A remark on papers of G. Pólya and P. K. Kamthan (Q1078350)

The author considers an entire Dirichlet series: \(f(s)=\sum^{\infty}_{n=1}a_ n \exp (s\lambda_ n).\) Let \(\{b_ n\}\) be a complex sequence such that \(\log | b_ n| =o(\lambda_ n),\) and \(n\to \infty\). It is shown that the Hadamard composition \(g(s)=\sum^{\infty}_{n=1}a_ n b_ n \exp (s\lambda_ n)\) is an entire Dirichlet series with the same (p,q) order, (p,q) type, proximate order and generalized (p,q) type as that of f(s).