Entire functions, analytic continuation, and the fractional parts of a linear function

Abstract

The main result of the paper is as follows.Theorem. Suppose that G(z) is an entire function satisfying the following conditions: 1) the Taylor coefficients of the function G(z) are nonnegative: 2) for some fixed C>0 and A>0 and for |z|>R₀, the following inequality holds:

$$|G(z)|< \exp \left( {C\frac{{|z|}}{{1n^A |z|}}} \right) \cdot $$

Further, suppose that for some fixed α>0 the deviation D_N of the sequence x_n={αn}, n=1, 2, ..., as N→∞ has the estimate D_N=0(ln^B N/N). Then if the function G(z) is not an identical constant and the inequality B+1<A holds, then the power series$\sum\nolimits_{n = 0}^\infty {G([\alpha n])z^n }$ converging in the disk |z|<1 cannot be analytically continued to the region |z|>1 across any arc of the circle |z|=1.