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Connections between connected topological spaces on the set of positive integers

100%
Szczuka P.
Open Mathematics
|
2013
|
tom 11
|
nr 5
876-881
EN
In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (ℕ, T) and (ℕ, T′).
2
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Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

100%
Hong S., Hu S., Hong S.
Open Mathematics
|
2016
|
tom 14
|
nr 1
146-155
EN
Let f be an arithmetic function and S = {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S1, …, Sk with k ≥ 1 being an integer and S1, …, Sk being gcd-closed sets such that (lcm(Si), lcm(Sj)) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.
3
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The closures of arithmetic progressions in the common division topology on the set of positive integers

84%
Szczuka P.
Open Mathematics
|
2014
|
tom 12
|
nr 7
1008-1014
EN
In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.
4
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On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

84%
Kühleitner M., Nowak W.
Open Mathematics
|
2013
|
tom 11
|
nr 3
477-486
EN
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.
5
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On certain arithmetic functions involving the greatest common divisor

84%
Krätzel E., Nowak W., Tóth L.
Open Mathematics
|
2012
|
tom 10
|
nr 2
761-774
EN
The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.
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