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 n e z λ n, z ∈ C, 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 ) = max { | a n | e R ( z λ 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 : R ( z λ n ) > 0 }, N 2 ( z ) := { n : R ( z λ n ) < 0 } and α ( 1 ) ( θ ) := lim –––– n → + ∞ n ∈ N 1 ( e i θ ) − ln | a n | R ( e i θ λ n ), α ( 2 ) ( θ ) := ¯¯¯¯¯¯¯¯ lim n → + ∞ n ∈ N 2 ( e i θ ) − ln | a n | R ( e i θ λ 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 { t e i θ : t ∈ ( α ( 2 ) ( θ ), α ( 1 ) ( θ ) ) } ⊂ G μ ( F ) as well as { t e i θ : t ∈ ( − ∞, α ( 2 ) ( θ ) ) ∪ ( α ( 1 ) ( θ ), + ∞ ) } ∩ G μ ( F ) = ∅ . 2 ) G μ ( F ) = ⋃ θ ∈ [ 0, π ) { z = t e i θ : t ∈ ( α ( 2 ) ( θ ), α ( 1 ) ( θ ) ) } . 3 ) If h := lim –––– n → + ∞ − ln | a n | ln n ∈ ( 1, + ∞ ), then ( h h − 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. [ABSTRACT FROM AUTHOR]