An example in Beurling's theory of primes

Beurling integers with RH and large oscillation

“…Note that (1.4) implies the Beurling number system satisfies the RH, that is, its zeta function analytically extends to Re s > 1/2, except for a simple pole located at s = 1, and has no zeros in this half-plane. In this regard, it is worthwhile to compare our generalized number system from Theorem 1.1 with earlier examples by E. Balanzario [2] and Zhang [19]. On the one hand, in Balanzario's example 1 the oscillation estimate (1.5) holds for N, but π only satisfies the weaker asymptotic relation (1.3).…”

“…For each r < η/ log τ , there is at least one solution φ near π/2 and at least one solution near −π/2 of the equation Im g m (re iφ ) = 0. For example, by selecting ε sufficiently small and τ large enough, there is a solution for φ in (2π/5, 3π/5) and in (−3π/5, −2π/5).This guarantees the existence of the path of steepest descent in the range |θ| ≤ η/2 say (since t m = t(2)…”

“…Balanzario constructs a 'continuous' example in[2], but F. A. Al-Maamori has recently shown[1] via probabilistic arguments that the example of Balanzario can be discretized.2 A large number of instances has been collected in the monograph[11]; see also the recent work[8].…”

Using.Liouville’s function for Creating a weird numbers from Reals

“…This exactly demonstrates the existence a discrete system of Beurling's prime (π p , N p ) in which π p relies on K and N p relies on the integers Beurling derives from K. Balanzario, now, define a continuous Beurling counting function p , according to the literature in his article (see [23]) defined a as follows:…”