Values of Entire Functions Represented by Gap Dirichlet Series

Introduction. In two previous notes [9],1 I stated some results which, in a general way, may be expressed as follows: If the Taylor series which represents an entire function satisfies a certain gap condition (which depends only on the order of F(z)), the zeros of F(z) —f(z) are not exceptional with respect to the proximate order of F(z) by any meromorphic function f(z) ^ w of lower order. On the other hand, Mandelbrojt (see for instance [3, Theorem XXV]) proves that an entire function represented by a Dirichlet series takes each value a j¿ «>, except at most one, in any horizontal strip of width greater than a quantity which depends only on the order (R) and on the upper density of the sequence of exponents of the series. In the present note we shall prove some results closely related to those of my above mentioned notes and to that of Mandelbrojt just quoted. These results may briefly be stated as follows: If an entire function F(s) can be represented by a Dirichlet series satisfying certain gap conditions, the zeros of F(s)—a cannot be exceptional with respect to the proximate order (R) of F(s) in any strip of width greater than a quantity, determined by the order, for every value of o^«> without exception. In §1 we shall deal with functions of finite order (R); in §2, we shall give results concerning functions of infinite order (R). I think it will be of interest to remark that the theorems of the type of the well known Hadamard's theorem (which asserts that a Taylor series satisfying a specific gap condition cannot be continued analytically outside its circle of convergence) together with the results concerning relationship between gap properties and the position of the Julia lines, and again together with the results of my above mentioned notes and those contained in one of my papers [lO], enable us to state the following general principle (without pretending that it holds for every case). By means of gap conditions in the Taylor series the disappearance of the possibility of the existence of the exceptional cases can be affirmed. Therefore, the content of this paper may be regarded as a link