On the stability of the maximum term of the Dirichlet series

We study the behavior of the maximum term of the modified Dirichlet series with positive exponents, the sum of which is an entire function. In this article, we prove a criterion for the equivalence of the logarithm of the maximum term of the original series and of the logarithm of the maximum term of the the modified series on the asymptotic set for the class of entire Dirichlet series defined by some convex majorant of growth. The corresponding problem on the stability of the maximum term for entire Dirichlet series of arbitrary growth was studied by the first author in connection with the Polya problem on the asymptotic behavior of entire transcendental functions on curves going to infinity.