Universality and distribution of zeros and poles of some zeta functions

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the Dirichlet series $\sum_{n=1}^{\infty} \chi(n) n^{-s}$ can be continued meromorphically to the half-plane $\operatorname{Re} s>\alpha$, and denote by $\zeta_{\chi}(s)$ the corresponding meromorphic function in $\operatorname{Re} s>\sigma(\chi)$. We construct $\zeta_{\chi}(s)$ that have $\sigma(\chi)\le 1/2$ and are universal for zero-free analytic functions on the half-critical strip $1/2<\operatorname{Re} s <1$, with zeros and poles at any discrete multisets lying in a strip to the right of $\operatorname{Re} s =1/2$ and satisfying a density condition that is somewhat stricter than the density hypothesis for the zeros of the Riemann zeta function. On a conceivable version of Cram\'{e}r's conjecture for gaps between primes, the density condition can be relaxed, and zeros and poles can also be placed at $\beta+i \gamma$ with $\beta\le 1-\lambda \log\log |\gamma|/\log |\gamma|$ when $\lambda>1$. Finally, we show that there exists $\zeta_{\chi}(s)$ with $\sigma(\chi) \le 1/2$ and zeros at any discrete multiset in the strip $1/2<\operatorname{Re} s \le 39/40$ with no accumulation point in $\operatorname{Re} s >1/2$; on the Riemann hypothesis, this strip may be replaced by the half-critical strip $1/2 < \operatorname{Re} s < 1$.Comment: This is the final version of the paper which has been accepted for publication in Journal d'Analyse Math\'{e}matiqu

Interpolation and sampling in small Bergman spaces

Carleson measures and interpolating and sampling sequences for weighted Bergman spaces on the unit disk are described for weights that are radial and grow faster than the standard weights $(1-|z|)^{-\alpha}$, $0<\alpha<1$. These results make the Hardy space $H^2$ appear naturally as a "degenerate" endpoint case for the class of Bergman spaces under study

Approximation numbers of composition operators on the H2H^2 space of Dirichlet series

By a theorem of Gordon and Hedenmalm, $\varphi$ generates a bounded composition operator on the Hilbert space $\mathscr{H}^2$ of Dirichlet series $\sum_n b_n n^{-s}$ with square-summable coefficients $b_n$ if and only if $\varphi(s)=c_0 s+\psi(s)$, where $c_0$ is a nonnegative integer and $\psi$ a Dirichlet series with the following mapping properties: $\psi$ maps the right half-plane into the half-plane $\operatorname{Re} s >1/2$ if $c_0=0$ and is either identically zero or maps the right half-plane into itself if $c_0$ is positive. It is shown that the $n$th approximation numbers of bounded composition operators on $\mathscr{H}^2$ are bounded below by a constant times $r^n$ for some $0<r<1$ when $c_0=0$ and bounded below by a constant times $n^{-A}$ for some $A>0$ when $c_0$ is positive. Both results are best possible. The case when $c_0=0$, $\psi$ is bounded and smooth up to the boundary of the right half-plane, and $\sup \operatorname{Re} \psi=1/2$, is discussed in depth; it includes examples of non-compact operators as well as operators belonging to all Schatten classes $S_p$. For $\varphi(s)=c_1+\sum_{j=1}^d c_{q_j} q_j^{-s}$ with $q_j$ independent integers, it is shown that the $n$th approximation number behaves as $n^{-(d-1)/2}$, possibly up to a factor $(\log n)^{(d-1)/2}$. Estimates rely mainly on a general Hilbert space method involving finite linear combinations of reproducing kernels. A key role is played by a recently developed interpolation method for $\mathscr{H}^2$ using estimates of solutions of the $\bar{\partial}$ equation. Finally, by a transference principle from $H^2$ of the unit disc, explicit examples of compact composition operators with approximation numbers decaying at essentially any sub-exponential rate can be displayed.Comment: Final version, to appear in Journal of Functional Analysi

Bioactive forage and phytotherapy to cure and control endo-parasite diseases in sheep and goat farming systems – a review of current scientific knowledge

Infestation with gastro-intestinal nematodes (GIN) in small ruminants can cause severe economic losses and endanger animal welfare. The development of organic farming systems, the increased public awareness for drug residues in agricultural products and the development of resistant strains of parasites have enforced the search for sustainable alternatives. The aim of this review is to summarise the current scientific knowledge of alternative strategies to prevent and control endo-parasitic diseases in organic sheep and goat farming systems. The conducted literature evaluation has revealed the major potential to be within the field of bioactive forages, phytotherapy, homeopathy and copper-oxide wire particles. Alternative management pattern like grazing management, nematophagous fungi, improved fodder and breeding are not considered. The administration and cultivation of bioactive forages and phytotherapy are displaying promise potential for endo-parasite control in organic sheep and goat farming. Scientific research has mainly concentrated on the extracts of the plant species chicory, Birdsfoot trefoil (Lotus corniculatus), Sainfoin (Onobrychis viciifolia), Sulla (Hedysarum coronarium) and Quebracho (Schinopsis ssp.). The analysis of these plants showed all plants to have some positive potential, but also highlighted individual limitations in application. However from the results of this literature review none of the investigated plants have been researched sufficiently in on farm experiments to recommend any for implementation at this stage. No concrete recommendation for a single plant can be given, further research on promising species for the commercial use is strongly recommended, as is the review of the law concerning the appliance of plant based remedies

Integral means and boundary limits of Dirichlet series

We study the boundary behavior of functions in the Hardy spaces HD^p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in HD^\infty, i.e., for ordinary Dirichlet series in H^\infty of the right half-plane. We discuss an important embedding problem for HD^p, the solution of which is only known when p is an even integer. Viewing HD^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.Comment: 13 page

Alternative strategies to prevent and control endoparasite diseases in organic sheep and goat farming systems – a review of current scientific knowledge

Infestation with gastro-intestinal nematodes in small ruminants can cause server economic losses and endanger animal welfare. The development of organic farming systems, the increased public awareness for drug residues in agricultural products and the development of resistant strains of parasites have enforced the search for sustainable alternatives. The aim of this paper was to provide information about endoparasite infecting small ruminants, to give an overview of the legal background and to investigate alternative control strategies and treatments, discussing them on overall viability. The main section has been divided into a part of non-chemotherapeutical control strategies and alternative anthelmintic treatments. The conducted research has revealed the major potential to be within the field of non-chemical options. Biological control, effective pasture management, selective breeding, enhanced nutrition and the administration of bioactive forages were discussed and found to hold numerous options. The investigation of alternative anthelmintic treatments reviewed phytotherapy, homeopathy and copper-oxide wire particles. Phytotherapy was examined at in detail and found to hold future potential, indicating a strong need for scientific verification of the potential of many plants. In conclusion this paper shows possibilities and limitations in the area of alternative anthelmintic treatments as well as in non-chemical control options and outlines future research fields