Beurling integers with RH and large oscillation

Abstract

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 $\pi (x)=\mathrm{Li}(x)+O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x)=\rho x+{\Omega }_{\pm }(x\exp (-c\sqrt{\log x\log \log x}))$ for some $c>0$, where $\rho >0$ 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.

arXiv comment31 pages, 2 figures

References in corpus (1)

• Some examples in the theory of Beurling's generalized prime numbers

Cited by in corpus (7)

• A new generalized prime random approximation procedure and some of its applications
• Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling
• A Riemann-von Mangoldt-type formula for the distribution of Beurling primes
• The optimal Malliavin-type remainder for Beurling generalized integers
• Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder
• Some examples of well-behaved Beurling number systems
• A note on Bohr's theorem for Beurling integer systems