REGULAR GROWTH OF DIRICHLET SERIES OF THE CLASS 𝐷(Φ) ON CURVES OF BOUNDED 𝐾-SLOPE

Abstract

We study the asymptotic behavior of the sum of en- tire Dirichlet series with positive exponents on curves of a bounded slope going in a certain way to infinity. For entire transcendental functions of finite order, Polia showed that if the density of the sequence of exponents is equal to zero, then for any curve going to infinity there is an unbounded sequence of points on which the logarithm of the modulus of the sum of the series is equivalent to the logarithm of the maximum of the modulus. Later, these results were completely transferred by I. D. Latypov to entire Dirichlet series of finite order and finite lower order by Ritt. Further gener- alization was obtained in the works of N. N. Yusupova–Aitkuzhina to the more general dual classes of Dirichlet series defined by the convex majorant. In this paper, we obtain necessary and sufficient conditions for the exponents under which the logarithm of the mod- ulus of the sum of any Dirichlet series from one such class on a curve of bounded slope is equivalent to the logarithm of the maximum term on an asymptotic set whose upper density is not less than a positive number depending only on the curve.

Keywords

• dirichlet series
• maximal term
• the curve of a bounded slope
• asymptotic set