On the domain of convergence of general Dirichlet series with complex exponents
Let (λ_n) be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series F(z)=_n=0^+∞ a_ne^zλ_n, zC, we denote G_μ(F), G_c(F), G_a(F) the domains of the existence, of the convergence and of the absolute convergence of maximal term μ(z,F)={|a_n|e^(zλ_n) : n≥ 0}, respectively. It is well known that G_μ(F), G_a(F) are convex domains. Let us denote N_1(z):={n : (zλ_n)>0}, N_2(z):={n : (zλ_n)<0} and \[α^(1)(θ) :=_0pt2n→ +∞nN_1(e^iθ) -|a_n|(e^iθλ_n),α^(2)(θ) :=_0pt2n→ +∞nN_2(e^iθ) -|a_n|(e^iθλ_n).\] Assume that a_n→ 0 as n→ +∞. In the article, we prove the following statements. 1) If α^(2)(θ)<α^(1)(θ) for some θ[0,π) then \[{te^iθ : t(α^(2)(θ),α^(1)(θ))}G_μ(F)\] as well as \[{te^iθ : t(-∞,α^(2)(θ))(α^(1)(θ),+∞)}G_μ(F)=.\] 2) G_μ(F)=_θ[0,π){z=te^iθ : t(α^(2)(θ),α^(1)(θ))}. 3) If h:=_n→ +∞-|a_n|n(1,+∞), then \[(hh-1· G_a(F))G_μ(F)G_c(F).\] If h=+∞ then G_a(F)=G_c(F)=G_μ(F), therefore G_c(F) is also a convex domain. , number=2, journal=Carpathian Mathematical Publications, author=Kuryliak, M.R. and Skaskiv, O.B., year=2023, month=Dec., pages=594–607