Some examples of well-behaved Beurling number systems

Abstract

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called [ α , β ] [\alpha ,\beta ] -systems, where α \alpha and β \beta are connected to the optimal power saving in the prime and integer-counting functions. It is known that every [ α , β ] [\alpha ,\beta ] -system satisfies max { α , β } ≥ 1 / 2 \max \{\alpha ,\beta \}\ge 1/2 . In this paper we show there are [ α , β ] [\alpha ,\beta ] -systems for each α ∈ [ 0 , 1 ) \alpha \in [0,1) and β ∈ [ 1 / 2 , 1 ) \beta \in [1/2, 1) . Assuming the Riemann hypothesis, we also construct certain families of [ α , β ] [\alpha ,\beta ] -systems with β > 1 / 2 \beta >1/2 .