We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies π(x) = Li(x) + O(√ x), while its integer counting function satisfies the oscillation estimate N (x) = ρx + Ω ± x exp(−c √ log x log log x) for some c > 0, where ρ > 0 is its asymptotic density. The construction is inspired by a classical example of H. Bohr for optimality of the convexity bound for Dirichlet series, and combines saddle-point analysis with the Diamond-Montgomery-Vorhauer probabilistic method via random prime number system approximations.

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].

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We study the family of Fourier-Laplace transforms F α,β (z) = F. p. ∞ 0 t β exp(it α − izt) dt, Im z < 0, for α > 1 and β ∈ C, where Hadamard finite part is used to regularize the integral when Re β ≤ −1. We prove that each F α,β has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems.

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).