Zeta functions and asymptotic additive bases with some unusual sets of primes

Fix delta is an element of (0, 1], sigma(0) is an element of [0, 1) and a real-valued function epsilon(x) for which (lim) over bar (x -> infinity) epsilon(x) <= 0. For every set of primes P whose counting function pi(P)(x) satisfies an estimate of the form pi(P)(x) = delta pi(x) + O(x(sigma 0) (+) (epsilon(x))), we define a zeta function zeta(P)(s) that is closely related to the Riemann zeta function zeta(s). For sigma(0) <= 1/2, we show that the Riemann hypothesis is equivalent to the non-vanishing of zeta(P)(s) in the region {sigma > 1/2}. For every set of primes P that contains the prime 2 and whose counting function satisfies an estimate of the form pi(P)(x) = delta pi(x) + O((log log x)(epsilon(x))), we show that P is an exact asymptotic additive basis for N, i. e. for some integer h = h(P) > 0 the sumset hP contains all but finitely many natural numbers. For example, an exact asymptotic additive basis for N is provided by the set {2, 547, 1229, 1993, 2749, 3581, 4421, 5281 ...}, which consists of 2 and every hundredth prime thereafter.