Entire Dirichlet series with monotonous coefficients and logarithmic h-measure
Let $F$ be an entire function represented by absolutely convergent for all $z\in\mathbb{C}$ Dirichlet series of the form $ F(z) = \sum\nolimits_{n=0}^{+\infty} a_{n}e^{zλ_{n}},$\ where a sequence $(λ_n)$ such that $λ_n\in\mathbb{R}\ \ (n\geq0)$, $λ_n\not=λ_k$ for any $n\not=k$ and $(\forall n\geq 0):\ 0\leqλ_n<β:=\sup\{λ_j:\ j\geq0\}\leq +\infty.$ {Let $h$ be non-decrease positive continuous function on $[0,+\infty)$ and $Φ$ increase positive continuous on $[0,+\infty)$ function.} In this paper we {find} the condition {on} $(μ_n)$ and $(λ_n)$ {such that} the relation $F(x+iy)=(1+o(1))a_{ν(x, F)}e^{(x+iy)λ_{ν(x, F)}} $ holds as $x\to +\infty$\ outside some set $E$ of finite logarithmic $h$-measure uniformly in $y\in\mathbb{R}$.
math.CV↗