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 lim sup (ψ(x) - x)/(x exp{-2√log x}) = 2 and lim inf (ψ(x) - x)/(x exp{-2√log x}) = -2. A sharpened form of the Diamond-Montgomery-Vorhauer random approximation and elements of analytic number theory are used in the construction. © Springer-Verlag 2007.