Borel type asymptotic relation for entire Dirichlet series and $h$-measure of an exceptional sets - Kuryliak, A. O.; Salo, T. M. (Matematychni Studii, 2026)
There are presented sufficient conditions for the entire Dirichlet series with monotonically increasing  sequence of exponents providing validity of Borel-type relation outside some set $E$ of finite $h$-measure, i.e. $m_h E=\int_E dh(x)<+\infty$ with a positive continuously differentiable on $[0,+\infty)$ function $h,$ whose derivative $h'$ increases to infinity.The corresponding Borel-type relation states that the logarithm of supremum of the Dirichlet series along an imaginary line behaves like as logarithm of maximal term of the Dirichlet series.The conditions are given as the convergence of some auxiliary series constructed from the values of the function $h'$ and the sequence of exponents.