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The proximate order of entire Dirichlet series. (English) Zbl 0602.30032

The notion of proximate order for an entire Dirichlet series with index- pair (p,q) is introduced and its existence theorem is proved by an elementary technique. The (p,q)-proximate order for a class of entire Dirichlet series is constructed under certain conditions. Also, the lower (p,q)-proximate order is defined and its existence theorem is stated. The authors remark that there exist two proximate orders of a given entire Dirichlet series with index-pair (p,q) and present conditions under which both comparison functions coincide. In the last section, the possibility of proximating a generalized proximate order by some other proximate order is searched.

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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