Behavior on the real line of entire functions represented by Dirichlet series with complex exponents

Abstract

It is shown, in particular, that if λn ≠ λk when n ≠ k, Re λn > 0, and\(\sum\limits_{n = 1}^\infty { (1 + \operatorname{Re} \lambda _n )} /|\lambda {}_n|^2< \infty\), then an entire function F that is bounded on the real line and represented by a Dirichlet series\(F(z) = \sum\limits_{n = 1}^\infty {}\) dn exp (λnz) that is uniformly and absolutely convergent on each compactum in ℂ is identically zero.