, elementary estimates immediately yield (cf. [23])
G(n)qn
nexp
X
m≤nX
d|m
dP(d)q−m
where the constants involved in only depend on c1and c2.
1.3 Convolution
In this section, we begin the study of general arithmetical functions, complex-valued
functions on a given additive arithmetical semigroup G.
A complex-valued function ˜
f(a) defined for all a∈Gis an arithmetical function
on G. The set of all arithmetical functions on Gwill be denoted by Dir(G). This
set can be made into a complex vector space (of infinite dimension) by means of the
point-wise operations
(˜
f+ ˜g)(a) = ˜
f(a) + ˜g(a),
(λ˜
f)(a) = λ˜
f(a)
for ˜
f, ˜g∈Dir(G), a∈Gand λ∈C. Further, this vector space becomes an associa-
tive algebra, which is the Dirichlet-algebra of G, under the convolution operation ∗
defined by
(1.24) ( ˜
f∗˜g)(a) = X
d|a
˜
f(d)˜g(a/d)
for ˜
f, ˜g∈Dir(G) and a∈G. Unless otherwise stated ˜
fdenotes an arithmetical
function on a given additive arithmetical semigroup G. It is easy to see that the
convolution is commutative and associative. Also, the convolution and addition are
distributive in the sense that
(˜
f+ ˜g)∗˜
h= ( ˜
f∗˜
h) + (˜g∗˜
h).
Moreover, for λ∈C,
λ(˜
f∗˜g) = (λ˜
f)∗˜g=˜
f∗(λ˜g).
The function
(1.25) ˜(a) = 1 for a= 1G
0 otherwise
is the neutral element for the convolution, that is
˜∗˜
f=˜
f∗=˜
f
11
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
for the complex-valued function ˜
fon G. Therefore, the arithmetical functions un-
der the addition, scalar multiplication, and convolution form a commutative algebra.
Although we shall make no use of the fact, it may be interesting to point out that
Dir(G) is actually a unique factorization domain (see [29]).
If there exists an arithmetical function ˜
f−1on Gsuch that ˜
f∗˜
f−1=˜
f−1∗˜
f= ˜,
then ˜
fis invertible and ˜
f−1is the convolution inverse of ˜
f. The equation ˜
f∗˜
f−1= ˜
is equivalent to the system of equations
1 = ˜
f(1G)˜
f−1(1G),
0 = X
d|a
˜
f(d)˜
f−1(a/d),(a6= 1G).
Hence ˜
f(1G)6= 0 is necessary for the existence of an inverse ˜
f−1. This condition is
also sufficient. Actually, if ˜
f(1G)6= 0, then the first equation of the above system
gives the value ˜
f−1(1G). First, we obtain ˜
f−1(p) = 1/˜
f(p) for all p∈P, and there-
after we get the values for ˜
f−1(a) for all a∈G.
Therefore, an arithmetical function ˜
fon Gis invertible if and only if ˜
f(1G)6= 0.
If Bis a subset of G, then the characteristic function of Bis given by
(1.26) 1B(a) = 1 for a∈B
0 otherwise
and the inverse of 1Bexists if and only if 1G∈B. In particular, 1Gis invertible and
this leads to
(1.27) ˜µ∗1G= ˜
where ˜µis the M¨obius function on G. This relation is sometimes also called the
M¨obius inversion formula on G.
As for the natural numbers, the M¨obius function on Gsatisfies
(1.28) ˜µ(a) = 0 if p2|afor some p∈P
(−1)˜ω(a)otherwise
hereby ˜ω(a) is
(1.29) ˜ω(a) := X
p|a
1
the prime divisor function on G.
12
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
Besides the M¨obius function on Gthe von Mangoldt’s function ˜
Λ plays an important
role in the investigations of additive arithmetical semigroups. We define ˜
Λ by
(1.30) ˜
Λ(a) = ∂(p),if ais a prime power pr6= 1G,
0,otherwise.
Let a∈Ghas the prime factorization a=pα1
1pα2
2. . . pαk
k,(k∈N). Then the following
holds
X
d|a
˜
Λ(d) =
k
X
i=1 X
d|pαi
i
˜
Λ(d)
=
k
X
i=1
∂(pi)αi=
k
X
i=1
∂(pαi
i)
=∂(a).
Therefore the von Mangoldt’s function ˜
Λ satisfies
(1.31) ˜
Λ = ˜µ∗∂.
1.4 Generating functions
For each function ˜
f, we associate a power series ˆ
F, the generating function ˆ
Fof ˜
f.
We define ˆ
Fby
(1.32) ˆ
F(z) = X
a∈G
˜
f(a)z∂(a)=
∞
X
n=0
X
a∈G
∂(a)=n
˜
f(a)
zn
for z∈C.
In particular, the generating function of 1G, is just the zeta function ˆ
ZGof G, since
(1.33) ˆ
ZG(z) := ˆ
Z(z) =
∞
X
n=0
X
a∈G
∂(a)=n
1G(a)
zn=
∞
X
n=0
G(n)zn.
For the special functions ˜,∂and 1Pthat we defined in the previous sections, we
can give the associate generating functions as follows:
ˆ
E(z) :=
∞
X
n=0
X
a∈G
∂(a)=n
˜(a)
zn= 1,
13
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
(1.34) ˆ
D(z) :=
∞
X
n=0
X
a∈G
∂(a)=n
∂(a)
zn=
∞
X
n=0
nG(n)zn=zˆ
Z0(z)
and the prime generating function of Gis
(1.35) ˆ
P(z) :=
∞
X
n=0
X
a∈G
∂(a)=n
1P(a)
zn=
∞
X
n=0
P(n)zn.
Since we assume G(n)qnn%with some real constants δand q,q > 1, the functions
ˆ
Z, ˆ
E, ˆ
Dand ˆ
Pare holomorphic for |z|< q−1.
The generating function of the convolution of the functions ˜
fand ˜g
∞
X
n=0
X
a∈G
∂(a)=n
(˜
f∗˜g)(a)
zn=
∞
X
n=0 X
a∈G
∂(a)=n
X
d|a
˜
f(d)˜g(a/d)
zn
=
∞
X
n=0
X
b,d∈G,
∂(b)+∂(d)=n
˜
f(d)˜g(b)
zn
=
∞
X
n=0
n
X
k=0 X
d∈G
∂(d)=k
˜
f(d)X
b∈G
∂(b)=n−k
˜g(b)
zn
=
∞
X
n=0
X
a∈G
∂(a)=n
˜
f(a)
zn
∞
X
n=0
X
a∈G
∂(a)=n
˜g(a)
zn
is the pointwise product of their generating functions. If the inverse function ˜
f−1of
˜
fexists then its generating function satisfies
ˆ
F(z)ˆ
F−1(z)=1,
which implies
ˆ
F−1(z) = 1
ˆ
F(z).
14
CHAPTER 1. BASIC DEFINITIONS AND FACTS ABOUT ADDITIVE
ARITHMETICAL SEMIGROUPS
For instance, the generating function ˆ
Mof the M¨obius function of Ghas the follow-
ing form
(1.36) ˆ
M(z) =
∞
X
n=0
X
a∈G
∂(a)=n
˜µ(a)
zn=1
ˆ
Z(z).
For the von Mangoldt’s function ˜
Λ = ˜µ∗∂, we obtain
ˆ
Λ(z) :=
∞
X
n=0
Λ(n)zn=
∞
X
n=0
X
a∈G
∂(a)=n
˜
Λ(a)
zn
(1.37)
=
∞
X
n=0
X
a∈G
∂(a)=n
(µG∗∂)(a)
zn
=zˆ
Z0(z)
ˆ
Z(z),
where the coefficients Λ(n) of the power series ˆ
Λ(z) are the von Mangoldt’s coeffi-
cients.
We can write the zeta function of an additive arithmetical semigroup in several
forms. We will summarize the most important ones, as we shall be making use
of these representations in our proofs. The zeta function has an Euler product
representation (cf. (1.2))
(1.38) ˆ
Z(z) =
∞
Y
n=1
(1 −zn)−P(n),
in the disc {z:|z|< q−1}, using (1.2) and (1.37) that can be written in the form
(1.39) ˆ
Z(z) = exp
∞
X
n=1
1
nX
d|n
dP(d)zn
= exp ∞
X
n=1
Λ(n)
nzn!.
In particular, ˆ
Z(z)6= 0 for |z|<1/q.
Using (1.39) we obtain that the coefficients Λ(n) and P(n) are related by
Λ(n) = X
d|n
dP(d)
and by the M¨obius inversion formula on the positive integers N
(1.40) nP(n) = X
d|n
Λ(d)µn
d.
15
Chapter 2
Investigation of arithmetical
functions
This chapter contains some basic definitions followed by the latest mean-value theo-
rems for multiplicative functions on additive arithmetical semigroups. We begin by
summarizing mean-value theorems for multiplicative functions of modulus ≤1, and
proceed to introduce our new mean-value theorems for unifomly summable mul-
tiplicative functions with nonzero mean-value. The necessary assumptions in our
mean-value theorems on the semigroup are weaker than the conditions of the previ-
ous results. Afterwards, we give a characterization of uniformly summable functions
on additive arithmetical semigroups. For the proof of our theorems we introduce
the new method developed by Indlekofer, that compares the coefficients of power
series. Later, we prove some lemmas and give the proof of our theorems.
Let us start with some basic definitions in this topic. An arithmetical function ˜
fon
Gis additive if
(2.1) ˜
f(ab) = ˜
f(a) + ˜
f(b) for all coprime a, b ∈G.
An arithmetical function ˜
fon Gis multiplicative if
(2.2) ˜
f(ab) = ˜
f(a)˜
f(b) for all coprime a, b ∈G.
An arithmetical function ˜
fon Gis completely multiplicative if
(2.3) ˜
f(ab) = ˜
f(a)˜
f(b) for all a, b ∈G.
If ˜
fis a multiplicative function on G, then P
a∈G
∂(a)=0
˜
f(a) = 1 (6= 0), therefore its
16
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
generating function ˆ
Fsatisfies
ˆ
F(z) =
∞
X
n=0
X
a∈G
∂(a)=n
˜
f(a)
zn
(2.4)
=Y
p 1 +
∞
X
k=1
˜
f(pk)zk∂(p)!
=: exp ∞
X
m=1
Λf(m)
mzm!.
In this thesis we investigate the mean-value of an arithmetical function on G.
Definition 2.1. Let ˜
f:G7→ C. We define the average value of ˜
fon elements of
degree nby
M(n, ˜
f) :=
1
G(n)P
a∈G
∂(a)=n
˜
f(a),if G(n)6= 0,
0,if G(n) = 0.
If lim
n→∞ M(n, ˜
f) exists, it is called the mean-value of ˜
fand is denoted by M(˜
f).
The mean-value M(1S) of a characteristic function 1Bof a set B⊆Gis called the
asymptotic density of S.
2.1 Mean-value theorems for multiplicative func-
tions of modulus ≤1
The first mean-value theorems for multiplicative functions on additive arithmetical
semigroups considered functions ˜
fof modulus ≤1 (i.e. with |˜
f| ≤ 1). Several
mathematicians, such as Bar´at, Indlekofer, Manstaviˇcius, Warlimont, Wehmeier
and Zhang, made contributions to this subject. In the results of Indlekofer and
Manstaviˇcius (see [24] and [25]) the authors required Axiom A#and proved ana-
logues of the results of Delange, Wirsing and Hal´asz on N, which describe the mean-
value of a multiplicative function of modulus ≤1 (cf. chapter 6 in [9]).
In this section, we compare two typical mean-value theorems for multiplicative func-
tions of modulus ≤1 on additive arithmetical semigroups, where the circle of con-
vergence of the zeta function is a natural boundary. The first result is a theorem by
Zhang and the second theorem was proven by Bar´at and Indlekofer. These results
are typical in the sense that Zhang made assumption on the coefficients G(n) of the
zeta function, whereas Bar´at and Indlekofer used analytical conditions on the zeta
function.
Zhang proved the following result for additive arithmetical semigroups satisfying
Axiom A(see [32], Theorem 6.3.1).
17
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Proposition 2.2. Suppose that either
(i) Gis an additive arithmetical semigroup satisfying the Chebyshev upper esti-
mate and ∞
X
n=1 |G(n)q−n−A|<∞
or
(ii) ∞
X
n=1
sup
n≤m|G(m)q−m−A|<∞,
holds.
Let ˜
fbe a multiplicative function with |˜
f(a)| ≤ 1for all a∈G. If there exists a real
number τ0such that the series
(2.5) X
p∈P
q−∂(p)1−Re(˜
f(p)e−iτ∂(p))
converges for τ=τ0, then
M(n, ˜
f) = Aqn(1+iτ0)Y
∂(p)≤n1−q−∂(p) 1 +
∞
X
k=1
q−k∂(p)(1+iτ0)˜
f(pk)!+o(1)
as n→ ∞. On the other hand, if there exists no such τ, then
M(n, ˜
f) = o(1).
Remark 2.3. We note, from Theorem 3.2.1 in [32], that an additive arithmetical
semigroup satisfying condition (ii) of Proposition 2.2 satisfies the Chebyshev upper
estimate.
Bar´at and Indlekofer formulated conditions on ˆ
Hwhich leads to a proof of the results
in [24] (see Theorem 2 in [2]). These conditions essentially imply the estimate
(2.6) X
n≤NΛ(n)q−n2=O(N) as N→ ∞
which is weaker than the Chebyshev upper estimate Λ(n)q−n=O(1) as n→ ∞.
Putting z=q−1yin (1.8) we define Z(y) := ˆ
Z(q−1y) and H(y) := ˆ
H(q−1y) and
obtain
(2.7) Z(y) = H(y)
1−yfor |y|<1,
18
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
and assume that H(y) is bounded in the disc |y|<1 satisfying
(2.8) lim
y→1−H(y) = A > 0.
Put H(y) =
∞
X
n=0
h(n)yn. Then the following holds (see [2], Theorem 1).
Theorem 2.4. Let H(y)be continuous for |y| ≤ 1and satisfy (2.8). If
(2.9)
∞
X
n=1
n2h2(n)r2n=O1
1−ras 0 < r < 1, r →1
then (2.6) holds.
The main result of Bar´at and Indlekofer ([2], Theorem 2) is the following mean-value
theorem
Theorem 2.5. Let Gbe an additive arithmetical semigroup satisfying (2.6) and let
H∈H∞(i.e. His bounded in |y|<1) satisfy (2.8). Further, let ˜
fbe a completely
multiplicative function, |˜
f| ≤ 1. Then the following two assertions hold.
(i) If the series (2.5) diverges for each τ∈(−π, π], then
M(n, ˜
f) = o(1)
as n→ ∞.
(ii) If the series (2.5) converges for some τ=τ0∈(−π, π], then
M(n, ˜
f) =Aqn(1+iτ0)Y
∂(p)≤n1−q−∂(p) 1 +
∞
X
k=1
q−k∂(p)(1+iτ0)˜
f(pk)!+o(1)
=cL(n) + o(1)
as n→ ∞, where cis an appropriate real constant, and L(y)is a slowly
oscillating function.
Theorem 2.5 supersedes Proposition 2.2, the corresponding result of Zhang. His
assumption
(2.10)
∞
X
n=1 G(n)q−n−A<∞
implies, since h(n) = G(n)q−n−G(n−1)q−n+1 the absolute convergence
∞
X
n=0 |h(n)|<∞,
19
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
and thus His continuous on the closed disc ¯
D={y:|y| ≤ 1}. Similarly, the
condition (ii) of Proposition 2.2
∞
X
n=0
sup
n≤m|G(m)q−m−A|<∞
yields
(2.11)
∞
X
n=0 |h(n)|<∞
and h(n) = o(n−1). This can easily be seen from (1.8) in section 1.2. From our
Remark 1.2 we obtain also, that (ii) of Proposition 2.2 yields the Chebyshev upper
estimate.
In Theorem 2.5, we have assumed only H∈H∞and (2.6), that follow from the
condition (2.11) and the Chebyshev upper estimate, but conversely H∈H∞and
(2.6) do not yield (2.11) and the Chebyshev upper estimate in general.
In [2], Bar´at and Indlekofer also gave an example (Example 1), which satisfies the
assumption (2.6) but does not satisfy the Chebyshev upper estimate Λ(N)qN.
The mean-value theorems so far assumed that the zeta function has the form ˆ
Z(z) =
ˆ
H(z)(1 −qz)−δwith δ≥1 (see for example [44]). In this thesis we deal with a more
general case:
(2.12) ˆ
Z(z) = ˆ
H(z)(1 −qz)−δ
with δ > 0. We use a method different from the one that have been used to prove
the results which we have described in this section.
In his recent work Indlekofer also formulated a general mean-value theorem for
multiplicative functions of modulus ≤1 on additive arithmetical semigroups under
the above mentioned general condition (2.12) (see Theorem 4, [23]).
Proposition 2.6. Let (G, ∂)be an additive arithmetical semigroup such that
ˆ
Z(z) =
∞
X
n=0
G(n)zn= exp ∞
X
m=1
¯
Λ(m)
mzm!=ˆ
H(z)
(1 −qz)δ
where ˆ
H(z) = O(1) for |z|< q−1,limz→1
q
−ˆ
H(z) = A > 0and δ > 0. Assume that
¯
Λ(m) = O(qm)and G(n)qnnδ−1. Suppose |˜
f(g)| ≤ 1for all g∈Gand either
(i) ˜
fis a completely multiplicative function on G, or
(ii) ˜
fis a multiplicative function such that ˜
f(pk)=0for each prime power pkwith
∂(p)≤log 2
log q.
20
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
If there exists a real number asuch that the series (2.5) converges for τ=a, then
X
g∈G
∂(g)=n
˜
f(g) = qina Y
∂(p)≤n
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)(1+ia)!G(n) + o(G(n)).
If (2.5) diverges for all τ∈Rthen
X
g∈G
∂(g)=n
˜
f(g) = o(G(n)).
2.2 New mean-value theorems for uniformly
summable multiplicative functions
We begin by introducing some definitions, and so on to present our main theorems
about mean-values of uniformly summmable multiplicative functions on additive
arithmetical semigroups.
The class of uniformly summable multiplicative functions has been defined by In-
dlekofer (see [17]) for functions defined on N. Indlekofer proved mean-value theorems
for the class of functions Lαand for uniformly summable functions.
In the case of additive arithmetical semigroups, the main contributions, to date,
are by Wehmeier (see [39]) and Zhang (see for example [44]). Therefore we restrict
ourselves to comparing their most recent results with our new results. Our new
mean-value theorems are more general than the ones proposed before. We prove our
results for a larger class of functions and under a weaker condition on the additive
arithmetical semigroups.
In the previous results in this field the class of arithmetical functions Lαwas of of
great interest. We introduce this class of functions as follows
Definition 2.7. Let ˜
f:G7→ C. If 1 ≤α < ∞, then ˜
fis said to be in Lαif
||˜
f||α:= (lim sup
n→∞
M(n, |˜
f|α))1/α
is finite.
After the classes Lαfor 1 ≤α < ∞the development continued and motivated by
Indlekofer’s results in the classical number theory (see [17]), the class of uniformly
summable functions became very important for additive arithmetical semigroups.
Wehmeier initiated the investigations in this direction for additive arithmetical semi-
groups. In our thesis we extend the previous results. Therefore we introduce the
following:
21
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Definition 2.8. Let ˜
f:G7→ C. Let K∈R; then we define ˜
fKby
˜
fK(a) = ˜
f(a),if |˜
f(a)| ≥ K,
0,otherwise.
˜
fis called uniformly summable if
lim
K→∞ sup
n≥1
M(n, |˜
fK|)=0.
We denote the set of all uniformly summable functions by L∗.
Remark 2.9. To emphasize the importance of the class L∗we remark, that for
α > 1
Lα$L∗$L1
holds.
Now, we introduce our results. Let ˜
fbe a multiplicative function on G. The idea is
to use Indlekofer’s latest approach which we summarize in section 2.4. His method
differs from the methods used before since he does not prove an asymptotic formula
for P
a∈G
∂(a)=n
˜
f(a)nvia Cauchy’s theorem
nX
a∈G
∂(a)=n
˜
f(a) = 1
2πZ|z|=r<q−1
ˆ
F0(z)
zn+1 dz
but compare P
a∈G,∂(a)=n
˜
f(a)nwith nG(n)
|nX
a∈G
∂(a)=n
˜
f(a)−AnnG(n)|=X
m≤n
Λf(m){X
a∈G
∂(a)=n−m
˜
f(a)−AnG(n−m)}
+AnX
m≤n
(Λf(m)−Λ(m))G(n−m)
(2.13)
X
m≤n|X
a∈G,∂(a)=m
˜
f(a)−AnG(m)|(2.14)
+|An|X
m≤n|Λf(m)−Λ(m)|G(n−m)
This leads via Parseval’s equality to an estimate of the distance between P
a∈G
∂(a)=n
˜
f(a)n
and AnnG(n), a procedure which is also effective for quantitative investigations of
occuring remainder terms.
Motivated by the above mentioned method and the corresponding papers of In-
dlekofer about mean-value theorems for multiplicative functions defined on N(see
Indlekofer [17], [18] and [19]) we formulate our main theorems.
22
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Theorem 2.10. Let (G, ∂)be an additive arithmetical semigroup such that
ˆ
Z(z) =
∞
X
m=0
G(m)zm= exp ∞
X
n=1
Λ(n)
nzn!=ˆ
H(z)
(1 −qz)δ
where ˆ
H(z) = O(1) for |z|< q−1,lim
z→1
q
−
ˆ
H(z) = A > 0and δ > 0. Assume that
Λ(m) = O(qm). Let ˜
fbe a multiplicative function and α≥1. If G(n)qnnδ−1and
˜
f∈L∗∩Lαand if M(˜
f)exists and is nonzero, then the following series
(2.15) X
p∈P
˜
f(p)−1
q∂(p)
(2.16) X
p∈P
|˜
f(p)|≤3/2
|˜
f(p)−1|2
q∂(p)
(2.17) X
p∈P;n≥2
|˜
f(pn)|λ
(q∂(p))n.
(2.18) X
p∈P
|| ˜
f(p)|−1|>1/2
|˜
f(p)|λ
q∂(p)
converge for 1≤λ≤α, and for each prime p
(2.19)
∞
X
n=1
˜
f(pn)
qn∂(p)+ 1 6= 0.
In the converse direction we have two cases: 1 ≤δand 0 < δ < 1. If the above
mentioned series of Theorem 2.10 converge, then we can prove the following:
Theorem 2.11. Let an additive arithmetical semigroup fulfill the conditions of The-
orem 2.10, with G(n)qnnδ−1for 1≤δ. Further we assume that G(n−1)
G(n)=
q−1+o(1) as n→ ∞.
Let ˜
fbe a multiplicative function and α≥1. If the series (2.15)-(2.18) converge,
then M(˜
f)exists, ˜
f∈L∗∩Lαand M(|˜
f|λ)exists for 1≤λ≤α. If in addition
(2.19) holds then M(˜
f)6= 0 and M(|˜
f|λ)6= 0 for 1≤λ≤α.
For 0 < δ < 1 we need a further assumption on our multiplicative function ˜
fin
order to prove the existence of the mean-value of ˜
f.
23
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Theorem 2.12. Let an additive arithmetical semigroup fulfill the conditions of
Theorem 2.10, with G(n)qnnδ−1for 0< δ < 1. Let ˜
fbe a multiplicative function
and α≥1. Further we assume that G(n−1)
G(n)=q−1+o(1) as n→ ∞ and
(2.20) ∀ε > 0 : ∃K > 0 : ∀n∈N:∀S⊆G:
S={a∈G:pk||a, p ∈P;|˜
f(pk)|α> K} ⇒ M(n, 1S|˜
f|α)< ε
holds. If the series (2.15)-(2.17) converge, then M(˜
f)exists, ˜
f∈L∗∩Lαand
M(|˜
f|λ)exists for 1≤λ≤α. If in addition (2.19) holds then M(˜
f)6= 0 and
M(|˜
f|λ)6= 0 for 1≤λ≤α.
The class of uniformly summable functions was defined by Indlekofer (see [17])
for functions defined on N, and he has proved mean-value theorems for uniformly
summable multiplicative functions.
In the case of additive arithmetical semigroups, as mentioned before, it was mainly
Wehmeier (see [39]) and Zhang (see for example [44]) who have made contributions
to this subject before this thesis. Zhang’s most recent results in this topic appeared
in 2008 [44]. In his paper he has presented mean-value theorems for functions of the
class Lα(α > 1) and for the case 1 ≤δapplying the methods which have been used
in the proofs in the classical case also.
Wehmeier’s most recent results in this field appeared in his PhD thesis in 2005
(see [39], chapter 6). Wehmeier has proven his mean-value theorems for uniformly
summable multiplicative functions applying the methods of Indlekofer’s proof for
the natural numbers Nbut only for the case δ= 1 .
In this thesis we prove our mean-value theorems in the case 0 < δ and for L∗∩Lα
(α≥1), which is a larger class of functions using other methods.
2.3 Characterization of uniformly summable func-
tions
In this section we give a characterization of uniformly summable functions on G. We
summarize the equivalent properties of L∗in the following lemma, that corresponds
to Lemma 1 in [17]. For additive arithmetical semigroups Wehmeier also proved the
equivalence of the assertions 1.-3. of the subsequent lemma (see Lemma 6.4 in [39]).
We shall apply in particular the first property of this lemma in our proof.
Lemma 2.13. The following statements are equivalent:
1. ˜
f∈L1and
∀ε > 0 : ∃γ > 0 : ∀n∈N:∀S⊆G: (M(n, 1S)< γ ⇒M(n, 1S|˜
f|)< ε);
2. ˜
f∈L∗;
24
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
3.
∀ε > 0∃α∈R: (||(|˜
f|−α)+||1< ε)
where (|˜
f|−α)+= max((|˜
f|−α),0) is the positive part of |˜
f|−α.
4. There exists a monotonic function ϕ:R7→ Rsuch that
(i) ϕ≥0,
(ii) ϕ(t)/t → ∞ as t→ ∞,
(iii) ϕ◦˜
f∈L1.
Proof. 1. ⇒2.: Let ε > 0. To prove that ˜
f∈L∗, we have to find K0such that
M(n, |˜
fK|)< ε for all K≥K0,n∈N.
Apply the assumption 1. to obtain γsuch that M(n, 1S|˜
f|)< ε if M(n, 1S)< γ.
Choose K0:= 2||˜
f||1/γ. Since ˜
f∈L1we know that M(n, |˜
f|)<2||˜
f||1<∞for all
n∈N. Let
S:= {a∈G;|˜
f(a)| ≥ K},
then ˜
fK=1S˜
f. For K≥K0it yields
M(n, 1S)≤M(n, 1S|˜
f|)/K ≤M(n, |˜
f|)
K0≤γM(n, |˜
f|)
2||˜
f||1
< γ.
Hence M(n, 1S|˜
f|) = M(n, |˜
fK|)< ε for all K≥K0,n∈N.
2. ⇒3.: Let ε > 0. There exists Ksuch that ||˜
fK||1< ε. Since max(|˜
f|−α, 0) <
|˜
f|αit suffices to set α:= K.
3. ⇒4.: There exist real numbers nk% ∞ such that
sup
n≥1
1
G(n)X
∂(a)=n
(|˜
f(a)|−nk)+<2−k.
Define ϕ:R7→ Rby ϕ(t) :=
∞
P
k=1
(m−nk)+if m≤t<m+ 1 where m∈Z. Then
ϕ≥0 and ϕis monotonic. Further, ϕ(m)/m =
∞
P
k=1
(1 −nk/m)+→ ∞ as m→ ∞
25
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
and therefore (ii) of 4. holds. Now,
1
G(n)X
∂(a)=n
ϕ(|˜
f(a)|) = 1
G(n)X
m∈Z
∞
X
k=1
(m−nk)+X
∂(a)=n
m≤| ˜
f(a)|<m+1
1
=1
G(n)
∞
X
k=1 X
m∈Z
(m−nk)+X
∂(a)=n
m≤| ˜
f(a)|<m+1
1
≤1
G(n)
∞
X
k=1 X
∂(a)=n
|˜
f(a)|≥nk
(m−nk)
≤1
G(n)
∞
X
k=1 X
∂(a)=n
(|˜
f(a)|−nk)+
≤1.
Therefore ϕ◦˜
f∈L1, which shows the assertion 4.
4. ⇒3.: Let ε > 0 and put c:= sup
n≥1
1
G(n)P
∂(a)=n
ϕ(|˜
f(a)|). Choosing α > 0 such that
ϕ(t)/t ≥c/ε for all t≥αwe have
1
G(n)X
∂(a)=n
(|˜
f(a)|−α)+≤1
G(n)X
∂(a)=n
|˜
f(a)|>α
|˜
f(a)|
≤ε
c
1
G(n)X
∂(a)=n
|˜
f(a)|>α
ϕ(|˜
f(a)|)
≤ε.
3. ⇒1.: For ε > 0 the assumption 3. yields that there exists a real number α1
such that
||˜
f||1=||(|˜
f|−α1) + α1||1≤ε+ 2α1
since ||˜
f|−α1|(a)≤max((|˜
f|−α1)+(a), α1).Hence ˜
f∈L1.
Apply now the assumption 3. to ε/2. It yields that there are α2∈Rand N∈N
such that
n≥N⇒M(n, (|˜
f|−α2)+)< ε/2.
Choose γso small that G(n)>1/γ implies n≥N, and that γ < ε
2α2.
Let S⊆Gand n∈Nsuch that M(n, 1S)< γ. We have to show that
M(n, 1S|˜
f|)< ε.
26
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
If Gn∩S=∅, then M(n, 1S|˜
f|) = 0. Otherwise it follows that n≥N. The
inequality |˜
f(a)| ≤ (|˜
f|−α2)+(a) + α2holds for all a∈G. Hence
M(n, 1S|˜
f|)≤M(n, (|˜
f|−α2)+) + M(n, 1Sα2)≤ε/2 + α2γ < ε.
This ends the proof of our lemma.
2.4 Indlekofer’s method
In this section we summarize the ideas and main results of [23].
To ease notational difficulties we make a variable transformation and put
(2.21) y=qz, λ(m) = q−mΛ(m) and γ(n) = q−nG(n)
which leads to
(2.22) Z(y) := ˆ
Z(yq−1) =
∞
X
n=0
γ(n)yn= exp ∞
X
m=1
λ(m)
mym!.
Then Z(y) is holomorphic for |y|<1.
The same transformation yields for the generating function ˆ
Fof an arithmetical
function ˜
fthe following
(2.23) F(y) := ˆ
F(yq−1) =
∞
X
n=0
X
a∈G
∂(a)=n
˜
f(a)
q−nyn.
For an arithmetical function ˜
fon Gwe define f:N07→ Cby
(2.24) f(n) := q−nX
a∈G
∂(a)=n
˜
f(a)
and call it the summatory function of ˜
f.
It may be observed, that
M(n, ˜
f) = P
a∈G
∂(a)=n
˜
f(a)
G(n)=f(n)
γ(n).
For example, the investigation of the mean-value of ˜
f=˜
Λ leads to defining the
asymptotic behaviour of Λ(n)
γ(n), which corresponds to the prime number theorem. In
this section we consider functions f:N0→Cwith f(0) = 1 and γ(n)≥0 for n∈N
and γ(0) = 1.
27
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
The transformed generating function Fof an arithmetical function ˜
fwith summa-
tory function fsatisfying f(0) = 1 can be written as
(2.25) F(y) :=
∞
X
n=0
f(n)yn= exp ∞
X
m=1
λf(m)
mym!.
The basic conditions in this section will be
(2.26) 0 ≤λ(m) = O(1) (m∈N)
and
(2.27) |Z(y)| Z(|y|)
1−|y|
1−y
ε
(|y|<1)
for some ε > 0. Then we assume that
(2.28) nγ(n)exp X
m≤n
λ(m)
m!
and
(2.29) exp X
k≤m
λ(k)
k!=o exp X
k≤n
λ(k)
k!! if m=o(n) (n→ ∞).
Definition 2.14. We say that the function Zgiven in (2.22) belongs to the exp −log
class Fin case (2.26)-(2.29) hold.
We notice that the definition of the functions Z∈ F does not require any analytic
continuation of Z(y) over the boundary |y|= 1.
We assume that λfsplits into
(2.30) λf=λf,1+λf,2
such that
(2.31) |λf,1(m)| ≤ λ(m) (m≤n) and
∞
X
m=1
|λf,2(m)|
m=c1<∞.
We may assume that λf,1(m) = 0 if m > n since these values do not influence f(n).
We can formulate the following (Theorem 2 in [23])
Theorem 2.15. Let Zbe an element of the exp-log class Fand let F(y)in (2.25)
satisfy (2.30) and (2.31) with
λf(m) = O(1),|λf,1(m)| ≤ λ(m)for all m ∈N
28
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
and ∞
X
m=1
|λf,2(m)|
m<∞.
If
F(y) = FI(y)FII(y)
where
FI(y) := exp ∞
X
m=1
λf,1(m)
mym!, FII(y) := exp ∞
X
m=1
λf,2(m)
mym!
for |y|<1, then the following two assertions hold.
(i) Let
(2.32)
∞
X
m=1
λ(m)−Re λf,1(m)eima
m
converge for some a∈R. Put
An= exp −ina +X
m≤n
λf,1(m)eima −λ(m)
m!FII(1).
It yields
f(n) = Anγ(n) + o(γ(n)) as n→ ∞.
(ii) Let (2.32) diverge for all a∈R. Then
(2.33) f(n) = o(γ(n)) as n→ ∞.
It may be observed, that we do not require analytic continuation of the generating
functions outside the disk of convergence.
2.5 Lemmata
We prove some lemmata that we shall use in the proofs of our theorems. The proof
of the first lemma follows the lines of Elliott [9], the second and the third lemma
were proven by Indlekofer in [23], therefore we omit their proofs here. Afterwards,
we present a mean-value theorem for multiplicative functions which are bounded
on the prime powers. Thereafter, we prove the fifth lemma, that is a tauberian
theorem, of which the first part was a problem proposed by Schur ([37]).
We shall use the following definition for our results. Put Gn:= {a∈G:∂(a) = n}
then
29
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Definition 2.16. A function ˜
h:G7→ Ris called finitely distributed if there exists
a sequence of integers (n1, n2, ...) and a subset H⊆Gsuch that for every nl,
#(H∩Gnl)≥cG(nl) and |˜
h(a1)−˜
h(a2)|< C for all a1, a2∈H∩Gnlwith some
parameters c > 0, C > 0.
We can describe a finitely distributed additive function on Gas follows
Lemma 2.17. If an additive function ˜gon Gis finitely distributed, then there is an
additive function ˜
hon Gand a constant c,c∈Rso that
˜g(a) = c∂(a) + ˜
h(a)
where both the series
(2.34) X
p
|˜
h(p)|>1
1
q∂(p)X
p
|˜
h(p)|<1
˜
h(p)2
q∂(p)
converge.
Proof. Since ˜gon Gis an additive function, therefore the function exp(it˜g) is a
multiplicative function of modulus 1 on Gfor any real number t. Define the real-
valued function las follows
l(t) = lim
n→∞
1
G(n)X
∂(a)=n
exp(it˜g(a))
.
The function lis well-defined, since the existence of this limit is guaranteed by
Proposition 2.6.
Further, we define the function
D(Θ) = sin πΘ
πΘ2,if Θ 6= 0,
1,if Θ = 0.
Then, for each real number y,
Z∞
−∞
e2πiΘyD(Θ) dΘ = 1−|y|,if |y| ≤ 1,
0,otherwise.
Interchanging summation and integration shows that for a positive α
Z∞
−∞
αX
∂(a)=n
exp(it˜g(a))
2
D(αt)dt =X
a1,a2∈Gn
|˜g(a1)−˜g(a2)|≤α
(1 −α−1|˜g(a1)−˜g(a2)|).
30
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
We divide by G(n), and let n→ ∞, and apply Lebesgue’s theorem on dominated
convergence. Since ˜gis a finitely distributed additive function and if αis sufficiently
large, then Z∞
−∞
αl(t)2D(αt)dt > 0.
To put it more precisely, if ˜gsatisfies the condition given in the definition of a finitely
distributed additive function, and if α≥2C, then the value of this integral is at
least as large as c2/2.
As a consequence, there is a set E, of positive Lebesgue measure, on which l(t)>0.
If, for some value of t, we have l(t)>0, then according to Indlekofer’s theorem (see
Proposition 2.6) on the multiplicative function exp(it˜g) of modulus 1 it yields that
there is a unique real number τ=τ(t), so that the series
(2.35) X
p∈P
q−∂(p)(1 −Re eit˜g(p)q−iτ∂(p))
converges. The convergence of this series is equivalent to that of the series
L(t, τ) = X
p∈P
q−∂(p)sin21
2t˜g(p)−1
2τ∂(p).
Such a number τmay be found for each member tof E. Indeed, there is a number
K, and a subset Fof E, of positive measure, so that whenever tbelongs to Fthe
inequality
(2.36) L(t, τ)≤K
is satisfied. Steinhaus proved the following: The differences generated by a set of
a real numbers of positive Lebesgue measure, cover an open interval around the
origin (for the proof see [9]). Therefore there is a proper interval around the origin,
(−2δ, 2δ) say, each point wof which has a representation w=t1−t2, with both t1
and t2belonging to the set F. In view of the inequality
sin2(x±y)≤2 sin2x+ 2 sin2y,
which is valid for all real numbers xand y, by (2.36) we see that
L(w, τ(t1)−τ(t2)) ≤2L(t1, τ(t1)) + 2L(t2, τ(t2)) ≤4K.
In particular, τ(w) exists and has the value
τ(w) = τ(t1−t2) = τ(t1)−τ(t2).
A simple extension of this argument shows that L(t, τ) is defined (and finite) for
every real number t, and that for every rational number r, the relation τ(rt) = rτ(t)
31
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
holds. For w=j
kδ(k∈N, j = 1, . . . , k) it yields τ(w) = τ(j
kδ) = j
kτ(δ). Since
j
kδ∈(−2δ, 2δ) we also get the inequality
Lj
kδ, τ j
kδ≤4K
which holds uniformly for each positive integer kand j= 1,2, ..., k.
For our next step we shall need the inequality
(2.37) 1
k
k
X
j=1
(1 −cos jy)≥1
2,
that is certainly valid when kis an integer, k≥2, and yis a real number in the
range π/k ≤ |y| ≤ π. This inequality may be deduced from the identity
1
k
k
X
j=1
(1 −cos jy) = 1 + 1
2k−sin((2k+ 1)y/2)
2ksin(y/2) ,
by means of the inequality
|2ksin(y/2)| ≥ 2k|y|/π.
We set c:= τ(δ)/δ, define ˜
h(p) = ˜g(p)−c∂(p), and deduce that for the argument
of the sinus function in the series Lj
kδ, τ j
kδyields
1
2
j
kδ˜g(p)−1
2τj
kδ∂(p) = jδ
k(˜g(p)−c∂(p)) = jδ
k˜
h(p).
Using the cosinus addition formula it follows
2sin21
2
j
kδ˜g(p)−1
2τj
kδ∂(p)= 1 −cos (jδh(p)/k))
By the inequality (2.36) we get
(2.38)
0
X
p
q−∂(p)(1 −cos(jδ˜
h(p)/k)) ≤8K
where ’ indicates that the summation runs over those primes for which
πδ−1≤ |˜
h(p)| ≤ πkδ−1.
32
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Utilising the inequality (2.37) and (2.38) we deduce that
0
X
p
1
q∂(p)≤2
0
X
p
1
q∂(p)
1
k
k
X
j=1
(1 −cos(jδ˜
h(p)/k))
≤21
k
k
X
j=1
0
X
p
1
q∂(p)(1 −cos(jδ˜
h(p)/k))
≤2
k
k
X
j=1
8K
≤16K,
and, since kmay be chosen arbitrarily large,
X
|˜
h(p)|≥πδ−1
1
q∂(p)≤16K.
It yields the convergence of the first series in (2.34). Also,
δ2
π2X
|˜
h(p)|≤πδ−1
q−∂(p)˜
h2(p)≤X
|˜
h(p)|≤πδ−1
q−∂(p)sin2(δ˜
h(p)/2)
≤L(δ, τ(δ)) ≤4K.
This completes the proof of Lemma 2.17.
Indlekofer’s method can be used to deal with multiplicative functions which are
bounded on prime powers. For this let ˜
f:G7→ Cbe multiplicative such that, for
some constant c > 1,
(2.39) |˜
f(pk)| ≤ cfor all prime powers pk.
Referring to the proof in [23] we recall
Lemma 2.18. Let (G, ∂)be an additive arithmetical semigroup satisfying G(n)
nρqnwhere q > 1and ρ∈R. If ˜
fis multiplicative satisfying (2.39), then there
exists m0∈Nsuch that
Π(y) = Y
p
∂(p)≥m0
1 +
∞
X
k=1
˜
f(pk)q−k∂(p)yk∂(p)!
= exp ∞
X
m=m0
λf(m)
mym!(|y|<1)
where λf(m)
m=X
p
∂(p)=m
˜
f(p)q−m+Oq−m
4
as m→ ∞.
33
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Motivated by the results for multiplicative functions on N(see [17]) we shall assume
that
(2.40) X
p
(|˜
f(p)|−1)2
q∂(p)<∞
and ˜
f∈ L1, i.e.
(2.41) M(n, |˜
f|)1.
We define the multiplicative function ˜
f1by
(2.42) ˜
f1(pk) = ˜
f(pk),if ∂(p)≥m0,
0,if ∂(p)< m0.
Clearly M(n, |˜
f1|)≤M(n, |˜
f|), and (2.41) implies
∞
X
n=0 X
g∈G
∂(g)=n
|˜
f1(g)|q−n|y|nZ(|y|)
Since P(d)≤G(d)qddδ−1,
(2.43)
Λ(m) = mP(m) + O mG m
2X
r≤m
1
r!=mP(m) + Omqm
2m
2δ−1log m.
By Lemma 2.18 we get
(2.44) X
p
|˜
f(p)|−1
q∂(p)r∂(p)≤c1with some c1>0
uniformly as r→1−.
Under these conditions the following holds
Lemma 2.19. Let (G, ∂)be an additive arithmetical semigroup such that
ˆ
Z(z) =
∞
X
n=0
G(n)zn= exp ∞
X
m=1
Λ(m)
mzm!=ˆ
H(z)
(1 −qz)δ
where ˆ
H(z) = O(1) for |z|< q−1,limz→1
q
−ˆ
H(z) = A > 0and δ > 0. Assume that
Λ(m) = O(qm)and G(n)qnnδ−1. Let ˜
fbe multiplicative and assume (2.40) and
(2.41). If ˜
f1satisfy (2.42) and (2.44), then, as n→ ∞,
M(n, |˜
f1|) = c2exp
X
m0≤∂(p)≤n
|˜
f(p)|−1
q∂(p)
+o(1)
with some positive constant c2.
34
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Proof. For further details see Theorem 6 in [23].
Our next lemma can be proven similarly as Proposition 2.6 (cf. Theorem 4 in [23]).
Lemma 2.20. Let (G, ∂)be an additive arithmetical semigroup such that
ˆ
Z(z) =
∞
X
n=0
G(n)zn= exp ∞
X
m=1
¯
Λ(m)
mzm!=ˆ
H(z)
(1 −qz)δ
where ˆ
H(z) = O(1) for |z|< q−1,limz→1
q
−ˆ
H(z) = A > 0and δ > 0. Assume
that ¯
Λ(m) = O(qm)and G(n)qnnδ−1. Suppose ˜
fis a multiplicative function
such that |˜
f(pk)|< K for each prime power pkwith ∂(p)≤log 2
log q, and the series
P
p∈P
(|˜
f(p)|−1)q−∂(p)converges. If there exists a real number asuch that the series
(2.5) converges for τ=a, then
X
g∈G
∂(g)=n
˜
f(g) = qina Y
∂(p)≤n
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)(1+ia)!G(n) + o(G(n)).
If (2.5) diverges for all τ∈Rthen
X
g∈G
∂(g)=n
˜
f(g) = o(G(n)).
In the proof of our mean-value theorems we often compute the coefficients of power
series, that can be written as a product of two other power series. Under certain
conditions we can compute the required coefficients. We describe our result in the
following lemma.
Lemma 2.21. Let C(z) = A(z)B(z)where the power series are defined as A(z) =
P∞
n=1 anzn,B(z) = P∞
n=1 bnzn,C(z) = P∞
n=1 cnzn. Assume that for a real number ρ
the relation bn−1/bn=ρ+o(1) holds as n→ ∞ and assume that P∞
n=1 |an||ρ|n<∞.
Let the radius rof converge of A(z)satisfy r≥ |ρ|. If
(i) |ρ|< r, or
(ii) |ρ|=rand bm=O(|bn||ρ|n−m)for all m≤n
then
cn∼A(ρ)bn(n→ ∞).
Proof. Let ε > 0 be so small that |ρ|+ε < r. Then there exists a constant A
independent of nand νsuch that
bn−ν
bn
=
bn−ν
bn−ν+1 ·. . . ·
bn−1
bn
< A(|ρ|+ε)ν, ν = 0,1, . . . , n;n= 0,1,2, . . .
35
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
For n>mwe obtain
(2.45) cn
bn−A(ρ) =
m
X
ν=0
aνbn−ν
bn−ρν+
n
X
ν=m+1
aν
bn−ν
bn−
∞
X
ν=m+1
aνρν=: Σ1+Σ2−Σ3.
In the case (i) there exists a positive integer m0such that
Σ2
∞
X
ν=m+1 |aν|(|ρ|+ε)ν≤ε
for m≥m0, and
Σ3≤
∞
X
ν=m+1 |aν||ρ|ν≤ε
for m≥m0.
In the case (ii), we obtain
Σ2
n
X
ν=m+1 |aν||ρ|ν≤ε
since |bn−ν|
|bn|=O(|ρ|ν) holds by our assumption in (ii). Further, we know that
Σ3≤
∞
X
ν=m+1 |aν||ρ|ν≤ε.
Thus in both cases the sum Σ2and Σ3are absolutely smaller than εfor m≥m0.
Choose mso large that these two terms are smaller than ε. For a fixed mwe
can choose nin both cases (i) and (ii) so that the first sum Σ1in (2.45) becomes
absolutely smaller than ε. It yields
cn∼A(ρ)bn
as (n→ ∞). This ends the proof of Lemma 2.21.
2.6 Proof of the new mean-value theorems
First we prove Theorem 2.10.
Proof. We assume that M(˜
f)6= 0 and ˜
f∈L1. Then there exists a natural number
n0and constants 0 < c1, c2<∞such that
(2.46) 0 < c1≤1
G(n)X
a∈G
∂(a)=n
|˜
f(a)| ≤ c2<∞
36
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
for all n≥n0. Then, for ε > 0 that is small enough,
1
G(n)X
a∈G,∂(a)=n
|˜
f(a)|≤ε
|˜
f(a)| ≤ c1
4
and with suitable K > 0,
1
G(n)X
a∈G,∂(a)=n
|˜
f(a)|>K
|˜
f(a)| ≤ c1
4
because ˜
fis uniformly summable. Thus
c2≥1
G(n)X
a∈G,∂(a)=n
ε<|˜
f(a)|≤K
|˜
f(a)| ≥ c1−c1
4−c1
4=c1
2>0.
It follows also, that c2
ε≥1
G(n)X
a∈G,∂(a)=n
ε<|˜
f(a)|≤K
1≥c1
2K>0,
i.e. 1
G(n)X
a∈G,∂(a)=n
ε<|˜
f(a)|≤K
11.
We define an additive function ˜gby
˜g(pk) = log |˜
f(pk)|,if ˜
f(pk)6= 0,
1,otherwise.
It yields
(2.47) 1
G(n)X
a∈G,∂(a)=n
log ε<˜g(a)≤log K
11,
therefore ˜gis finitely distributed. Then by Lemma 2.17 we deduce
(2.48) ˜g(a) = c∂(a) + ˜
h(a)
where ˜
hsatisfies (2.34).
Our next step is to prove c= 0. With (2.48) we derive
(2.49) 0 6=|˜
f(p)|=ec∂(p)e˜
h(p).
37
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
By Lemma 2.17
(2.50) X
p∈P
|˜
h(p)|>C
1
q∂(p)<∞.
for all C > 0. If |e˜
h(p)−1|> η1for some η1,0< η1<3/4,then |˜
h(p)| ≥ min{log(1+
η1),−log(1 −η1),1}hence with (2.50)
(2.51) X
p∈P
|e˜
h(p)−1|>η1
1
q∂(p)<∞.
We define
(2.52) P1:= {p∈P;e˜
h(p)<1−η1}
and
(2.53) P2:= {p∈P;e˜
h(p)>1 + η1}
with 0 < η1<3/4. Let
S1:= {a∈G;∃p∈P1∪P2:p|a, ∂(p)≥n0}
S2:= {a∈G;∃p∈P:p2|a, ∂(p)≥n0}
and
S3:= {a∈G;∃p∈P:pk|a, k ≥k0, ∂(p)≤n0}.
Put
S:= S1∪S2∪S3.
We apply Proposition 2.6 on the multiplicative function 1G\S. Therefore we can
choose n0and k0such that for all n0≤n
(2.54) M(n, 1S)< γ.
Then Shas a density less than γ, hence lim sup
n→∞
M(n, 1S˜
f)< ε and therefore
lim inf
n→∞ M(n, 1G\S˜
f)> ε,
by our choice of ε.
38
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Since G(n)qnnδ−1, we have
X
∂(a)=n
˜ω(a) = X
p
∂(p)=n
1 + X
p
∂(p)≤n−1
G(n−∂(p))
P(n) + X
p
∂(p)≤n−1
qn−∂(p)(n−∂(p))δ−1
qn
n+qn
n−1
X
m=1
q−mP(m)(n−m)δ−1
=qn
n+qnX
n
2≤m≤n−1
q−mP(m)(n−m)δ−1+qnX
1≤m<n
2
q−mP(m)(n−m)δ−1
qn
n+qn
nX
1≤m≤n
mδ−1+qnnδ−1X
1≤m≤n
2
1
m
qn
n+qn
nnδ+qnnδ−1log n
and therefore
(2.55) M(n, ˜ω) = O(log n).
By our assumption log(|˜
f(a)|) = c∂(a) + ˜
h(a). Hence
|˜
f(a)|=ec∂(a)e˜
h(a).
Since ˜
f∈L∗
(2.56) 1 M(n, |˜
f|1G\S) = ecn 1
G(n)X
a∈G\S
∂(a)=n
e˜
h(a).
We show that chas to be zero. For this we estimate
1
G(n)X
a∈G
∂(a)=n
e˜
h(a).
By the definition of the set Swe have
(2.57) a∈G\S, p|a⇒˜
h(pk) = O(1),
and it yields
(2.58) |˜
h(a)| ≤ C˜ω(a) for a∈G\S.
39
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Therefore
(2.59) 1
G(n)X
a∈G\S
∂(a)=n
e˜
h(a)≤1
G(n)X
a∈G\S
∂(a)=n
e|˜
h(a)|.
Put
(2.60) ˜g˜
h(a) := e|˜
h(a)|.
Then ˜g˜
h(a) is a multiplicative function with e|˜
h(pk)|=O(1) by (2.57). Thus, there
exists a real constant c3such that
(2.61)
∞
X
n=0
g˜
h(n)znexp ∞
X
m=1
c3Λ(m)
mzm!.
It yields for 0 <|z|=r < q−1there exist positive constants c4and c5so that
(2.62) g˜
h(n)rn
qnexp X
m≤n
c4Λ(m)
mrm!exp(c5log n).
Hence there exists a constant c6with
(2.63) g˜
h(n)
qnnδ−1exp(c6log n),
it means
(2.64) 1
G(n)X
a∈G\S
∂(a)=n
e˜
h(a)exp(c6log n).
On the other hand, the equations (2.55) and (2.58) with the Jensen-inequality (see
[36]) yields
1
G(n)X
a∈G\S
∂(a)=n
e˜
h(a)≥1
G(n)X
a∈G\S
∂(a)=n
e−|˜
h(a)|
≥1
G(n)X
a∈G\S
∂(a)=n
e−C˜ω(a)
exp
−c4
G(n)X
a∈G\S
∂(a)=n
˜ω(a)
exp(−c3log n)
40
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Collating our last results we arrive at the conclusion that there exists a real constant
c7such that
(2.65) 1 M(n, |˜
f|1G\S)ecne±c7log nas n→ ∞.
Hence cmust be zero, and it follows that |˜
f(a)|=e˜
h(a), for all a∈G, for which
|˜
f(a)| 6= 0.
By Lemma 2.17 the series
(2.66) X
p∈P
|˜g(p)|<1
(˜g(p))2
q∂(p)and X
p∈P
|˜g(p)|>1
1
q∂(p)
converge. If ||˜
f(p)|−1| ≤ η1, then the series expansion of the logarithm yields
log |˜
f(p)|= log(1 + (|˜
f(p)|−1)) = |˜
f(p)|−1 + O((|˜
f(p)|−1)2)
so that for η1= 1/2
||˜
f(p)|−1| ≤ 2|log |˜
f(p)|| = 2|˜g(p)|
and
|˜g(p)| ≤ 2||˜
f(p)|−1| ≤ 1.
Therefore
X
p∈P
|˜
f(p)|<1/2
(|˜
f(p)|−1)2
q∂(p)X
p∈P
|˜g(p)|>log(1/2)
1
q∂(p)<∞
and
X
p∈P
1/2≤| ˜
f(p)|≤3/2
(|˜
f(p)|−1)2
q∂(p)X
p∈P
|˜q(p)|≤1
(˜g(p))2
q∂(p)<∞.
Thus the series
X
p∈P
|˜
f(p)|≤3/2
(|˜
f(p)|−1)2
q∂(p)
converges. Furthermore
(2.67) |˜
f(p)−1|2= (|˜
f(p)|−1)2+ 2(|˜
f(p)|−1) −2(Re (˜
f(p)) −1)
where the series over the first term on the right hand side converges (see above).
Choose K > 0 large enough and let k0, n0be the parameters, which we have chosen
such that M(n, 1S)< γ holds.
41
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
We show that the series over the second term on the right hand side of (2.67) is
bounded. Let the multiplicative function ˜
f∗be defined as
(2.68) ˜
f∗:= ˜
f1G\S.
Then the function ˜
f∗is bounded on the set of the prime powers. Since M(˜
f) exists
and is nonzero there exists a natural number n1,n1≥n0such that
(2.69) |M(n, ˜
f)| 1 for all n≥n1.
Our assumption ˜
f∈L1yields
(2.70) M(n, |˜
f|)1 for all n≥n1.
Since ˜
f∈L∗we obtain also
(2.71) |M(n, |˜
f∗|)| 1 for all n≥n1.
For the moment put 0 <|z|=tq−1<1 with 0 < t < 1. The assertion (2.71) and
the definition of ˜
f∗yield
(2.72) 1
∞
P
n=1 P
a∈G
∂(a)=n
|˜
f∗(a)|tnq−n
ˆ
Z(tq−1)exp
X
n≥n0X
p
∂(p)=n
|˜
f∗(p)|−1
q∂(p)t∂(p)
.
Put
an=X
p
n0≤∂(p)=n
|˜
f∗(p)|−1
qn.
From our assumption Λ(n) = O(qn) follows P(n) = O(qn
n), therefore we obtain
an=O1
n.
We have ∞
X
n=0
antn=O(1) for t→1.
We show that
(2.73) X
n≤N
an=O(1).
42
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
If we put t= 1 −1
N, then
∞
X
n=0
antn−X
n≤N
an≤X
n≤N
an(tn−1)
+X
n>N
antn
≤X
n≤N|an|
exp nlog 1−1
N−1
+O1
N∞
X
n=0
tn
≤X
n≤N
O1
nOn
N+O1
NN
=O(1).
Hence (2.73) is satisfied and thus
X
n≤NX
p
n0≤∂(p)=n
|˜
f∗(p)|−1
qn=O(1).
There exist only finitely many terms with ∂(p)< n0, therefore
(2.74) X
n≤NX
p,∂(p)=n
|˜
f(p)|≤K
|˜
f(p)|−1
q∂(p)=O(1).
Next, we prove that the sum over the third term on the right hand side of (2.67) is
bounded, and with this we find that (2.16) converges, as has been claimed.
Let εbe an arbitrary positive number. By the definition of ˜
f∗and the formula
(2.68) we deduce
AG(n)∼G(n)M(n, ˜
f) = G(n)M(n, ˜
f∗) + G(n)ϑε (as n→ ∞)
with 0 ≤ |ϑ| ≤ 1. Therefore
Aˆ
Z(z)∼ˆ
F(z) = ˆ
F∗(z) + ϑε ˆ
Z(z)
where ˆ
F∗denotes the generating function of ˜
f∗. Dividing by ˆ
Z(z) and utilising the
formula (2.69) it follows
ˆ
F∗(z)
ˆ
Z(z)1.
Using here, further, the notation 0 <|z|=tq−1<1 with 0 <t<1, we obtain as
by (2.72) that
1exp
X
n≥n0X
p
∂(p)=n
Re ˜
f∗(p)−1
q∂(p)r∂(p)
.
43
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Then with analogue tauberian arguments as above we obtain
X
n≤NX
p∈P
∂(p)≥n0,|˜
f(p)|≤3/2
Re (˜
f(p)) −1
q∂(p)=O(1),
i.e that the partial sums of the series over the third term on the right hand side of
(2.67) are bounded. Putting our results together in (2.67) we obtain the convergence
of the series
X
p∈P
|˜
f(p)|≤3/2
|˜
f(p)−1|2
q∂(p),
i.e the convergence of (2.16).
Next we prove the convergence of the series (2.18). Let
S4:= {a∈G;∃p∈P:p|a;||˜
f(p)|−1|>1/2, ∂(p)≥n0}.
Since M(˜
f) exists and is nonzero and ˜
f∈L1there exists a natural number n1,
n1≥n0such that (2.69) and (2.70) hold. By our assumption ˜
f∈L∗, thus
(2.75) M(n, |˜
f|1G\S4)1 for all n≥n1.
Put in what follows 1 < λ ≤αand β∈Rwith 1
λ+1
β= 1.
Then H¨older’s inequality yields
11
G(n)X
a∈G
∂(a)=n
|˜
f(a)| ≤ 1
G(n)
X
a∈G
∂(a)=n
|˜
f(a)|λ
1
λ
G(n)1
β
=G(n)1−1
λ
G(n)
X
a∈G
∂(a)=n
|˜
f(a)|λ
1
λ
=
1
G(n)X
a∈G
∂(a)=n
|˜
f(a)|λ
1
λ
=M(n, |˜
f|λ)1
λ
1
since ˜
f∈Lα. Hence
M(n, |˜
f|λ)1 for all n≥n1.
44
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
By the formula (2.75) we obtain similarly
M(n, |˜
f|λ1G\S4)1 for all n≥n1.
For 0 < r =|z|<1/q we obtain
1
ˆ
Z(r)
∞
P
n=0
P
a∈G\S4
∂(a)=n
|˜
f(a)|λ
rn
ˆ
Z(r)
∞
P
n=0
P
a∈G
∂(a)=n
|˜
f(a)|λ
rn
=Y
p∈P,∂(p)≥n0
|| ˜
f(p)|−1|>1/2
1 +
∞
X
k=1 |˜
f(pk)|λrk∂(p)!−1
.(2.76)
Remark 2.22. Consider an infinite product
∞
Q
n=1
(1 + bn), where bn≥0 is satisfied.
Then ∞
Y
n=1
(1 + bn) = lim
N→∞ Y
n≤N
(1 + bn)≤lim
N→∞ exp X
n≤N
bn!,
where we have made use of the inequality 1 + x≤exp(x), which is valid for x≥0.
On the other hand, we know
lim
N→∞ X
n≤N
bn≤lim
N→∞ Y
n≤N
(1 + bn).
Thus the product
∞
Q
n=1
(1 + bn) is convergent if and only if
∞
P
n=1
bnis convergent.
The last product in (2.76) has the form
∞
Q
n=1
(1+bn), where bn≥0. Therefore Remark
2.22 yields that there exists a real constant c8such that for all r∈R
X
p;|| ˜
f(p)|−1|>1/2|˜
f(p)|λr∂(p)≤c8<∞.
Thus for r→1/q
X
p;|| ˜
f(p)|−1|>1/2
|˜
f(p)|λ
q∂(p)<∞.
which yields the convergence of the series (2.18) for all 1 ≤λ≤α.
Next, we prove the convergence of the series (2.17). Put
S5:= {a∈G;∃p∈P:pk|a;k≥2, ∂(p)≥n0}.
45
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Then, analogous to what we have seen above, there exists n1∈Nsuch that for
n≥n1
M(n, |˜
f|λ)1 and M(n, |˜
f|λ1G\S5)1.
For 0 < r =|z|<1/q the following holds
1
∞
P
n=0
P
a∈G\S5
∂(a)=n
|˜
f(a)|λ
rn
∞
P
n=0
P
a∈G
∂(a)=n
|˜
f(a)|λ
rn
=Y
p∈P,k≥2
∂(p)≥n0
1 +
∞
X
k=1 |˜
f(pk)|λrk∂(p)!−1
.
Using Remark 2.22 it follows that there exists a real constant c9such that for all
r∈RX
p∈P,k≥2
∂(p)≥n0
|˜
f(pk)|λrk∂(p)≤c9<∞.
Thus for r→1/q
X
p∈P;k≥2
|˜
f(pk)|λ
qk∂(p)<∞.
holds, and therefore the series (2.17) converges for all 1 ≤λ≤α.
Next, we show the validity of (2.19) for every p∈P. By the convergence of the
series (2.16) and (2.18) for |z|=r < q−1we have
Y
p;∂(p)≥n0 1 +
∞
X
k=1
˜
f(pk)zk∂(p)!≤Y
p;∂(p)≥n0 1 +
∞
X
k=1
˜
f(pk)zk∂(p)!
exp
X
p;∂(p)≥n0
(|˜
f(p)|−1)rk∂(p)
ˆ
Z(r)
ˆ
Z(r).
Suppose now, that for some p1with ∂(p1)< n0we have
1 +
∞
X
k=1
˜
f(pk)q−k∂(p)= 0.
Hence 1 +
∞
P
k=1
˜
f(pk
1)zk∂(p1)=o(1) as r→q−1. Thus, as r→q−1
Y
p∈P 1 +
∞
X
k=1
˜
f(pk)zk∂(p)!=o(1)O(ˆ
Z(r)) = o(ˆ
Z(r))
46
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
and we achieve a contradiction to ˆ
F(z)∼cˆ
Z(z) for z→q−1with c6= 0.
1 +
∞
X
k=1
˜
f(pk)q−∂(p)6= 0,
which is then true for all primes p∈P.
Finally, we prove the convergence of the series (2.15). By the convergence of (2.18)
and the condition (2.19), there exists some number m0sufficiently large such that
|˜
f(p)q−∂(p)|<1
4and
(2.77)
1 +
∞
X
k=1
˜
f(pk)(q−1eiΘ)k∂(p)
>1
2
for all pwith ∂(p)≥m0and all real Θ with |Θ| ≤ π. We write
ˆ
F(r)
ˆ
Z(r)=Y
p,∂(p)<m0
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!Y
p,∂(p)≥m0
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!
=: Π1(r)Π2(r),
where the first product Π1(r) is continuous by (2.18). We now estimate the second
product Π2(r):
Π2(r) = Y
p,∂(p)≥m0
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!
=Y
p,∂(p)≥m0 1+(˜
f(p)−1)r∂(p)+
∞
X
k=2
(˜
f(pk)−˜
f(pk−1))rk∂(p)!
=Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!Y
p,∂(p)≥m0
|˜
f(p)−1|>1
2
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!
=:Π3(r)Π4(r).
By the convergence of the serie (2.18) the product Π4(r) of the last line is continuous
for r≤q−1.
47
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
We can write the first product Π3(r) of the last line as follows
Π3(r) = Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
1−˜
f(p)r∂(p)
1−˜
f(p)r∂(p)
=Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −˜
f(p)r∂(p))−1
×Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −r∂(p)) 1 +
∞
X
k=1
˜
f(pk)rk∂(p)!(1 −˜
f(p)r∂(p))
=Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −˜
f(p)r∂(p))−1×Y
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
1−r∂(p)−(˜
f(p)r∂(p))2+˜
f(p)r2∂(p)
+
∞
X
k=2
(˜
f(pk)−˜
f(pk−1))rk∂(p)(1 −˜
f(p)r∂(p))!
=:Π5(r)Π6(r).
The convergence of the series (2.16) yields that the second product Π6(r) of the
above line is continuous for r≤q−1. Also
Π5(r) = exp
X
p,∂(p)≥m0
|˜
f(p)−1|≤ 1
2
log(1 −(˜
f(p)−1)r∂(p))
.
After the power series expansion of the logarithm we can summarize our last results
and write
(2.78) ˆ
F(r)
ˆ
Z(r)=F1(r) exp
−X
∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −˜
f(p))r∂(p)
.
Here F1(r) is continuous for r≤q−1. If now the condition (2.19) is satisfied, then
F1(q−1)6= 0. It follows from (2.78) that
lim
r→q−1X
∂(p)≥m0
|˜
f(p)−1|≤ 1
2
(1 −˜
f(p))r∂(p)
48
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
exists. Appealing directly to our condition P(m) = O(qm
m) and to the well-known
tauberian theorem of Ingham (see Theorem 106 in [16]),
X
m≥m0
q−mX
p,∂(p)=m
|˜
f(p)−1|≤ 1
2
(1 −˜
f(p))
converges and then the series
X
p∈P
˜
f(p)−1
q∂(p)
converges, too. Hence (2.15) converges, as it has been claimed. This ends the proof
of Theorem 2.10.
Next, we prove Theorem 2.11
Proof. First we prove that M(˜
f) exists. By the convergence of (2.18) and the condi-
tion (2.19), there exists some number m0sufficiently large such that |˜
f(p)q−∂(p)|<1
4
and (2.77) holds for all pwith ∂(p)≥m0and all real Θ with |Θ| ≤ π. We write
ˆ
F(z) = Y
p,∂(p)<m0 1 +
∞
X
k=1
˜
f(pk)zk∂(p)!Y
p,∂(p)≥m0
|˜
f(p)|<K
1 +
∞
X
k=1
˜
f(pk)zk∂(p)!
×Y
p,∂(p)≥m0
|˜
f(p)|≥K
1 +
∞
X
k=1
˜
f(pk)zk∂(p)!
=: Π1(z)Π2(z)Π3(z),
where the first product Π1(z) is absolutely convergent for |z| ≤ q−1, since each factor
of the finite product Π1(z) is convergent by (2.18). The third product Π3(z) is also
absolutely convergent for |z| ≤ q−1. We now estimate the second product Π2(z):
Π2(z) = Y
p,∂(p)≥m0
|˜
f(p)|<K
1 +
∞
X
k=2
˜
f(pk)zk∂(p)!1−˜
f(p)z∂(p)
1−˜
f(p)z∂(p)
=Y
p,∂(p)≥m0
|˜
f(p)|<K
(1 −˜
f(p)z∂(p))−1Y
p,∂(p)≥m0
|˜
f(p)|<K
1 +
∞
X
k=2
˜
f(p)( ˜
f(pk)−˜
f(pk−1))zk∂(p)!
=:Π4(z)Π5(z).
By the convergence of the series (2.18) the second product Π5(z) of the last line is ab-
solutely convergent for |z| ≤ q−1. We apply Lemma 2.20 to the product Π4(z), that is
49
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
a generating function of a completely multiplicative function ˜
f1, where ˜
f1(p) = ˜
f(p)
for ∂(p)≥m0and |˜
f(p)|< K, and ˜
f1(p) = 0 otherwise. We obtain
X
a∈G,∂(a)=n
˜
f1(a) = Y
p∈P
(1 −q∂(p))(1 −˜
f(p)q−∂(p))−1G(n) + o(G(n)).
Thus we can write
(2.79) ˆ
F(z) = Π4(z)(Π1(z)Π5(z)Π3(z)) =: Π4(z)A(z)
where A(z) is absolutely convergent for |z|=q−1. Applying Lemma 2.21 it follows
M(˜
f) = M(˜
f1)A(q−1).
Hence M(˜
f) exists and has the above form. If in addition (2.19) holds, then
M(˜
f)6= 0 as it has been claimed.
If α > 1 and ||˜
f(p)|−1|<1/2, then
|˜
f(p)|α−1 = α(|˜
f(p)|−1) + O((|˜
f(p)|−1)2)
and
(|˜
f(p)|α−1)2=O((|˜
f(p)|−1)2) = O(|˜
f(p)−1|2),
and the corresponding series converge (cf (2.67)).
Therefore, in the same way as above we deduce that M(|˜
f|λ) exists for 1 ≤λ≤α
and ˜
f∈Lα. If in addition (2.19) holds, then M(|˜
f|λ)6= 0 for 1 ≤λ≤αas it has
been claimed.
Next, we prove that ˜
f∈L∗. Using the equation (2.79) we can write the multiplica-
tive function ˜
fas the convolution
(2.80) ˜
f=˜
f1∗˜
f2,
where ˜
f1is the completely multiplicative function defined above; and ˜
f2is a mul-
tiplicative function, such that its generating function A(z) is absolutely convergent
for |z| ≤ q−1. It yields
(2.81) Σ∗:= X
m∈NX
b∈G,∂(b)=m|˜
f2(b)|q−∂(b)<∞.
Hence for an arbitrary ε∗there exists a natural number m0such that
X
m≥m0X
b∈G,∂(b)=m|˜
f2(b)|q−∂(b)<ε∗
2.
50
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Using the convergence of the series (2.15)-(2.18) we deduce by Lemma 2.19 that
M(|˜
f1|) and M(|˜
f1|2) exist.
Let ε > 0 be arbitrary and fixed. We prove that there exists K0such that
X
a∈G,∂(a)=n|˜
fK0(a)|< εG(n)
holds for all n∈N.
X
a∈G,∂(a)=n|˜
fK0(a)|=X
a,b∈G
|˜
f1(a)|| ˜
f2(b)|≥K0
∂(a)+∂(b)=n
|˜
f1(a)||˜
f2(b)|
=X
a,b∈G
|˜
f1(a)|| ˜
f2(b)|≥K0
|˜
f2(b)|≥K1,∂(a)+∂(b)=n
|˜
f1(a)||˜
f2(b)|+X
a,b∈G
|˜
f1(a)|| ˜
f2(b)|≥K0
|˜
f2(b)|<K1,∂(a)+∂(b)=n
|˜
f1(a)||˜
f2(b)|
=:Σ1+ Σ2,
where the parameter K1is chosen such that ∂(b)≥m0as |˜
f2(b)| ≥ K1. Let us now
estimate Σ1. By our assumption G(n)qnnδ−1(1 ≤δ) we obtain
Σ1=X
b∈G
|˜
f2(b)|≥K1
∂(b)≤n
|˜
f2(b)|X
a∈G
∂(a)=n−∂(b)
|˜
f1(a)|
≤X
b∈G
m0≤∂(b)≤n
|˜
f2(b)|X
a∈G
∂(a)=n−∂(b)
|˜
f1(a)| X
b∈G
m0≤∂(b)≤n
|˜
f2(b)|q−∂(b)G(n)
<ε
2G(n),
whereby we have used the following
G(n−∂(b)) qn−∂(b)(n−∂(b))δ−1=qnnδ−1(1 −∂(b)/n)δ−1q−∂(b)q−∂(b)G(n).
51
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
Afterwards, we estimate Σ2. We use (2.81) and our assumption G(n)qnnδ−1to
obtain the following
Σ2=X
a,b∈G
|˜
f2(b)|<K1
|˜
f1(a)|| ˜
f2(b)|≥K0,∂(a)+∂(b)=n
|˜
f1(a)||˜
f2(b)|
=X
b∈G,|˜
f2(b)|<K1X
a∈G
|˜
f1(a)|| ˜
f2(b)|≥K0
∂(a)=n−∂(b)
|˜
f1(a)|2
|˜
f1(a)|
≤X
b∈G,|˜
f2(b)|<K1
|˜
f2(b)||˜
f2(b)|
K0X
a∈G
∂(a)=n−∂(b)
|˜
f1(a)|2
K1
K0X
b∈G|˜
f2(b)|G(n−∂(b)) ≤ε
2G(n),
since M(|˜
f1|2) exists.
Therefore ˜
f∈L∗. This ends the proof of Theorem 2.11.
Now we prove Theorem 2.12
Proof. Let ε > 0 be arbitrary and fixed. Then by (2.20) there exists K > 0 with
S={a∈G:pk|a, p ∈P, k ≥1,|˜
f(pk)|> K}
such that
M(n, |˜
f|1S)< ε.
Let such a Kbe fixed. It yields
1
G(n)X
a∈G
∂(a)=n
˜
f(a)−1
G(n)X
a∈G\S
∂(a)=n
˜
f(a)
< ε.
By Lemma 2.20 we obtain
1
G(n)X
a∈G\S
∂(a)=n
˜
f(a)→M(1G\S˜
f) = Y
p∈P
|˜
f(pk)|≤K
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!
52
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
as n→ ∞. Therefore
M(1G\S˜
f) = Y
p∈P
|˜
f(p)|≤K2
|˜
f(pk)|≤K,K=2,3,...
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!
×Y
p∈P
K≥| ˜
f(p)|>K2
|˜
f(pk)|≤K
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!
=:Π1,KΠ2,K.
The product Π2,K is absolutely convergent for |z| ≤ q−1and
lim
K→∞ Π2,K =Y
p∈P
|˜
f(p)|>K2
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!.
Estimating the product Π1,K we deduce
Π1,K =Y
p∈P
|˜
f(p)|≤K2
(1 −q−∂(p))(1 + ˜
f(p)q−∂(p))Y
p∈P
|˜
f(p)|≤K2
|˜
f(pk)|≤K
(1 + ˜
f(p)q−∂(p))−1 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!
=:Π1Π3,K.
We derive
Π1,K =Y
p∈P,∂(p)≤m0
|˜
f(p)|≤K2
|˜
f(pk)|≤K
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!
×Y
p,∂(p)>m0
|˜
f(p)|≤K2
(1 −q−∂(p))(1 + ˜
f(p)q−∂(p))Y
p∈P,∂(p)>m0
|˜
f(p)|≤K2
|˜
f(pk)|≤K
(1 + ˜
f(p)q−∂(p))−1 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!
=:Π4,KΠ5Π6,K.
Therefore
lim
K→∞ Π4,K =Y
p∈P,∂(p)≤m0
|˜
f(p)|≤K2
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!,
53
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
since Π4,K is a finite product. Since
(1 −q−∂(p))(1 + ˜
f(p)q−∂(p)) = 1 −q−∂(p)+˜
f(p)q−∂(p)−˜
f(p)q−2∂(p)
the product Π5is convergent because of the convergence of the series (2.15) and
(2.16). Hence for a given positive real number ε1there exists a natural number n0
such that
1
G(n)X
a∈G\S
∂(a)=n
˜
f(a)−M(1G\S˜
f)
< ε1
holds for all n≥n0. Thus
(2.82) lim sup
n→∞
1
G(n)X
a∈G,∂(a)=n
˜
f(a)−M(1G\S˜
f)
< ε +ε1.
Considering the limit for K→ ∞ we obtain
lim
K→∞ M(1G\S˜
f) = Y
∂(p)∈P
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!.
where the last product does not depend on K. Therefore (2.82) yields
M(˜
f) = Y
∂(p)∈P
(1 −q−∂(p)) 1 +
∞
X
k=1
˜
f(pk)q−k∂(p)!.
It means that the mean-value M(˜
f) exists and has the above form. If (2.19) holds,
we obtain also, that M(˜
f) is nonzero.
The existence of the mean-value M(|˜
f|λ) for 1 ≤λ≤αfollows in the same way,
since the series corresponding to (2.15)-(2.18) for |˜
f|λare convergent, then ˜
f∈Lα
holds.
Finally, we prove that ˜
f∈L∗. For a real number K,K > 0 it yields
(2.83) X
a∈G
|˜
f(a)|>K
∂(a)=n
|˜
f(a)|=X
a∈G\S
|˜
f(a)|>K
∂(a)=n
|˜
f(a)|+X
a∈S
|˜
f(a)|>K
∂(a)=n
|˜
f(a)|
where the second sum on the right hand side is < ε/2 and tends to zero as K→ ∞.
Put ˜
f3=˜
f1G\S. Then ˜
f3is a multiplicative function with |˜
f3(pk)| ≤ Kand Lemma
54
CHAPTER 2. INVESTIGATION OF ARITHMETICAL FUNCTIONS
2.18 yields M(n, ˜
f3) = O(1). Therefore
X
a∈G\S
|˜
f(a)|>K
∂(a)=n
|˜
f(a)| ≤ X
a∈G\S
|˜
f(a)|>K
∂(a)=n
|˜
f(a)||˜
f(a)|
K
=1
KX
a∈G
|˜
f3(a)|>K
∂(a)=n
|˜
f3(a)|2< G(n)ε/2
if Kis large enough. By (2.83) it follows that ˜
f∈L∗.
This ends the proof of Theorem 2.12.
55
Chapter 3
Applications
In this chapter we give two probabilistic applications of our main results. Let now
˜g:G→Rbe a (real-valued) additive function defined on G. Then, by the continuity
theorem of L´evy, the distribution functions
(3.1) Gn(x) := 1
G(n)#{a∈G:∂(a) = n, ˜g(a)≤x}
tend to a limit distribution G(x),
(3.2) Gn⇒ G,
if and only if there exists a function ϕ(t) which is continuous at t= 0 such that
1
G(n)X
a∈G
∂(a)=n
eit˜g(a)→ϕ(t)
as n→ ∞ for t∈R. Moreover, ϕ(t) is the characteristic function of G(x). We note
that the function ˜
f(a) := eit˜g(a)is multiplicative and |˜
f(a)|= 1 since ˜gis real-valued
and additive.
3.1 Finitely distributed additive functions
In this section we characterize all additive functions ˜gon Gwhich, after a suitable
translation, possess a limiting distribution. In order that there exists a sequence
{α(n)},n∈N, for which the frequencies
1
G(n)#{a∈G, ∂(a) = n: ˜g(a)−α(n)≤x}
converge to a weak limit as n→ ∞ we give here the necessary and sufficient condi-
tions.
56
CHAPTER 3. APPLICATIONS
Theorem 3.1. Suppose that an additive arithmetical semigroup (G, ∂)fulfills the
assumptions of Theorem 2.11. Let ˜gbe a real-valued additive function on G. Then
the following assertions hold
(i) If, for some α(n)the frequencies
1
G(n)#{a∈G, ∂(a) = n: ˜g(a)−α(n)≤x}
converge to a weak limit as n→ ∞, then ˜gis finitely distributed.
(ii) If ˜gis finitely distributed, then it has a decomposition ˜g(a) = c∂(a)+˜
h(a)with
a real constant cand an additive function ˜
hwhere both the series
(3.3) X
p
|˜
h(p)|>1
1
q∂(p)X
p
|˜
h(p)|<1
˜
h(p)2
q∂(p)
converge.
(iii) If ˜ghas a representation c∂ +˜
h, where the series (3.3) both converge, and if
we define
α(n) = cn +X
∂(p)≤n,|˜
h(p)|≤1
˜
h(p)
q∂(p)(n≥1),
then the frequencies
1
G(n)#{a∈G, ∂(a) = n: ˜g(a)−α(n)≤x}
converge to a weak limit as n→ ∞.
Proof of (i). If the number wis chosen sufficiently large, and such that ±ware
continuity points of the limiting distribution of ˜g(a)−α(n), then
(3.4) lim
n→∞
1
G(n)#{a∈G, ∂(a) = n: ˜g(a)−α(n)≤w}>1
2.
Moreover, if a1and a2are any two elements in Gwhich are counted in a typical
frequency,
(3.5) |˜g(a1)−˜g(a2)| ≤ |˜g(a1)−α(n)|+|α(n)−˜g(a2)| ≤ 2w,
from which it is clear that ˜gis finitely distributed.
The assertion (ii) is contained in Lemma 2.17.
Proof of (iii). Consider the characteristic function
ψ(n, t) = 1
G(n)exp(−itα(n)) X
a∈G,∂(a)=n
exp(it˜g(a)).
57
CHAPTER 3. APPLICATIONS
In the same way as above, we can show, that ψ(n, t) converges for n→ ∞ using
Proposition 2.6 and taking in account the property G(n−1)
G(n)=q−1+o(1) as n→ ∞.
Therefore the frequencies
1
G(n)#{a∈G, ∂(a) = n: ˜g(a)−α(n)≤x}
converge to a weak limit as n→ ∞.
This ends the proof of Theorem 3.1.
3.2 Three-series theorem
In this section we present our version of the well-known Three-series theorem under
weak conditions about the additive arithmetical semigroups. We remark here, that
in a paper of Bar´at, Indlekofer and Kaya (see [3]), the authors prove the Two-
series theorem in additive arithmetical semigroups and pull together the properties
of finitely distributed functions and the characterisation of essentially convergent
series in the Stone-Cech compactification of G. Some ideas of the construction of
the described Stone-Cech compactification in [3] were motivated by the construction
described by Bar´at and Indlekofer (see [2]).
Theorem 3.2. (Three-series theorem).
Suppose that an additive arithmetical semigroup (G, ∂)fulfills the assumptions of
Theorem 2.11. A real-valued additive function ˜gon Ghas a limit distribution func-
tion G(x)if and only if the three series
(3.6) X
|˜g(p)|≥1
q−∂(p),X
|˜g(p)|<1
˜g(p)q−∂(p),X
|˜g(p)|<1
˜g2(p)q−∂(p)
all converge. Moreover, the limit distribution function G(x)has the characteristic
function
(3.7) φ(t) = Y
p1−q−∂(p) 1 +
∞
X
k=1
q−k∂(p)eit˜g(pk)!,
where the infinite product is taken over all p∈Pin ascending order of ∂(p).
Proof. Assume that real-valued additive function ˜gon Ghas a limit distribution
function G(x). Assertions (i) and (ii) of Theorem 3.1 yields that the first and the
third series of (3.6)
X
|˜g(p)|≥1
q−∂(p),X
|˜g(p)|<1
˜g(p)q−∂(p)
58
CHAPTER 3. APPLICATIONS
converge. By our assumption ˜ghas a limit distribution function G(x), on the other
hand, (iii) yields that 1
G(n)#{a∈G, ∂(a) = n: ˜a−α(n)≤x}tends to a weak limit,
too. Then:
α(n) = cn +X
∂(p)≤n,|˜g(p)|≤1
˜g(p)
q∂(p)
converges and, since
X
∂(p)≤n,|˜g(p)|≤1
˜g(p)
q∂(p)=O(log n)
cmust be zero, which implies the convergence of
X
p,|˜g(p)|<1
˜g(p)q−∂(p).
Put
α(n) = X
∂(p)≤n,|˜g(p)|≤1
˜g(p)
q∂(p).
Then (iii) of Theorem 3.1 yields, that the frequencies
1
G(n)#{a∈G, ∂(a) = n: ˜g(a)−α(n)≤x}
has a limit distribution D(x) as n→ ∞; and lim
n→∞ α(n) = αexists, then ˜galso has
a limit distribution D(x−α).
This ends the proof of Theorem 3.2
59
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63
Index
additive arithmetical semigroup, 4
additive function, 16
asymptotic density, 17
Axiom A#, 6
Axiom ¯
A#, 6
Axiom A, 8
Axiom A1, 8
Axiom A2, 9
characteristic function
of a limit distribution, 56
of a set, 12
Chebyshev upper estimate, 10
completely multiplicative function, 16
convolution, 11
degree mapping, 4
distribution function, 56
Euler product formula, 15
finitely distributed function, 29, 30
generating function, 13
of G(zeta function of G), 5
prime, 14
M¨obius function, 12
mean-value of a function, 17
multiplicative function, 16
summatory function, 27
uniformly summable function, 22
von Mangoldt’s
coefficients, 15
function, 13
64
