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On a theorem of Pólya on entire functions with real Taylor coefficients. (English. Russian original) Zbl 0868.30002

Sib. Math. J. 38, No. 1, 37-46 (1997); translation from Sib. Mat. Zh. 38, No. 1, 46-55 (1997).
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 \(r \to \infty \).).

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
30D10 Representations of entire functions of one complex variable by series and integrals
30D15 Special classes of entire functions of one complex variable and growth estimates
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References:

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