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Maximum term and its rank of an entire Dirichlet series. (English) Zbl 0666.30018

Let \(f(s)=\sum^{\infty}_{n=1}a_ n \exp (s\lambda_ n)\) where \(s=\sigma +it\), \(0\leq \lambda_ 1<\lambda_ n<\lambda_{n+1}\to \infty\) as \(n\to \infty\) and \(\lim_{n\to \infty}\log n/\lambda_ n=D<\infty,\) be an entire Dirichlet series of (p,q)-order \(\rho\).
Let \(u(\sigma)=\max_{n\geq 1}\{| a_ n| e^{\sigma \lambda_ n}\}\) and \(\nu (\sigma)=\max \{n|\) \(u(\sigma)=| a_ n| e^{\sigma \lambda_ n}\}\) denote the maximum term and rank of f(s). The authors obtain some relations involving the maximum terms of f(s) and its derivatives. They also construct a proximate order, in terms of u(\(\sigma)\), for entire Dirichlet series of (p,q)-order \(\rho\).
Reviewer: O.P.Juneja

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30B50 Dirichlet series, exponential series and other series in one complex variable
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