T. M. Salo | Searcharxiv

arXiv2015

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↗

arXiv2015

The minimum modulus of gap power series and h-measure of exceptional sets

For entire Dirichlet series of the form $F(z)=\sum\limits_{n=0}^{+\infty} a_{n}e^{zλ_n},\ 0\leλ_n\uparrow+\infty\ (n\to+\infty)$, we establish conditions under which the relation $$ F(x+iy)=(1+o(1))a_{ν(x,F)}e^{(x+iy)λ_{ν(x,F)}} $$ is true as $x\to+\infty$ outside some set $E$ such that $\text{ h-meas }(E)=\int_{E}dh(x)<+\infty$ uniformly in $y\in\Bbb{R}$, where $h(x)$ is positive continuous function increasing to $+\infty$ on $[0,+\infty)$ with non-decreasing to $+\infty$ derivative.

math.CV↗