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be a power series with the radius of convergence 1 . We say that it has Ostrowski gaps p if there exists a p<1 and a pair of infinite sequences mk and nk, with mk <nk and lira nk/mk> 1, such that I a„ I < pn for mk _<_ n < nk . It has infinite Ostrowski gaps p (p < 1) if to every p'> p there corresponds a pair of infinite sequences Ink and nk (depending on p') with mk <nk and lim nk/mk= oo such that I an I < p' n for Mk < n < nk . We denote by A (n, r) the number of roots of f(x) = 1 + a1x + + anxn within the circle of radius r. It is well known that every overconvergent power series has Ostrowski gaps, and that every power series with Ostrowski gaps is overconvergent in a domain of which every regular point of the circle of convergence is an interior point . We are going to prove the following theorems :

On the connection between gaps in power series and the roots of their partial sums