We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $\alpha \in (0,1]$ and $c>0$ (with $c\leq 1$ if $\alpha =1$), a generalized number system is constructed with Riemann prime counting function $ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), $ and whose integer counting function satisfies the extremal oscillation estimate $N(x)=\rho x + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$ for any $c'>(c(\alpha +1))^{\frac {1}{\alpha +1}}$, where $\rho>0$ is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].

Let $1\,<\,{{p}_{1}}\,\le \,{{p}_{2}}\,\le \,{{p}_{3}}\,\le \,...$ be an infinite sequence $P$ of real numbers for which ${{p}_{i}}\,\to \,\infty $, and associate with this sequence the Beurling zeta function$\zeta P\left( s \right)\,:=\,{{\prod\nolimits_{i=1}^{\infty }{\left( 1\,-\,p_{i}^{-s} \right)}}^{-1}}$. Suppose that for some constant $A\,>\,0$, we have $\zeta P\left( s \right)\tilde{\ }A/\left( s-1 \right),\ \text{as}\,s\,\downarrow \,1$. We prove that $P$ satisfies an analogue of a classical theorem of Mertens: ${{\prod{_{{{p}_{i}}\le x}\left( 1\,-\,{1}/{{{p}_{i}}}\; \right)}}^{-1}}\,\sim \,A{{\text{e}}^{\gamma }}\,\log \,x$, as $x\,\to \,\infty $. Here $\text{e}\,\text{=}\,\text{2}\text{.71828}...$ is the base of the natural logarithm and $\gamma \,=\,0.57721...$ is the usual Euler–Mascheroni constant. This strengthens a recent theorem of Olofsson.

We axiomatize the main properties of the classical Turán Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0–1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first–order 0–1 law.

If the condition is satisfied (and hence we have a first-order 0–1 law) we give a natural model of the limit law theory; and show that the limit law theory is decidable if the theory of the directly indecomposables is decidable. Using asymptotic methods from the partition calculus a useful test is derived to show several admissible classes have a first–order 0–1 law.

Using Feferman-Vaught techniques we show a certain property of the fine spectrumof an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order law. We present three conditions for verifying that the above property actually holds.

The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition.

The second condition is similar to the first condition, but designed to handle the discrete case, i.e., when the sizes of the structures in an admissible class K are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher's Axiom A#.

The third condition is also for the discrete case, when there is a uniform bound on the number of K-indecomposables of any given size.