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.
1991 Mathematics Subject Classification: 11N45, 11T55, 40E05
Key words: Chebyshev-type upper estimate, additive arithmetic semigroup, approximate convolution inverse
(Received 31 March 1999; in revised form 21 October 1999)