The main purpose of this paper is to extend the study of various annihilator conditions on polynomials to formal power series in which addition and substitution are used as operations. This process is not as routine as in the ring case because the substitution of one formal power series into another may not be well defined. Two approaches are introduced to solve this problem. A result of Armendariz on the polynomial extension of a reduced Baer ring is extended to the study of entire functions. It is shown that nearrings of entire functions satisfy certain annihilator conditions. Our results are applied to obtain connections between the multiplicative and substitution structures of various formal power series rings.