Lower Bounds for Absolutely Convergent Dirichlet Series and Pits Property

This article investigates the lower bounds of analytic functions defined by absolutely convergent Dirichlet series in the left half-plane and establishes conditions under which such functions exhibit the pits property. The study extends classical results for entire Dirichlet series and lacunary power series by refining assumptions on the sequence of exponents and resolving gaps in earlier proofs found in the literature. Central to the analysis is the introduction of a modified function k(σ), whose behavior determines the size of exceptional sets surrounding the zeros of the series. Under suitable growth conditions on the exponents, the authors prove that outside small disks centered at the zeros, the modulus of the Dirichlet series admits explicit lower bounds involving its maximum term. Several auxiliary lemmas provide sharp estimates for the maximum modulus, the distribution of zeros, and the behavior of truncated Dirichlet polynomials. The main theorem demonstrates that the pits property holds uniformly in vertical strips approaching the imaginary axis. The paper concludes with a discussion of how the imposed step-size condition on exponents might be weakened, formulating a conjecture regarding a more general condensation index.