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

Dirichlet series with real coefficients

“…where ( ; ) is the number of the points in the segment {ℎ : |ℎ − | }. We note that in the case ( ) = ln , condition (7) holds true immediately (it was shown in [4]).…”

“…Lemma 1 [5]. Let Φ ∈ and inverse for function Φ satisfies condition (4). Let also ( ) be a non-decaying positive and continuous on [0, ∞) function and…”

“…As ∆ = 0, identity (2) is valid also for the function of finite lower order [3]. In [4] there were found the best possible conditions for function ( ) (being weaker than the condition ∆ = 0) under those for each function of finite order (finite lower order) defined by series (1) the asymptotic identity…”

“…exp ⏟ ⏞ ( ) ( ≥ 1), then as = 1, class (Φ) comprises Dirichlet series of finite Ritt lower order. This is why the appropriate results in [3]- [4] are corollaries of Theorem 1.…”