A Chebyshev Type Upper Estimate for Prime Elements in Additive Arithmetic Semigroups

 Let G(n) and Λ(n) be two sequences of nonnegative numbers which satisfy G(0)=1 and an additive convolution equation . A Chebyshev-type upper estimate for prime elements in an additive arithmetic semigroup is essentially a tauberian theorem on Λ(n) and G(n). Suppose with real constants . The theorem proved here states that and that holds with an explicit function R(n) of order <1 in n. This theorem is sharp. It has several applications.