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

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: 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 converging in the disk |z|<1 cannot be analytically continued to the region |z|>1 across any arc of the circle |z|=1. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 540–550, October, 1999.