Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems

Abstract

In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the Erd\H{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one \v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the Erd\H{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.