Beurling primes with RH and Beurling primes with large oscillation

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 and its prime counting function satisfies π(x) = li (x) + O(x ^1/2). The associated zeta function of the second has infinitely many zeros on the curve σ = 1−1/log t and no zeros to the right of the curve and the Chebyshev function ψ(x) of its primes satisfies

$$ \limsup\, (\psi(x)-x)/(x\exp\{-2\sqrt{\log\,x}\})=2$$

and

$$ \liminf\, (\psi(x)-x)/(x\exp\{-2\sqrt{\log\,x}\})=-2. $$

A sharpened form of the Diamond–Montgomery–Vorhauer random approximation and elements of analytic number theory are used in the construction.