A Chebyshev type upper estimate for prime elements in additive arithmetic semigroups
Let \(G(n)\) and \(\Lambda(n)\) be two sequences of nonnegative numbers which satisfy \(G(0)=1\) and the convolution equation \(\sum_{k=0}^n\Lambda(k)G(n-k)=nG(n)\), \(n=0,1,2,\ldots\). A Chebyshev-type upper estimate for prime elements in an additive arithmetic semigroup \(\mathfrak G\) is proved essentially by a Tauberian argument on \(\Lambda\) and \(G(n)\), where \(G(n)\) denotes the total number of elements of degree \(n\) in \(\mathfrak G\). The estimate requires a very weak condition, i.e. suppose \[ \sum_{n=1}^\infty \sup_{n\leq m}\left|G(m)q^{-m}-\sum_{\nu=1}^r A_\nu m^{\rho_\nu-1}\right|<\infty \] with real constants \(0\leq \rho_1<\cdots<\rho_r\), \(\rho_r\geq 1\), \(A_1,\ldots,A_r\), \(A_r>0\). Then the estimation \(\Lambda(n)\ll q ^n\) and \(\sum_{m=1}^n\Lambda(m)q^{-m}=\rho_rn+O(n^\alpha)\) with \(\alpha<1\) hold. The proof utilizes asymptotic analysis, convolution techniques and H. G. Diamond's idea of approximate convolution inverse.
scientific article; zbMATH DE number 1441488