Remarks on the Number Theory Over Additive Arithmetical Semigroups
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REMARKS ON THE NUMBER THEORY OVER ADDITIVE ARITHMETICAL SEMIGROUPS K.-H. Indlekofer1,2 and E. Kaya3
UDC 511
We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding ⇣-functions. We use these results to prove prime number theorems and a Selberg formula for semigroups of this kind.
1. Introduction Abstract analytic number theory first appeared as a generalization of the classical number theory on the (semigroup) N of natural numbers with the special emphasis on the derivation of the famous prime-number theorem: If ⇡(x) denotes the total number of positive rational x as x ! 1, primes x, then ⇡(x) ⇠ log x and of the classical Landau prime-ideal theorem, which extends the prime-number theorem to the (semigroup) GK of integral ideals in an algebraic number field K. There are numerous mathematical systems, especially in abstract algebra, which have elementary “unique factorization” properties similar to the properties of the positive integers. Many of these systems have an additional property, which makes them more “arithmetical” in a sense, and more treatable by the techniques of classical number theory. This property comes from the existence of a function measuring the “size” of an individual object (usually, the cardinality or “degree” of an object in a certain sense) with an essential attribute that there are only finitely many inequivalent objects whose “size” does not exceed any chosen bound. Motivated by these systems, Knopfmacher [17] introduced a formal concept of additive arithmetical semigroup, which he defined as 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 @ : G ! N [ {0} with the properties: (i) @(1) = 0 and @(p) > 0 for all p 2 P, (ii) @(ab) = @(a) + @(b) for all a, b 2 G, (iii) the total number G(n) of elements a 2 G of degree @(a) = n is finite for each n ≥ 0. Then (G, @) is called an additive arithmetical semigroup. Obviously, G(0) = 1 and G is countable. 1
Faculty of Computer Science, Electrical Engineering, and Mathematics, University of Paderborn, Paderborn, Germany; e-mail: k-heinz@ math.uni-paderborn.de. 2 Corresponding author. 3 Mersin University, Mersin, Turkey; e-mail: [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 3, pp. 371–390, March, 2020. Original article submitted January 5, 2017. 422
0041-5995/20/7203–0422
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Springer Science+Business Media, LLC
R EMARKS ON THE N UMBER T HEORY OVER A DDITIVE A RITHMETICAL S EMIGROUPS
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Let ⇡(n) := # {p 2 P : @(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 +
1 X
G(n)y n ,
n=1
then the following identity holds at least in the formal sense: F (y) = 1 +
1 X
G(n)y n =
n=1
1 Y
n=1
(1 − y n )−⇡(n) .
F (y) is called the generating function (or
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