Remarks on number theory over additive arithmetical semigroups
UDC 511 We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding $\zeta$-functions.  We use these results to prove prime number theorems and a Selberg formula for such semigroups.
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| author | Indlekofer, K.-H. Kaya, E. Indlekofer, K.-H. Kaya, E. |
| author_facet | Indlekofer, K.-H. Kaya, E. Indlekofer, K.-H. Kaya, E. |
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| description | UDC 511
We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding $\zeta$-functions.  We use these results to prove prime number theorems and a Selberg formula for such semigroups. |
| doi_str_mv | 10.37863/umzh.v72i3.6042 |
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UDC 511
K.-H. Indlekofer* (Faculty Comput. Sci., Electric. Eng. and Math. Univ. Paderborn, Germany),
E. Kaya** (Mersin Univ., Turkey)
REMARKS ON NUMBER THEORY
OVER ADDITIVE ARITHMETICAL SEMIGROUPS
ЗАУВАЖЕННЯ ДО ТЕОРIЇ ЧИСЕЛ
ДЛЯ АДИТИВНИХ АРИФМЕТИЧНИХ НАПIВГРУП
We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the
corresponding \zeta -functions. We use these results to prove prime number theorems and a Selberg formula for such semigroups.
Розглянуто додатковi арифметичнi напiвгрупи i наведено старi та новi доведення для розподiлу нулiв вiдповiдних \zeta -
функцiй. Цi результати використано для доведення теорем простих чисел i формули Сельберга для таких напiвгруп.
1. Introduction. Abstract analytic number theory has arisen first as a generalization
of the classical number theory on the (semigroup) \BbbN of natural numbers with the special emphasis
on the derivation of the famous Prime Number Theorem: If \pi (x) denotes the total number of positive
rational primes \leq x, then \pi (x) \sim x
\mathrm{l}\mathrm{o}\mathrm{g} x
as x \rightarrow \infty ,
and
of Landau’s classical Prime Ideal Theorem, which extends the Prime Number Theorem to the
(semigroup) GK of integral ideals in an algebraic number field K.
There are a large number of mathematical systems, particularly ones arising in abstract algebra,
which have elementary “unique factorization” properties analogous to those of the positive integers.
In the case of many of these systems there is an additional property, which make them more
“arithmetical” in a sense, and more treatable by techniques of classical number theory. This property
comes from the existence of a function measuring the “size” of an individual object (usually the
object’s cardinality or “degree” in some sense) with the essential attribute that there are only a finite
number of inequivalent objects whose “size” does not exceed any chosen bound.
Motivated by such systems Knopfmacher [17] introduced the formal concept of an additive
arithmetical semigroup, which he defined to be a commutative semigroup G with an (additive) degree
mapping on G. To be more precise, let G be a free commutative semigroup with identity element
1, generated by a countable set P of primes and admitting an integer valued degree mapping \partial :
G \rightarrow \BbbN \cup \{ 0\} with the properties
(i) \partial (1) = 0 and \partial (p) > 0 for all p \in P,
(ii) \partial (ab) = \partial (a) + \partial (b) for all a, b \in G,
(iii) the total number G(n) of elements a \in G of degree \partial (a) = n is finite for each n \geq 0.
Then (G, \partial ) is called an additive arithmetical semigroup. Obviously, G(0) = 1 and G is countable.
* K.-H. Indlekofer was supported by a grant of Deutsche Forschungsgemeinschaft (Project Number 289386657).
** E. Kaya was supported by the Research Fund of Mersin University in Turkey with Project Number MEU-2015-
COL-00167-LIREK.
c\bigcirc K.-H. INDLEKOFER, E. KAYA, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3 371
372 K.-H. INDLEKOFER, E. KAYA
Let
\pi (n) := \# \{ p \in P : \partial (p) = n\}
denote the total number of primes of degree n in G. If we consider G(n) together with its associated
enumeration function
F (y) := 1 +
\infty \sum
n=1
G(n)yn,
then the identity, at least in the formal sense,
F (y) = 1 +
\infty \sum
n=1
G(n)yn =
\infty \prod
n=1
(1 - yn) - \pi (n)
holds.
F (y) is called the generating function (or zeta function) of the additive arithmetical semigroup G.
Remark 1. It is worthwhile to mention here some properties of the sequences \gamma (n), \beta (n) \in \BbbR ,
which are formally related by
1 +
\infty \sum
n=1
\gamma (n)yn =
\infty \prod
n=1
(1 - yn) - \beta (n)
and which have been used in [14, p. 448]:
\beta (n) \in \BbbZ for all n \in \BbbN \leftrightarrow \gamma (n) \in \BbbZ for all n \in \BbbN (1.1)
and
\beta (n) \geq 0 for all n \in \BbbN \Rightarrow \gamma (n) \geq \beta (n) for all n \in \BbbN . (1.2)
Condition (1.2) is obvious. Concerning (1.1) we assume that \gamma (n) is integer-valued for all n \in \BbbN .
Then
1 +
\infty \sum
n=1
\gamma (n)yn = (1 - y) - \beta (1)(1 + \alpha y2 + . . .) = 1 + \beta (1)y + . . .
and \beta (1) \in \BbbZ . Now, assume that k \in \BbbN \setminus \{ 1\} is the smallest number such that \beta (k) /\in \BbbZ . Then\Biggl(
1 +
\infty \sum
n=1
\gamma (n)yn
\Biggr) \Biggl(
k - 1\prod
m=1
(1 - ym)\beta (m)
\Biggr)
=
= (1 - yk) - \beta (k)(1 + \alpha \prime yk+1 + . . .) = 1 + \beta (k)yk + . . . . (1.3)
The left-hand side of (1.3) is a power series with integer coefficients, and, thus, \beta (k) \in \BbbZ . This
contradiction proves \beta (n) \in \BbbZ for all n \in \BbbN . The implication of the opposite direction in (1.1) is
obvious. A corresponding result for the formal representation
1 +
\infty \sum
n=1
\gamma (n)yn =
\infty \prod
n=1
(1 + \varrho (n)yn)
may be found in [15] (Proposition 1).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 373
The following examples show that degree functions arise in various forms.
Example 1 (monic polynomials over GF (q)). A simple example of an additive arithmetical
semigroup is provided by the multiplicative semigroup Gq of all monic polynomials in one inde-
terminate X over the finite field GF (q) with q elements and with \partial (a) = \mathrm{d}\mathrm{e}\mathrm{g}(a) for a \in Gq, where
\mathrm{d}\mathrm{e}\mathrm{g}(a) denotes the degree of the polynomial a. Here G(n) = qn, and the generating function may
be written as
F (y) =
\infty \sum
n=0
qnyn =
\infty \prod
n=1
(1 - yn) - \pi q(n) ,
and
\pi q(n) =
1
n
\sum
r| n
\mu (r)qn/r
can be deduced as an algebraic consequence of the Euler product for F (y).
Example 2 (multisets). Let P be a finite or denumerable set. Following Flajolet and Sedgewick
[6], we use the notation
SEQ(P ) :=
\bigl\{
(p1, . . . , pl) : l \geq 0, pi \in P, i = 1, . . . , l
\bigr\}
,
where the element for l = 0 corresponds to the identity element 1 and the size | a| of an object
a \in SEQ(P ) is a nonnegative integer and is to be taken as the sum of the sizes of its components,
a = (p1, . . . , pl) \mapsto \rightarrow | a| = | p1| + . . .+ | pl| .
Then we define the multiset MSET (P ) as the quotient
MSET (P ) := SEQ(P )/R,
where the equivalence relation R is defined by (p1, . . . , pr)R(q1, . . . , qr) if there exists some arbi-
trary permutation \tau of 1, . . . , r such that qj = p\tau (j) for all j (see [6, p. 26]). Obviously, a multiset
MSET (P ) (together with a size function | \cdot | ) is nothing else but an additive arithmetical semigroup
G with G = MSET (P ), where P denotes the set of primes, and the degree \partial (p) is given by the
size | p| . Then \pi (n) is the number of objects in P that have size n.
Example 3 (partitions). As a special example of a multiset we choose P = \BbbN with \partial (j) = j
for j \in \BbbN . If a = (n1, . . . , nl) \in MSET (\BbbN ), then \partial (a) = n1 + . . .+ nl. Obviously,
\pi (n) = \#\{ p \in P : \partial (p) = n\} = 1
and
G(n) = p(n) := number of partitions of n.
The generating function F (y) is given by
F (y) = 1 +
\infty \sum
n=1
p(n)yn =
\infty \prod
n=1
(1 - yn) - 1
and converges for | y| < 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
374 K.-H. INDLEKOFER, E. KAYA
Example 4 (monic polynomials over GF (q) in several indeterminates). For an integer k \geq 2,
let Gk,q be the multiplicative semigroup of all monic polynomials in k indeterminates X1, . . . , Xk
over a finite field GF (q), where \partial (f) is the degree of f \in Gk,q. The prime elements are the (monic)
irreducible polynomials in Gk,q. Carlitz [4] has proven that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\pi (n)
G(n)
= 1. (1.4)
This shows, that “almost all” elements of Gk,q are primes in the sense that G(n) > 0 for sufficiently
large n and \pi (n) \sim G(n) as n \rightarrow \infty . Further, by a result of Wright (see Theorem 3 in [20]), (1.4)
implies that the generating function
F (y) = 1 +
\infty \sum
n=1
G(n)yn =
\infty \prod
n=1
(1 - yn) - \pi (n)
diverges for all y \in \BbbC .
In order to develop an arithmetical theory we assume that
G(n) \ll qnn\varrho with some q > 1, \varrho \in \BbbR ,
so that F (y) is holomorphic for | y| < q - 1. The logarithmic derivate of F is given by
F \prime (y)
F (y)
=
\infty \sum
n=1
\Biggl( \sum
d| n
d\pi (d)
\Biggr)
yn - 1. (1.5)
Putting
\lambda (n) =
\sum
d| n
d\pi (d),
gives
F (y) = 1 +
\infty \sum
n=1
G(n)yn = \mathrm{e}\mathrm{x}\mathrm{p}
\Biggl( \infty \sum
n=1
\lambda (n)
n
yn
\Biggr)
.
By Möbius inversion formula
n\pi (n) =
\sum
d| n
\lambda (d)\mu
\Bigl( n
d
\Bigr)
.
A straightforward calculation (see [9, p. 86]) shows that
\lambda (n) = n\pi (n) +O
\Bigl(
nqn/2
\Bigl( n
2
\Bigr) \varrho
\mathrm{l}\mathrm{o}\mathrm{g} n
\Bigr)
.
An abstract prime number theorem (for additive arithmetical semigroups) is a theorem about the
asymptotic behavior of \pi (n) and \lambda (n), respectively.
Further, we shall concentrate on additive arithmetical semigroups whose zeta function F can be
written in the form
F (y) =
H(y)
(1 - qy)\delta
with some \delta > 0,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 375
where H(y) is holomorphic for | y| < q - 1 and continuous for | y| \leq q - 1 and H(q - 1) \not = 0. By a
result of Indlekofer [11] this implies H(q - 1) > 0.
F (y) has no zeros in the disk \{ y \in \BbbC : | y| < q - 1\} but may have zeros on the circle \{ y \in \BbbC :
| y| = q - 1\} . If q - 1e2\pi it is a zero of F (y), the number
\alpha (t) := \mathrm{s}\mathrm{u}\mathrm{p}
\Biggl\{
\alpha : \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\nearrow q - 1
(q - 1 - r) - \alpha | F (re2\pi it)| < \infty
\Biggr\}
(1.6)
or, equivalently, following Beurling [3],
\alpha (t) := \mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\nearrow q - 1
\mathrm{l}\mathrm{o}\mathrm{g} | F (re2\pi it)|
\mathrm{l}\mathrm{o}\mathrm{g}(q - 1 - r)
(1.7)
is called, by definition, the order of the zero q - 1e2\pi it.
In this paper we deal with the “total number” of zeros of F (y), which is the key to the investi-
gation of the abstract prime number theorem. In the case \delta = 1, we assume in addition, that H(y)
is holomorphic for | y| < R, where R > q - 1. In the general case \delta > 0, we only assume, that H(y)
is holomorphic for | y| < q - 1 and continuous for | y| \leq q - 1, and H(q - 1) > 0.
As applications we present several abstract prime number theorems and a corresponding Selberg
formula.
Remark 2. In several papers we have investigated the mean behavior of additive and multi-
plicative functions on additive arithmetical semigroups (see [1, 2, 10, 16]). In this context it is
recommendable, to deal with probabilistic aspects, as it has been done, for example, in [12] and [13]
for functions defined on the (multiplicative) semigroup \BbbN of natural numbers.
2. Zeros of the \bfitzeta -function. We put
F (y) =
H(y)
(1 - qy)\delta
, \delta > 0, (2.1)
and consider the special case \delta = 1 separately from the general case \delta > 0.
2.1. The case \bfitdelta = \bfone . When Knopfmacher [17] introduced the concept of the additive arithmeti-
cal semigroup his investigations are based on the following axiom.
Axiom \bfscrA \# . There exist constants A > 0, q > 1 and \nu with 0 \leq \nu < 1 (all depending on G),
such that
G(n) = Aqn +O(q\nu n) as n \rightarrow \infty .
If G satisfies Axiom \scrA \# , then the generating function
F (y) =
\infty \sum
n=0
G(n)yn
is holomorphic in the disc | y| < q - \nu up to a pole of order one at y = q - 1, and we get
F (y) =
A
1 - qy
+H1(y),
where
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
376 K.-H. INDLEKOFER, E. KAYA
H1(y) =
\infty \sum
n=0
rny
n
with
rn := G(n) - Aqn.
Putting
H(y) := A+ (1 - qy)H1(y),
gives
F (y) =
H(y)
1 - qy
with H(0) = 1 and H(q - 1) = A. H and H1 are holomorphic for | y| < q - \nu .
Chapter 8 of [17] deals with the abstract prime number theorem:
If the additive arithmetical semigroup G satisfies Axiom \scrA \#, then
\lambda (n) = qn +O
\biggl(
qn
n\alpha - 1
\biggr)
, n \rightarrow \infty ,
or, equivalently,
\pi (n) =
qn
n
+O
\biggl(
qn
n\alpha
\biggr)
, n \rightarrow \infty ,
is true for any \alpha > 1.
Note that this result is only valid if F ( - q - 1) \not = 0. In [14], Indlekofer, Manstavičius and War-
limont gave (in a more general setting) much sharper results valid also in the case F ( - q - 1) = 0.
We write y = re2\pi it and assume that there is some R > q - 1 such that H(y) is holomorphic for
r < R and H(q - 1) \not = 0. If H(y) \not = 0 for | y| \leq q - 1, then there exists some r, q - 1 < r < R, such
that H(y) \not = 0 for | y| \leq r. This is contained in the following theorem.
Theorem 1 ([14], Theorem 1). If H(y) \not = 0 for | y| = q - 1, then there exists 0 < \vargamma < 1 such
that H(y) \not = 0 for | y| \leq q - \vargamma .
If F (y) has zeros on the circle \{ y \in \BbbC : | y| = q - 1\} , then the following two theorems hold.
Theorem 2 ([14], Theorem 1). If F (y) has zeros of modulus q - 1, then F has exactly one zero
in the disk \{ y \in \BbbC : | y| < R\} . It is located at y = - q - 1 and has order 1.
Theorem 3 ([14], Theorem 2). Asssume R \geq q - 1/2. If F ( - q - 1) = 0 we have
F (r)F ( - r) \gg
\bigl(
1 - q
1
2 r
\bigr) - 1
for r \nearrow q - 1/2. (2.2)
The main part of the proof of Theorem 3 is to show that
F (r)F ( - r) \geq F (r2) for r \nearrow q - 1/2, (2.3)
from which the assertion (2.2) follows immediately since H(q - 1) > 0. Furthermore, (2.3) implies
directly the following corollary.
Corollary 1 ([14], Corollary 2). Assume R \geq q - 1/2. If
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\nearrow q - 1/2
(1 - rq1/2)F (r)F ( - r) \leq 0,
then F (y) has no zeros for r = q - 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 377
The investigation continued with the restriction that H(y) is holomorphic for | y| < q - 1 and
continuous for | y| \leq q - 1. By a result in [11] this implies H(q - 1) > 0. Zhang required assumptions
on the coefficients of the \zeta -function (see [21]) whereas Indlekofer assumed in [8] a specific boundary
behavior of H. This reads as the following axiom.
Axiom \bfitA \bfone [8]. H(y) is continuous for | y| \leq q - 1, H \prime (y) is bounded for | y| < q - 1 .
Axiom A1 is much less restricted than Axiom A\# and seems to be the weakest condition known
today which ensures a Chebyshev type upper estimate \pi (n) \ll qn
n
. Some conclusions which can be
derived from Axioam A1 without appealing to Axiom A\# are given in the following theorem.
Theorem 4 ([8], Theorem 1). If Axiom A1 holds, then either H(y) \not = 0 for every | y| \leq q - 1 or
f(y) :=
H(y)
1 + qy
defines a function f, which is holomorphic for | y| < q - 1 and continuous and different from zero for
all | y| \leq q - 1 .
2.2. The case \bfitdelta > \bfzero . First, we prove the following theorem.
Theorem 5. Let
g(y) =
h(y)
(1 - y)\delta
, \delta > 0,
be holomorphic and different from zero for | y| < 1. Assume that h(y) is continuous for | y| \leq 1 and
h(1) > 0. Also assume that
\mathrm{l}\mathrm{o}\mathrm{g} g(y) =
\infty \sum
m=1
amym, | y| < 1, (2.4)
where the coefficients am are nonnegative. Let 0 < t1 < . . . < tn < 1 be given. Then
n\sum
j=1
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\nearrow 1 -
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi itj )|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
\leq \delta . (2.5)
Remark 3. This theorem has been proven by Zhang in [21] (Theorem 4.2). Here we give a
different proof which is shorter and uses ideas from Beurling’s paper [3].
Proof of Theorem 5. We start with the identity (0 < r < 1)
p\prod
i=1
1 - r2
1 - 2r \mathrm{c}\mathrm{o}\mathrm{s} yi + r2
=
\sum
\nu 1\in \BbbZ
. . .
\sum
\nu p\in \BbbZ
r| \nu 1| +...+| \nu p| \mathrm{c}\mathrm{o}\mathrm{s}(\nu 1y1 + . . .+ \nu pyp) (2.6)
(cf. [3, p. 281]). Put
\bfitnu := (\nu 1, . . . , \nu p) \in \BbbZ p,
\| \bfitnu \| :=
p\sum
i=1
| \nu i|
and let 0 < t1 < . . . < tn < 1. Consider a maximal subset of elements, which are linearly inde-
pendent over \BbbQ , of the set \{ t1, . . . , tn\} . Then there exist s1, . . . , sp, which are linearly independent
over \BbbQ , such that
tj =
p\sum
i=1
cjisi, j = 1, . . . , n, (2.7)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
378 K.-H. INDLEKOFER, E. KAYA
with cji \in \BbbZ , j = 1, . . . , n, i = 1, . . . , p. If the t1, . . . , tn are linearly independent then choose
sj = tj , j = 1, . . . , n, and
cji =
\left\{ 1, j = i,
0, j \not = i.
Put in (2.6)
yi = siy, i = 1, . . . , p.
Then for every \varrho , 0 < \varrho < 1, there exists N0 = N0(\varrho ), such that for every N \geq N0\sum
\| \bfitnu \| >N
\bfitnu \in \BbbZ p
\varrho \| \bfitnu \| \leq \varrho N
\sum
vvv\in \BbbZ p
\varrho \| \bfitnu \| \leq
\biggl(
1 - \varrho
1 + \varrho
\biggr) p
and, by (2.6),
0 \leq P (y) :=
\sum
\| \bfitnu \| \leq N
\varrho \| \bfitnu \| \mathrm{c}\mathrm{o}\mathrm{s}(\bfitnu , \bfits )y, (2.8)
where
\langle \bfitnu , \bfits \rangle = \nu 1s1 + . . .+ \nu psp.
Let y = 2\pi m with m \in \BbbN . We multiply (2.8) by amrm and sum over m. Then, by (2.4), for
0 < r < 1,
0 \leq
\sum
\| \bfitnu \| \leq N
\langle \bfitnu ,\bfits \rangle \in \BbbZ
\varrho \| \bfitnu \| \mathrm{l}\mathrm{o}\mathrm{g} g(r) +
\sum
\| \bfitnu \| \leq N
\langle \bfitnu ,\bfits \rangle /\in \BbbZ
\varrho \| \bfitnu \| \mathrm{l}\mathrm{o}\mathrm{g}
\bigm| \bigm| g(re2\pi i\langle \bfitnu ,\bfits \rangle )\bigm| \bigm| .
It follows that \sum
\| \bfitnu \| \leq N
\langle \bfitnu ,\bfits \rangle /\in \BbbZ
\varrho \| \bfitnu \|
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi i\langle \bfitnu ,\bfits \rangle )|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
\leq
\sum
\| \bfitnu \| \leq N
\langle \bfitnu ,\bfits \rangle \in \BbbZ
\varrho \| vvv\|
\mathrm{l}\mathrm{o}\mathrm{g} g(r)
\mathrm{l}\mathrm{o}\mathrm{g}
1
1 - r
. (2.9)
Now, define
T0 :=
\bigl\{
\bfitnu \in \BbbZ p : \langle \bfitnu , \bfits \rangle \in \BbbZ
\bigr\}
and, for j = 1, . . . , n,
Tj :=
\bigl\{
\bfitnu \in \BbbZ p : \langle \bfitnu , \bfits \rangle + tj \in \BbbZ
\bigr\}
.
Obviously, putting \bfitnu \bfitj = (cj1, . . . , cjp),
Tj := \{ \bfitnu + \bfitnu \bfitj : \bfitnu \in T0\} .
Note that T0, T1, . . . , Tn are mutually disjoint. From (2.9) we have
n\sum
j=1
\sum
\| \bfitnu \| \leq N
\bfitnu \in Tj
\varrho \| \bfitnu \|
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi itj )|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
+
\sum
\| \bfitnu \| \leq N
\bfitnu /\in T0\cup (
\bigcup n
j=1
Tj)
\varrho \| \bfitnu \|
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi i\langle \bfitnu ,\bfits \rangle )|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
\leq
\sum
\| \bfitnu \| \leq N
\bfitnu \in T0
\varrho \| \bfitnu \|
\mathrm{l}\mathrm{o}\mathrm{g} g(r)
\mathrm{l}\mathrm{o}\mathrm{g}
1
1 - r
.
Note that
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1 -
\mathrm{l}\mathrm{o}\mathrm{g} g(r)
\mathrm{l}\mathrm{o}\mathrm{g}
1
1 - r
= \delta
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REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 379
and
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow 1 -
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi i(\bfitnu ,\bfits ))|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
\geq 0
if \langle \bfitnu , \bfits \rangle /\in \BbbZ . This implies, letting N \rightarrow \infty ,
n\sum
j=1
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow 1 -
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi itj )|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
\sum
\bfitnu \in Tj
\varrho \| \bfitnu \| \leq \delta
\sum
\bfitnu \in T0
\varrho \| \bfitnu \| . (2.10)
Now, either
T0 = \{ \bfzero \}
or there exists \bfitnu \bfzero \in \BbbZ p such that
T0 = \{ \bfitnu = t\bfitnu \bfzero : t \in \BbbZ \} . (2.11)
For the proof of (2.11), assume that, for some \bfitnu \in T0, \langle \bfitnu , \bfits \rangle = a \in \BbbZ and a \not = 0. Then the set
A :=
\bigl\{
a : 0 < a = \langle \bfitnu , \bfits \rangle for some \bfitnu \in T0
\bigr\}
is non-empty and has a minimal element a0 := \mathrm{m}\mathrm{i}\mathrm{n}\{ a :
a \in A\} with a0 = \langle \bfitnu \bfzero , \bfits \rangle for some (unique) \bfitnu \bfzero \in \BbbZ p. Since s1, . . . , sp are linearly independent,
\bfitnu \in T0 if and only if there exist t \in \BbbZ such that \bfitnu = t\bfitnu \bfzero .
If T0 = \{ \bfzero \} , then, by (2.10),
n\sum
j=1
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}
r\rightarrow 1 -
\mathrm{l}\mathrm{o}\mathrm{g} | g(re2\pi itj )|
\mathrm{l}\mathrm{o}\mathrm{g}(1 - r)
\varrho \| \bfitnu \bfitj \| \leq \delta \varrho ,
and letting \varrho \rightarrow 1 - gives the assertion (2.5) of Theorem 5. In the case (2.11) we put\sum
0
:=
\sum
\bfitnu \in T0
\varrho \| \bfitnu \| =
\sum
t\in \BbbZ
\varrho | t| \| \bfitnu \bfzero \| , (2.12)
\sum
j
:=
\sum
\bfitnu \in Tj
\varrho \| \bfitnu \| =
\sum
t\in \BbbZ
\varrho \| \bfitnu \bfitj +t\bfitnu \bfzero \| (2.13)
and arrive at \sum
0
= 1 + 2
\sum
n\in \BbbN
\varrho n\| \bfitnu \bfzero \| =
1
1 - \varrho \| \bfitnu \bfzero \|
\bigl(
1 - \varrho \| \bfitnu \bfzero \| + 2\varrho \| \bfitnu \bfzero \| \bigr) .
In the same way we obtain
\varrho \| \bfitnu \bfitj \|
\sum
0
\leq
\sum
j
\leq \varrho - \| \bfitnu \bfitj \|
\sum
0
, j = 1, . . . , n,
which implies
\mathrm{l}\mathrm{i}\mathrm{m}
\varrho \rightarrow 1
\sum
j\sum
0
= 1. (2.14)
Then (2.14) proves, by (2.10), (2.12) and (2.13), the assertion of Theorem 5.
We apply Theorem 5 to the \zeta -function F (y). Observe that, if y = q - 1e2\pi it, 0 < t <
1
2
, is a
zero of F (y), then y = q - 1e - 2\pi it is a zero of F (y), too. By using the notations (1.6) and (1.7),
respectively, we arrive at the following theorem.
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380 K.-H. INDLEKOFER, E. KAYA
Theorem 6. Let the \zeta -function F (y) of G be given by (2.1), where H(y) is holomorphic for
| y| < q - 1 and continuous for | y| \leq q - 1, and H(q - 1) = 0. Then
\alpha
\biggl(
1
2
\biggr)
+
\sum
0<t< 1
2
\alpha (t) \leq \delta
or
2
\sum
0<t< 1
2
\alpha (t) \leq \delta
according as y = q - 1 is or is not a zero of F (y).
Theorem 7. Let k \in \BbbN be defined by
k :=
\left\{ [\delta ] + 1, if \delta /\in \BbbN ,
\delta , if \delta \in \BbbN .
Assume that H(k)(y) = O(1) for | y| < q - 1. Then the order \alpha of a zero of H is a positive integer.
Proof. All derivatives H(j)(y) with j < k are continuous for | y| \leq q - 1. Hence, we obtain by
Taylor’s formula for 0 < r < q - 1
H(re2\pi i\vargamma ) =
k - 1\sum
n=1
1
n!
H(n)
\bigl(
q - 1e2\pi i\vargamma
\bigr)
e2\pi in\vargamma (r - q - 1)n+
+
1
(k - 1)!
r\int
q - 1
H(k)
\bigl(
te2\pi i\vargamma
\bigr)
e2\pi ik\vargamma (r - t)k - 1dt. (2.15)
The last term of (2.15) can be estimated by O
\biggl(
1
k!
(r - q - 1)k
\biggr)
.
Suppose that \alpha is not an integer, i.e.,
m - 1 < \alpha < m \leq k.
Then by (2.15), H(j)
\bigl(
q - 1e2\pi i\vargamma
\bigr)
= 0 for j < m and
H(re2\pi i\vargamma )
(q - 1 - r)\alpha \prime = O
\Bigl(
(q - 1 - r)m - \alpha \prime
\Bigr)
as r \rightarrow q - 1
for every \alpha \prime , \alpha < \alpha \prime \leq m, which contradicts the assumption on \alpha . Therefore, \alpha must be a positive
integer.
By Theorems 6 and 7 we obtain the following corollaries.
Corollary 2. Let \delta = 1 and H \prime (y) = O(1) for | y| < q - 1. Then H(y) has either no zeros on the
circle | y| = q - 1, or exactly one zero y = - q - 1 of order one.
Corollary 2 is contained in Theorem 4.
Corollary 3. Let 0 < \delta < 1 and H \prime (y) = O(1) for | y| < q - 1. Then H(y) has no zeros on the
circle | y| = q - 1.
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REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 381
Corollary 4. Let 1 < \delta < 2 and H \prime \prime (y) = O(1) for | y| < q - 1. Then H(y) has either no zeros
on the circle | y| = q - 1, or exactly one zero y = - q - 1 of order one.
The condition H \prime (y) = O(1) implies that \alpha (\nu ) \geq 1 if y = q - 1e2\pi i\nu is a zero of H(y). Thus the
following result holds.
Corollary 5. Let 1 < \delta < 2 and H \prime (y) = O(1) for | y| < q - 1. Then H(y) has either no zeros
on the circle | y| = q - 1, or exactly one zero y = - q - 1 of order \leq \delta .
The following example could be considered in the context of Theorem 6 and Corollary 3, whereas
Example 6 illuminates the strength of the condition in Corollary 2.
Example 5. Let r1, s1, r2, s2 be positive integers such that \delta =
r1
s1
, \alpha =
r2
s2
\in \BbbQ with 0 < \alpha <
< \delta . Consider q = ms1s2, where m > 2
\delta
r1s2 - s1r2
+ 1 holds. Put
\lambda (n) = qn
\bigl(
\delta + ( - 1)n+1\alpha
\bigr)
, n = 1, 2, . . . .
Then the \lambda (n)’s are all positive. Furthermore, \pi (1) = \lambda (1) > 0 and, for n \geq 2,
n\pi (n) =
\sum
d| n
\lambda (d)\mu
\biggl(
n
d
\biggr)
\geq \lambda (n) -
\sum
1\leq d\leq n
2
\lambda (d) \geq
\geq qn(\delta - \alpha ) - 2\delta
\sum
1\leq d\leq n
2
qd \geq qn(\delta - \alpha ) - 2\delta
qq
n
2
q - 1
=
= q
n
2
\biggl(
q
n
2 (\delta - \alpha ) - 2\delta q
q - 1
\biggr)
> (\delta - \alpha )(q - 1) - 2\delta > 0. (2.16)
A straightforward calculation shows that
y \cdot F
\prime (y)
F (y)
=
\infty \sum
n=1
\lambda (n)yn =
\delta qy
1 - qy
+
\alpha qy
1 + qy
and
F (y) =
(1 + qy)\alpha
(1 - qy)\delta
, | y| < q - 1,
where 0 < \alpha < \delta . Then G(0) = 1 and
G(n) = qn
n\sum
k=0
\Biggl(
k + \delta - 1
\delta - 1
\Biggr) \biggl(
\alpha
n - k
\biggr)
, n = 1, 2, . . . ,
are positive integers. Therefore, by Remark 1 and (2.16), \pi (n) \in \BbbN .
Example 6. This example is motivated by Example 6.5 of [22]. Consider
\lambda (n) = (1 - \mathrm{c}\mathrm{o}\mathrm{s}n\theta ) qn, n = 1, 2, . . . , (2.17)
where q = 2 \cdot 54 and \mathrm{c}\mathrm{o}\mathrm{s} \theta =
4
5
. Since
\mathrm{s}\mathrm{i}\mathrm{n}n\theta = \mathrm{I}\mathrm{m}
\biggl( \biggl(
4 + 3i
5
\biggr) n\biggr)
= \mathrm{I}\mathrm{m}
\biggl( \biggl(
in
5n
\biggr)
(3 - 4i)n
\biggr)
=
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382 K.-H. INDLEKOFER, E. KAYA
=
\left\{
\pm 1
5n
\biggl(
3n -
\biggl(
n
2
\biggr)
3n - 242 + . . .
\biggr)
, if n is odd,
1
5n
\biggl( \biggl(
n
1
\biggr)
4n - 13 -
\biggl(
n
3
\biggr)
4n - 333 + . . .
\biggr)
, if n is even,
we see that \mathrm{s}\mathrm{i}\mathrm{n}n\theta \not = 0. Thus 1 - \mathrm{c}\mathrm{o}\mathrm{s}n\theta > 0. Furthermore, we have
1 - \mathrm{c}\mathrm{o}\mathrm{s}n\theta =
1
5n
(5n - \mathrm{R}\mathrm{e}(4 + 3i)n) \geq 1
5n
, n = 1, 2, . . . .
Since 5n - \mathrm{R}\mathrm{e}(4 + 3i)n \in \BbbZ we obtain that \lambda (n) is a positive integer for every n \in \BbbN . Now, we
shall show that
\pi (n) =
1
n
\sum
d| n
(1 - \mathrm{c}\mathrm{o}\mathrm{s} d\theta )qd\mu
\biggl(
n
d
\biggr)
is a positive integers for every n \geq 1, too. Obviously,
n\pi (n) =
\sum
d| n
\lambda (d)\mu
\biggl(
n
d
\biggr)
\geq (1 - \mathrm{c}\mathrm{o}\mathrm{s}n\theta )qn - 2
\sum
1\leq d\leq n
2
qd \geq
\geq q
3n
4 - 2q1+
n
2
q - 1
\geq q
3n
4 - 4q
n
2 > 0 (2.18)
since q = 2 \cdot 54. We observe that, by formulae (1.5) and (2.17), the generating function F (y) is given
by
F (y) =
(1 - ei\theta qy)
1
2 (1 - e - i\theta qy)
1
2
1 - qy
=
\infty \sum
n=0
G(n)yn, (2.19)
and has two zeros q - 1e\pm \theta . Squaring both sides of (2.19) we conclude
F 2(y) =
(1 - ei\theta qy)(1 - e - i\theta qy)
(1 - qy)2
=
\infty \sum
n=0
\left( \sum
m\leq n
G(m)G(n - m)
\right) yn.
By (6.15) in Example 6.5 of [22],\sum
m\leq n
G(m)G(n - m) = 2(1 - \mathrm{c}\mathrm{o}\mathrm{s} \theta )nqn. (2.20)
We show by induction that G(n) is an even integer number for all n \geq 1. Obviously,
G(1) = 2 \cdot 53.
Now, suppose that G(m) is even for 1 \leq m \leq n. Then\sum
m\leq n+1
G(m)G(n+ 1 - m) = G(0)G(n+ 1) +G(0)G(n+ 1) +
\sum
1\leq m\leq n
G(m)G(n+ 1 - m) \Rightarrow
\Rightarrow 2G(n+ 1) +
\sum
1\leq m\leq n
G(m)G(n+ 1 - m) = 2(1 - \mathrm{c}\mathrm{o}\mathrm{s} \theta )(n+ 1)qn+1 \Rightarrow
\Rightarrow 2G(n+ 1) \equiv 0 \mathrm{m}\mathrm{o}\mathrm{d} 4
and G(n+ 1) is an even integer, too. Then, by Remark 1 and (2.18), \pi (n) \in \BbbN for all n \geq 1.
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REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 383
3. Applications. 3.1. Abstract Prime Number Theorems. In the following we use the nota-
tion (2.1).
Theorem 8 ([14], Theorem 1). Let \delta = 1 in (2.1) and assume that H(y) is holomorphic for
| y| < R, where R > q - 1. If F has no zeros of modulus q - 1, then for every r,
1
q
< r < R, one has
\lambda (n)q - n = 1 -
l\sum
j=1
(qbj)
- n +O((qr) - n),
where bj , 1 \leq j \leq l = l(r), are the zeros of F with q - 1 < | bj | < r, counted according their
multiplicities.
Theorem 9 ([14], Theorem 1). Assume that the assumptions of Theorem 8 hold. If F ( - q - 1) =
= 0, then
\lambda (n)q - n = 1 - ( - 1)n +O
\bigl(
(qr) - n
\bigr)
for every r, q - 1 < r < R.
Example 7. In [14] we gave an example by choosing
\pi (n) =
\left\{ [2qn/n], if n is odd,
0, even.
Then F (y) is given by
F (y) =
1 + qy
1 - qy
\biggl(
1 + qy2
1 - qy2
\biggr) 1/2
H1(y),
where H1(y) is holomorphic and different from zero for | y| \leq q - 1/2.
In the case \delta = 1, we assume that Axiom \scrA 1 is fulfilled. Then we have the following theorem.
Theorem 10 ([8], Theorem 2). Let \delta = 1 in (2.1) and let Axiom A1 holds. If H( - q - 1) \not = 0,
then
\lambda (n)
qn
= 1 + o(1) as n \rightarrow \infty
and, if H( - q - 1) = 0, then
\lambda (n)
qn
+
\lambda (n - 1)
qn - 1
= 2 + o(1) as n \rightarrow \infty .
Furthermore, the asymptotic formula\sum
m\leq n
\lambda (m)q - m = n+ o(n1/2) as n \rightarrow \infty
holds.
Observe that Theorem 10 ensures only a Chebyshev type upper estimate \lambda (n) \ll qn.
A small change of Axiom \scrA 1 leads to the abstract prime number theorem and to the asymptotic
formula (3.3) with remainder term o(1). This modification is contained in the following axiom.
Axiom \bfscrA \bftwo [8]. The conditions of Axiom \scrA 1 hold, and in addition, the power series of H \prime
converges absolutely for | y| \leq q - 1.
An immediate consequence is given in the following theorem.
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384 K.-H. INDLEKOFER, E. KAYA
Theorem 11 ([8], Theorem 3). Let \delta = 1 in (2.1). If Axiom \scrA 2 holds and if H( - q - 1) \not = 0,
then
\lambda (n)
qn
= 1 + o(1) as n \rightarrow \infty (3.1)
and, if H( - q - 1) = 0, then
\lambda (n)
qn
= 1 - ( - 1)n + o(1) as n \rightarrow \infty . (3.2)
Furthermore, the asymptotic formula\sum
m\leq n
\lambda (m)q - m = n+
H \prime (q - 1)
qH(q - 1)
+ o(1) as n \rightarrow \infty (3.3)
holds.
Theorem 12. Let 0 < \delta < 1 in (2.1) and assume that (2.1) holds for 0 < \delta < 1, and H \prime (y) =
= O(1) for | y| < q - 1. Then
\lambda (n)
qn
= \delta + o(1), n \rightarrow \infty .
Proof. The equation (1.5) implies
\infty \sum
n=1
\lambda (n)q - n =
\delta
1 - qy
+ y
H \prime (y)
H(y)
.
Put
H(y) :=
\infty \sum
n=0
h(n)yn and
1
H(y)
=
\infty \sum
n=0
h1(n)y
n,
where
\sum \infty
n=0
| h(n)| < \infty . Since H(y) \not = 0 for | y| \leq q - 1 the series
\sum \infty
n=0
h1(n)y
n converges
absolutely for | y| \leq q - 1.
We conclude
q - n
\sum
m\leq n
mh(m)h1(n - m) =
\sum
m\leq n
mh(m)q - mh1(n - m)q - (n - m) \leq
\leq
\sum
m\leq n
2
mh(m)q - mh1(n - m)q - (n - m)
\underbrace{} \underbrace{} \sum
1
+
\sum
n
2
<m\leq n
mh(m)q - mh1(n - m)q - (n - m)
\underbrace{} \underbrace{} \sum
2
.
Obviously, \sum
1
\leq O(1)
\sum
0<m\leq n
2
h1(n - m)q - (n - m) = o(1), n \rightarrow \infty ,
and \sum
2
\leq o(1)
\sum
n
2
<m\leq n
h1(n - m)q - (n - m) = o(1), n \rightarrow \infty .
Thus, the assertion of Theorem 12 holds.
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REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 385
Following Theorem 12 we put
f(y) :=
\infty \sum
n=0
\gamma (n)yn =
H(y)
1 + qy
.
With this notation we prove the following theorem.
Theorem 13. Let (2.1) holds with \delta = 1. Assume that Axiom \scrA 1 holds and H( - q - 1) = 0.
Suppose that the power series of f converges absolutely for | y| \leq q - 1 and n\gamma (n)q - n = o(1) as
n \rightarrow \infty . Then
\lambda (n)
qn
= 1 - ( - 1)n + o(1), n \rightarrow \infty .
Proof. By
F (y) =
1 + qy
1 - qy
f(y)
we have
\infty \sum
n=1
(\lambda (n)q - n - ( - 1)n - 1)qnyn = y
f \prime (y)
f(y)
.
Now, consider
y
H \prime (y)
H(y)
=
\Biggl\{ \infty \sum
n=1
nh(n)yn
\Biggr\} \Biggl\{ \infty \sum
n=0
h1(n)y
n
\Biggr\}
=
\infty \sum
n=0
\left( \sum
m\leq n
mh(m)h1(n - m)
\right) yn.
Let
1
f(y)
=:
\infty \sum
n=0
\gamma 1(n)y
n.
Since f(y) \not = 0 for | y| \leq q - 1 the series
\sum \infty
n=0
\gamma 1(n)y
n converges absolutely for | y| \leq q - 1
and n\gamma (n)q - n = o(1) as n \rightarrow \infty , the proof of Theorem 12 immediately implies the assertion of
Theorem 13.
Remark 4. If Axiom \scrA 2 holds and H( - q - 1) = 0 is, then the power series of f converges
absolutely for | y| = q - 1 (see for details [8]). Furthermore, by the absolute convergence of the power
series of H \prime and f we obtain the absolute convergence of the power series of f \prime . This means that
Axiom \scrA 2 implies the assumptions of Theorem 13.
More generally, Indlekofer showed in his paper [9] the following quantitive results for (3.1)
and (3.2).
Theorem 14 ([9], Theorem 1). Let \delta = 1 in (2.1). Assume that, for some k \in \BbbN , the kth
derivative H(k)(y) of H(y) converges absolutely for | y| \leq q - 1. Then the following two assertions
hold:
(i) if H( - q - 1) \not = 0, then
\lambda (n)
qn
= 1 +O(n - (k+1)) +O
\Biggl(
n \mathrm{m}\mathrm{a}\mathrm{x}
n
4
\leq m\leq n
| h(m)| q - m
\Biggr)
as n \rightarrow \infty ;
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386 K.-H. INDLEKOFER, E. KAYA
(ii) if H( - q - 1) = 0, then
\lambda (n)
qn
= 1 - ( - 1)n +O
\left( n
\sum
n
8
\leq m
| h(m)| q - m
\right) as n \rightarrow \infty .
Observe that if
\infty \sum
n=1
nk| h(n)| q - n < \infty ,
then
n \mathrm{m}\mathrm{a}\mathrm{x}
n
4
\leq m\leq n
| h(m)| q - m = o(n1 - k)
and, obviously, \sum
m\geq n
| h(m)| q - m = o(n - k).
Now, we give abstract prime number theorems for additive arithmetical semigroups, which satisfy
the condition of Theorem 7 in the case 0 < \delta < 2. First of all, we have the following theorem.
Theorem 15. Assume (2.1) with 1 < \delta < 2. If H \prime \prime (y) = O(1) for | y| < q - 1, then the following
assertions hold:
(i) if H( - q - 1) \not = 0, then
\lambda (n)
qn
= \delta + o(n - 1) as n \rightarrow \infty ;
(ii) if H( - q - 1) = 0, then
\lambda (n)
qn
= \delta + ( - 1)n + o(n - 3/2) as n \rightarrow \infty .
Proof. Since H \prime \prime (y) = O(1) for | y| < q - 1 we conclude, by Parseval’s equation,
\infty \sum
n=1
n| h(n)| q - n \leq
\Biggl( \infty \sum
n=1
1
n2
\Biggr) 1/2\Biggl( \infty \sum
n=1
n4| h(n)| 2q - 2n
\Biggr) 1/2
< \infty (3.4)
and
| h(n)| q - n = o(n - 2) as n \rightarrow \infty . (3.5)
By (1.5) and (2.1), we have
\infty \sum
n=1
\lambda (n)yn = \delta
qy
1 - qy
+ y
H \prime (y)
H(y)
.
If H( - q - 1) \not = 0, then, by (3.4),
y
H \prime (y)
H(y)
=
\Biggl( \infty \sum
n=1
nh(n)yn
\Biggr) \Biggl( \infty \sum
n=0
h1(n)y
n
\Biggr)
=:
\sum
n=0
rny
n, (3.6)
where
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REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 387
1
H(y)
=
\infty \sum
n=0
h1(n)y
n
is absolutely convergent for | y| \leq q - 1. Obviously,
y
\biggl(
1
H(y)
\biggr) \prime
= - H \prime (y)
H2(y)
=
\infty \sum
n=1
nh1(n)y
n. (3.7)
Let us consider
1
H2(y)
=:
\infty \sum
n=1
h2(n)y
n.
Then
1
H2(y)
and
\biggl(
1
H2(y)
\biggr) \prime
are absolutely convergent for | y| \leq q - 1, and thus h2(n) =
= O(n - 1| h3(n)| ), where
\infty \sum
n=1
| h3(n)| q - n < \infty .
By (3.5) and (3.7), we conclude
n| h1(n)| \leq
\sum
m\leq n
2
m| h(m)| q - m| h2(n - m)| q - (n - m)+
+
\sum
n
2
<m\leq n
m| h(m)| q - m| h2(n - m)| q - (n - m) \ll
\ll \mathrm{m}\mathrm{a}\mathrm{x}
n
2
<m\leq n
| h2(m)| q - m + n \mathrm{m}\mathrm{a}\mathrm{x}
n
2
<m\leq n
h(m)q - m = o(n - 1) as n \rightarrow \infty .
From (3.6) we deduce in the same way
| rn| q - n = o(n - 1),
which proves (i) of Theorem 15.
In the other case, we have
\infty \sum
n=1
\lambda (n)yn = \delta
qy
1 - qy
+
qy
1 + qy
+ y
f \prime (y)
f(y)
,
where
f(y) =
H(y)
1 + qy
=
\infty \sum
n=0
\gamma (n)yn.
Then (see [8, p. 192])
| \gamma (n)| q - n \leq \delta 1(n) :=
\sum
n\leq m
| h(m)| q - m,
and by Parseval’s equation (cf. (3.4))
\delta 1(n) \leq
1
n
\sum
n\leq m
| h(m)| q - m = o(n - 3/2) as n \rightarrow \infty .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
388 K.-H. INDLEKOFER, E. KAYA
Thus
1
f(y)
is absolutely convergent for | y| \leq q - 1 and \gamma (n)q - n = o(n - 3/2). As in the proof of
Theorem 13 we obtain the assertion (ii), which ends the proof of Theorem 15.
3.2. Abstract Selberg formula.
Theorem 16. Let \delta = 1 in (2.1). Suppose that the power series of H \prime \prime (y) converges absolutely
for | y| \leq q - 1. Then the following assertions hold:
(i) if H( - q - 1) \not = 0, then
n\lambda (n) +
n - 1\sum
k=1
\lambda (k)\lambda (n - k) = 2nqn + c1q
n + o(qn) as n \rightarrow \infty , (3.8)
where
c1 = 2q - 1H
\prime (q - 1)
H(q - 1)
- 1;
(ii) if H( - q - 1) = 0, then
n\lambda (n) +
n - 1\sum
k=1
\lambda (k)\lambda (n - k) = 2nqn + c2q
n + o(qn) as n \rightarrow \infty ,
c2 =
- 1
2
+
3
2
( - 1)n + 2q - 1 f
\prime (q - 1)
f(q - 1)
+ q - 2H
\prime \prime ( - q - 1)
f( - q - 1)
( - 1)n.
(3.9)
Proof. The left-hand sides of (3.8) and (3.9) is the nth coefficients of the power series of
y
\biggl(
y
F \prime (y)
F (y)
\biggr) \prime
+
\biggl(
y
F \prime (y)
F (y)
\biggr) 2
.
The equation
y2
F \prime \prime (y)
F (y)
+ y
F \prime (y)
F (y)
= y
\biggl(
y
F \prime (y)
F (y)
\biggr) \prime
+
\biggl(
y
F \prime (y)
F (y)
\biggr) 2
describes the analogue of the Selberg identity of classical number theory. By (2.1) we deduce
y2
F \prime \prime (y)
F (y)
+ y
F \prime (y)
F (y)
=
2q2y2
(1 - qy)2
+
qy
1 - qy
+
+
2qy
1 - qy
y
H \prime (y)
H(y)
+ y
H \prime (y)
H(y)
+ y2
H \prime \prime (y)
H(y)
=:
=:
\sum
1
+
\sum
2
+
\sum
3
+
\sum
4
+
\sum
5
, (3.10)
where
\sum
1
+
\sum
2
+
\sum
3
+
\sum
4
+
\sum
5
denote the corresponding power series.
Obviously,
the nth (n \geq 2) coefficient of
\sum
1
+
\sum
2
is 2nqn - qn. (3.11)
Let us assume that H( - q - 1) \not = 0. Then the power series of
1
H(y)
(and H \prime (y)) converge absolutely
for | y| \leq q - 1 so that, as n \rightarrow \infty ,
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REMARKS ON NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS 389
the nth (n \geq 2) coefficient of
\sum
3
is 2qnq - 1H
\prime (q - 1)
H(q - 1)
+ o(qn). (3.12)
Further, since H \prime \prime (y) is absolutely convergent for | y| \leq q - 1
the nth (n \geq 2) coefficient for
\sum
4
+
\sum
5
is o(qn) as n \rightarrow \infty . (3.13)
Collecting (3.11), (3.12) and (3.13) gives assertion (i).
For the proof of (ii) assume that H( - q - 1) = 0. Then, putting H(y) = (1 + qy)f(y), by (3.10)
we obtain
y2
F \prime \prime (y)
F (y)
+ y
F \prime (y)
F (y)
=
2q2y2
(1 - qy)2
+
qy
1 - qy
+
2qy
1 - qy
\biggl(
qy
1 + qy
+ y
f \prime (y)
f(y)
\biggr)
+
+
qy
1 + qy
+ y
f \prime (y)
f(y)
+ y2
H \prime \prime (y)
(1 + qy)f(y)
=
=
2q2y2
(1 - qy)2
+
qy
1 - qy
+
2qy
1 - qy
qy
1 + qy
+
qy
1 + qy
+
+
2qy
1 - qy
y
f \prime (y)
f(y)
+ 8y
f \prime (y)
f(y)
+ y2
H \prime \prime (y)
(1 + qy)f(y)
. (3.14)
Putting f(y) =
\sum \infty
n=0
\gamma (n)yn we know
| \gamma (n)| q - n \leq \delta 1(n) =
\sum
n\leq m
| h(m)| q - m
which implies
\infty \sum
n=0
n| \gamma (n)| q - n \leq
\infty \sum
n=1
n
\sum
n\leq m
| h(m)| q - m \ll
\infty \sum
n=1
n2| h(n)| q - n < \infty .
Thus the power series of
1
f(y)
, f \prime (y) and H \prime \prime (y) are absolutely convergent for | y| \leq q - 1. By this
the sum of the nth coefficient of the power series for the right-hand side in (3.14) is given by
2(n - 1)qn + qn +
qn + ( - 1)nqn
2
+ ( - 1)nqn + 2qnq - 1 f
\prime (q - 1)
f(q - 1)
+ q - 2H
\prime \prime ( - q - 1)
f( - q - 1)
( - 1)nqn + o(qn)
which proves (ii) of Theorem 16.
Remark 5. In [18] (Chapter 3.7), Zhang proved a weaker form of Selberg’s formula
n\lambda (n) +
n - 1\sum
k=1
\lambda (k)\lambda (n - k) = 2nqn +O(qn)
under stronger conditions. He assumed that G(n) = Aqn + O(qnn - \gamma ) with \gamma > 3, which implies
| h(n)| q - n = O(n - \gamma ), so that the absolute convergence of H \prime \prime (y) follows immediately.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
390 K.-H. INDLEKOFER, E. KAYA
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Received 05.01.17
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| id | umjimathkievua-article-6042 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:43Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cf/4d82519ac4113db049dd5ef40abbe9cf.pdf |
| spelling | umjimathkievua-article-60422022-03-26T11:01:29Z Remarks on number theory over additive arithmetical semigroups Зауваження до теорiї чисел для адитивних арифметичних напiвгруп Indlekofer, K.-H. Kaya, E. Indlekofer, K.-H. Kaya, E. UDC 511 We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding $\zeta$-functions.&nbsp;&nbsp;We use these results to prove prime number theorems and a Selberg formula for such semigroups. UDC 511 Розглянуто додаткові арифметичні напівгрупи і наведено старі та нові доведення для розподілу нулів відповідних $\zeta$-функцій. Ці результати використано для доведення теорем простих чисел і формули Сельберга для таких напівгруп.&nbsp; Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6042 10.37863/umzh.v72i3.6042 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 3 (2020); 371-390 Український математичний журнал; Том 72 № 3 (2020); 371-390 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6042/8669 |
| spellingShingle | Indlekofer, K.-H. Kaya, E. Indlekofer, K.-H. Kaya, E. Remarks on number theory over additive arithmetical semigroups |
| title | Remarks on number theory over additive arithmetical semigroups |
| title_alt | Зауваження до теорiї чисел для адитивних арифметичних напiвгруп |
| title_full | Remarks on number theory over additive arithmetical semigroups |
| title_fullStr | Remarks on number theory over additive arithmetical semigroups |
| title_full_unstemmed | Remarks on number theory over additive arithmetical semigroups |
| title_short | Remarks on number theory over additive arithmetical semigroups |
| title_sort | remarks on number theory over additive arithmetical semigroups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6042 |
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