Estimates of indicators of an entire function with negative roots

The article continues the series of works by the authors devoted to the study of the relationship between the laws growth of an entire function and the features of the distribution of its roots. The asymptotic behavior of an entire function of finite non-integer order with a sequence of negative roots having the prescribed lower and upper densities is investigated. Particular attention is paid to the case when the sequence of roots has zero lower density. Accurate estimates for the indicator and lower indicator of such a function are given. The angles on the complex plane in which these characteristics are identically equal to zero are described. In some special cases explicit formulas for indicators are proved. Terms used, usual root sequence densities, are simple and illustrative, in contrast to many complicated integral constructions including root counting function that are typical for the growth theory of entire functions. The results are applied to the well-known problem of the extremal type of an entire function of order ρ∈(0,+∞)∖N with zeros on a ray. This problem has been studied in detail only in the case of ρ∈(0,1). For ρ>1, the exact formula for calculating the smallest possible type of such a function in terms of the densities of its roots is still unknown. For the mentioned extreme value, a new two-sided estimate is found that strengthens Popov's results (2009). The conjecture regarding the behavior of the extremal type for ρ→p∈N is formulated.The presentation is supplemented with a brief survey of classical results of Valiron, Levin, Goldberg and recent advances from the works of Popov and of the authors. Some problems on the topic under discussion are outlined.