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)

In classical prime number theory, several relations are considered to be equivalent to the Prime Number Theorem. For Beurling generalized numbers, some auxiliary conditions may be needed to deduce one relation from another one. We show conditions under which the Beurling analog of the sharp version of Mertens' sum formula does or does not hold.

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We introduce the classical theory of the interplay between group theory and topology into the context of operads and explore some applications to homotopy theory. We first propose a notion of a group operad and then develop a theory of group operads, extending the classical theories of groups, spaces with actions of groups, covering spaces and classifying spaces of groups. In particular, the fundamental groups of a topological operad is naturally a group operad and its higher homotopy groups are naturally operads with actions of its fundamental groups operad, and a topological K(π, 1) operad is characterized by and can be reconstructed from its fundamental groups operad. Two most important examples of group operads are the symmetric groups operad and the braid groups operad which provide group models for Ω ∞ Σ ∞ X (due to Barratt and Eccles) and Ω 2 Σ 2 X (due to Fiedorowicz) respectively. We combine the two models together to produce a free group model for the canonical stabilization Ω 2 Σ 2 X ֒→ Ω ∞ Σ ∞ X, in particular a free group model for its homotopy fibre.