On the behavior of power series with positive completely multiplicative coefficients

Abstract

We consider power series with positive completely multiplicative coefficients. We obtain a large family of power series that have the unit circle as natural boundary, as well as new $\mathrm{\Omega }$-theorems for power series with positive completely multiplicative coefficients when the argument tends to roots of unity. Also $\mathrm{\Omega }$-estimates for some partial sums of completely multiplicative functions are given.