On the convergence rate of the partial sums of positive entire Dirichlet series (Q1180811)

Let \(\Lambda=(\lambda_ n)\) be a sequence of nonnegative numbers monotonically increasing to \(\infty\), \(\lambda_ 0=0\), and let \(S(\Lambda)\) denote the class of entire Dirichlet series, that is, of absolutely convergent series of the form \[ F(z)=\sum^ \infty_{k=0}a_ k\exp\{z\lambda_ k\},\;a_ 0=1,\;z=x+iy. \] Some notations: \(F\in S_ +(\Lambda)\) iff \(F\in S(\Lambda)\) and \(a_ k>0\) \((k\in\mathbb{N})\); \(S_ n(x)=\sum^ n_{k=0}a_ k\exp(x\lambda_ k)\); \(\sigma_ n(F)=\max\{{1\over S_ n(x)}-{1\over F(x)}:x\in\mathbb{R}\}\). The main result is: If \(F\in S_ +(\Lambda)\) then \[ \limsup_{n\to\infty}\left({1\over\ln n}\ln{1\over\sigma_ n(F)}\right)=\infty\quad\Leftrightarrow\quad\sum^ \infty_{n=1}{1\over n\lambda_ n}<\infty. \]