Overconvergence Phenomena and Grouping in Exponential Representation of Solutions of Linear Di erential Equations of In nite Order

Introduction This paper tries to shed some light on the very classical topic of overconvergence for Dirichlet series, by employing results in the theory of innite order dier-ential operators with constant coecients ([3], [13]). The possibility of linking innite order dierential operators with gap theorems and related subjects such as overconvergence phenomena was rst suggested by Ehrenpreis in [9], but in a form which could not be fully exploited. On the other hand, the proof of Theorem 1 in [13] indicates that if some lacunary Dirichlet series happens to be analytically extended beyond a point on the line of absolute convergence, then it is automatically extended analytically to a wider region and, in particular , its abscissa of holomorphy may be dierent from its abscissa of absolute convergence. We note that the circle of convergence coincides with that of holo-morphy for a Taylor series. This fact indicates that the situation we discuss here is tied up with some subtle features of the behavior of the frequencies n of the Dirichlet series P a n e 0nz in question (see Remark 2 of Section 1 for the concrete description of the subleties). One natural question, then, might be whether we can describe the analytically continued function\concretely" in terms of the starting convergent Dirichlet series. Such a description, if it exists, may be highly transcendental; for example, let us consider the Dirichlet series (z) = +1 X n=1 (01) n01 n z which is actually equal to (1 0 2 10z)(z), with (z) denoting the zeta function of Riemann (cf.e.g. [10, page 30]). It is known that (z) determines an entire function and that its value outside the domain of convergence can be found through the functional equation it satises. In the situation we consider in this article, however, we can describe the analytically continued object in a straightforward manner; the description is given by an appropriate grouping of the summation of the Dirichlet series in question. Such a clear result is obtained just because we consider a lacunary series. On the other hand, enlarging the domain of holomorphy of a series by using a suitably grouped sum is exactly the classical subject of overconvergence studied by Ostrowski ([17]), Bernstein ([7]), and others. In a word, our main result (Theorem 2.1) asserts that, if the frequencies n of the Dirichlet series in question are positive and suciently lacunary, then the abscissa of …