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

A. I. Pavlov

Matematičeskie zametki, Tome 66 (1999) no. 4, pp. 540-550

Résumé

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 $|z|>R_0$, the following inequality holds: $$ |G(z)|\exp\biggl(C\frac{|z|}{\ln^A|z|}\biggr). $$ {\it Further, suppose that for some fixed $\alpha>0$ the deviation $D_N$ of the sequence $x_n=\{\alpha n\}$, $n=1,2,\dots$, as $N\to\infty$ has the estimate $D_N=O(\ln^BN/N)$. Then if the function $G(z)$ is not an identical constant and the inequality $B+1$ holds, then the power series $\sum_{n=0}^\infty G([\alpha n])z^n$ converging in the disk $|z|1$ cannot be analytically continued to the region $|z|>1$ across any arc of the circle $|z|=1$.}