Abstract
In this paper, we study sequences of complex numbers of refined order. Multiple terms are allowed in such sequences. We consider complex sequences with finite maximal density for a given refined order. We construct special coverings of multiple sets \(\{\lambda_k,n_k\}\) consisting of disks of special radii centered at the points \(\lambda_k\) . In particular, we construct coverings whose connected components have a relatively small diameter as well as coverings that are \(C_0\) -sets. These coverings act as exceptional sets for entire functions of finite refined order and completely regular growth. Outside these sets, we obtain a representation of the logarithm of the modulus of an entire function. Earlier, a similar representation was obtained by B. Ya. Levin outside the exceptional set with respect to which only its existence is asserted. In contrast, in this paper we present a simple constructive construction of the exceptional set.
