6 papers
Beurling integers with RH and large oscillation
Frederik Broucke, Gregory Debruyne, Jasson Vindas
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The pr…
An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions
Frederik Broucke, Gregory Debruyne, Jasson Vindas
We study the family of Fourier-Laplace transforms $$ F_{α,β}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^β\exp(\mathrm{i} t^α-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{I…
Beurling numbers whose number of prime factors lies in a specified residue class
Gregory Debruyne
We find asymptotics for , the number of positive integers below whose number of prime factors is . We study this question in the context of B…
Halász's theorem for Beurling generalized numbers
Gregory Debruyne, Frederick Maes, Jasson Vindas
We show that Halász's theorem holds for Beurling numbers under the following two mild hypotheses on the generalized number system: existence of a positive density for the generaliz…
Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth
Gregory Debruyne, David Seifert
We prove that a general version of the quantified Ingham-Karamata theorem for -semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results…
An abstract approach to optimal decay of functions and operator semigroups
Gregory Debruyne, David Seifert
We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of -semigroups, and in…