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. Mathematics Subject Classification (1991)

Beurling integers with RH and large oscillation

“…Several arguments from [10] have been sharpened by W.-B. Zhang in [19]. Interestingly, Zhang complemented these results by showing that there are also Beurling number systems for which, in contrast, the RH and the asymptotic estimate (1.1) both hold.…”

“…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).…”

Completely multiplicative functions with sum zero over generalised prime systems

“…hold. The existence of P Z was shown by Zhang [17]. The proof of the following theorem is roughly identical with the previous theorem except that we need a strictly stronger bound on Z P (s) than (21).…”

“…For the usual primes, (2) holds with β = 0 and if the RH is true, then (3) would hold for α = 1 2 . Such systems exist as was shown by Zhang [17]. Indeed, P Z (his system) satisfies these with α = β = 1 2 .…”

Mean-value theorems and extensions of the Elliott–Daboussi theorem on Beurling’s generalized integers (I)

“…Readers interested in the connection of Beurling's generalized integers with algebraic number fields may look at the recent paper [5] of Diamond, Montgomery, and Vorhauer and [16].…”