On a theorem of Pólya on entire functions with real Taylor coefficients (Q1360849)

The author studies entire functions represented by the Dirichlet series with real coefficients \[ F(s)=\sum\limits_{n=1}^{\infty }a_ne^{\lambda _ns}\quad (s=\sigma +it), \] where \(\{\lambda _n\}\) (\(0<\lambda _n\uparrow \infty \)) is a sequence satisfying some noncondensation conditions. Put \(M(\sigma )=\sup\limits_{{}t{}<\infty }F(\sigma +it){}\). Suppose \(F(s)\) has finite \(R\)-order (or finite lower \(R\)-order). Criteria for the asymptotic equality \[ \log M(\sigma )=(1+o(1))\log {}F(\sigma ){} \] to hold as \(\sigma \to \infty \) beyond some set \(E\subset [0,\infty )\) of zero lower density are obtained. (Here lower density of a measurable set \(E\subset [0,\infty )\) is defined to be the lower limit of the ratio \(\operatorname{mes} (E\cap [0,r])/ r\) as \(r \to \infty \).).