On the equivalence of the sum and the maximal term of the Dirichlet series absolutely convergent in the half-plane

Abstract

For absolutely convergent in the half-plane $\{z\colon {\rm Re\,}z<0\}$ Dirichlet series $F(z)=\sum\limits_{n=0}^{+\infty}a_ne^{z\lambda_n},$ where $0\leq\lambda_n\uparrow +\infty\ (0\leq n\uparrow +\infty),$ we establish conditions on the coefficients of its Newton majorant, sufficient for the relation $F(x+iy)=(1+o(1))a_{\nu(x)}e^{(x+iy)\lambda_{\nu(x)}}$ to hold as $x\to -0$ outside some set $E$ of zero logarithmic density in the point $0,$ uniformly by $y\in{\mathbb R}$.