Beurling primes with RH and Beurling primes with large oscillation (Q877165)

From the author's abstract: Two Beurling generalized number systems, both with \(N(x)=kx+O(x^{1/2}\exp\{c(\log x)^{2/3}\})\) and \(k>0\) are constructed. The associated zeta function of the first satisfies the RH (Riemann Hypothesis) and ist prime counting function satisfies \(\pi(x)=li(x)+O(x^{1/2}\). The associated zeta function of the second has infinitely many zeros on the curve \(\sigma=1-1/\log t\) and no zeros to the right of the curve and the Chebyshev function \(\psi(x)\) of its primes satisfies \[ \begin{aligned} &\limsup\bigl(\psi(x)-x\bigr)/\bigl(x\exp\{-2\sqrt{\log x}\,\}\bigr)=2 \qquad \text{and}\\ &\liminf\bigl(\psi(x)-x\bigr)/\bigl(x\exp\{-2\sqrt{\log x}\,\}\bigr)=-2. \end{aligned} \] A sharpened form of the Diamond-Montgomery-Vorhauer random approximation and elements of analytic number theory are used in the construction.