Mandelbrojt’s inequality and Dirichlet series with complex exponents.

Abstract

for n = 1, 2, . * , which implies (1.3) with D*?1/p. Dirichlet series with complex exponents have been investigated by Ritt [18], [19], Vaisali [22], Hille [5], Polya [16], Valiron [23], Leont'ev [7], Kahane [6], and the author [4]. Mandelbrojt's inequality [9, p. 77] gives an estimate of the coefficient cn in terms of the maximum modulus of the analytic continuation of a function f(z) to which the series converges (or which it merely represents asymptotically; this is of the greatest significance, although here we shall consider only convergent series). This generalization of Cauchy's estimate of the coefficients of a Taylor series has proved very fruitful in a number of seemingly diverse areas of the theory of functions [8; 9], one of which is the detection of singularities. However, the usual formulations of the inequality do not lead to the most delicate possible results in this last area of investigation, unless D* = 0, because one can not approach closer than ,jiD* to a singularity of f(z), where D7* is the mean upper density (see ?3 below) of the sequence {Xn}. To be more precise: it has usually been assumed that the series represents f(z) in a domain containing a disc of radius -7rD*, and that the analytic continuation of f(z) is along a curve each point of which is the center of a disc of radius 7rD* on which the continuation of f(z) is regular. It will be