ALGEBRAIC VALUES OF CERTAIN ANALYTIC FUNCTIONS DEFINED BY A CANONICAL PRODUCT

We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain $C(\log H)^{\unicode[STIX]{x1D702}}$ bounds for the number of algebraic points of height at most $H$ on certain subsets of the graphs of such functions. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ depend on data associated with the functions and can be effectively computed from them.

ON AN EXTREMAL PROBLEM RELATED TO THE COMPLETENESS OF A SYSTEM OF EXPONENTIALS IN THE DISK

In this paper, we pose several problems of finding the extremal ρ-type and the lower ρ-type of the canonical product with real positive zeros having prescribed averaged upper and/or lower ρ-density, ρ ∈ (0; 1). We obtain new results that can be considered as a development of the classical theorems on the connection between different growth characteristics of entire functions and densities of their distributions of zeros.