Abstract:
The article considers the behavior of the sum of the Dirichlet series $F(s)=\displaystyle \sum\limits_{n} a_ne^{\lambda_ns},$ $0<\lambda_{n}\uparrow\infty, $ which converges absolutely in the left half-plane $\Pi_0$, on a curve arbitrarily approaching the imaginary axis — the boundary of this half-plane. We have obtained a solution to the following problem: Under what additional conditions on $\gamma$ will the strengthened asymptotic relation be valid in the case when the argument $s$ tends to the imaginary axis along $\gamma$ over a sufficiently massive set.
Keywords:Dirichlet series, lacunary power series, maximal term, curve of bounded slope, convergence half-plane.
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