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Asymptotic analysis of non-self-adjoint Hill operators

100%
Veliev O.
Open Mathematics
|
2013
|
tom 11
|
nr 12
2234-2256
EN
We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator provided that the potential q satisfies sufficient conditions.
2
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On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

100%
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.
3
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On certain arithmetic functions involving the greatest common divisor

100%
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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