Dirichlet series with rarely alternating real coefficients (Q1358062)

Let \(F(s)=\sum_{n=1}^\infty a_{n}\exp(\lambda_{n}s)\) be a Dirichlet series that converges absolutely in \(\{\operatorname{Re}s<0\}\). Put \(M(\sigma):=\sup\{|F(\sigma+it)|\: t\in\mathbb R\}\). The author presents conditions under which \(\log M(\sigma)=(1+o(1))\log |F(\sigma)|\) as \(\sigma\longrightarrow0-\) outside a set \(E\subset[-1,0)\) with \(\limsup_{\sigma\to0-}(-1/\sigma)\int_{E\cap[\sigma,0)}dx=0\).