On the stability of entire multiple Dirichlet series

M. M. Dolynyuk, O. B. Skaskiv
Ivan Franko National University of Lviv
Let $D^p(\lambda)$ be the class of entire multiple Dirichlet series of the form $F(z)=\sum\nolimits_{\|n\|=0}^{+\infty}{a_{n}\textrm{e}^{(z,\lambda_{n})}}$, $z\in\mathbb{C}^{p},$\ $p\geq 1$, with exponents satisfying the conditions $\lambda_n=(\lambda^{(1)}_{n_1},\ldots,\lambda^{(p)}_{n_p})$, $n=(n_1,\ldots,n_p)\in\mathbb{Z}^{p}_{+}$, $0\le\lambda^{(j)}_{k}\leq\lambda^{(j)}_{k+1}\to +\infty$\ $(0\leq k\to +\infty)$;\ $w\colon [0,+\infty)\to [0,+\infty)$ {a} nondecreasing function, and $\nu_1(t)=\sum_{\|\lambda_n\|\leq t}e^{w(\|\lambda_n\|)}$, $\|a\|=a_1+\ldots+a_p$, $(a,b)=a_1b_1+\ldots+a_pb_p$, for $a=(a_1,\ldots, a_p),\ b=(b_1,\ldots, b_p)\in\mathbb{C}^p$.\ If $\int_{1}^{+\infty}t^{-1}d\ln\nu_1(t)$ is finite and $F_w\in D^p(\lambda)$,\ $F_w(z)=\sum\nolimits_{\|n\|=0}^{+\infty}{a_{n}\textrm{e}^{w(\|\lambda_n\|)+(z,\lambda_{n})}}$.\ {Then} $ \ln\max\{|a_n|e^{w(\|\lambda_n\|)+(\sigma,\lambda_n)}\colon n\in\mathbb{Z}^{p}_{+}\}\sim\ln\max\{|a_n|e^{(\sigma,\lambda_n)}\colon n\in\mathbb{Z}^{p}_{+}\} $ as $|\sigma| \to +\infty$\ $(\sigma\in K\setminus E)$, for {an} arbitrary cone $K$ in $\mathbb{R}^p_+$ with vertex at the point $O$ such that $\overline{K}\backslash\{O\}\subset\mathbb{R}^p_+,$ and {a} measurable set $E\subset \mathbb{R}_{+}^{p}$ such that $\tau_{p}(E\cap K)=\int\nolimits_{E\cap K}\frac{d\sigma_1\ldots d\sigma_p}{|\sigma|^{p-1}}$ is finite.
Dirichlet series; maximal term; stability; exceptional set; asymptotic estimate; Laplace-Stieltjes integral
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