We demonstrate that if a sequence of positive numbers increasing to infinity has S-density then it can be prolonged to some sequence that is the zero set of the Weierstrass product of regular behavior on the real axis.
We study the Dirichlet series that are convergent only in a half-plane and whose sequence of exponents extends to a "regular" sequence. We establish some unimprovable estimates for the order of the sum of the Dirichlet series in a half-strip whose width depends on the special distribution density of the exponents.Keywords: order of the Dirichlet series in a half-strip Let Λ = {λ n } (0 < λ n ↑ ∞) be a sequence satisfying the conditionStudying the entire functionsthat are defined by convergent Dirichlet series, Ritt introduced the notion of R-order. Let us give the definition of this quantity which is the broadest characteristic of growth for (2). Since (2) converges in the whole plane, by (1), it converges absolutely. Let M (σ) = sup |t|<∞ |F (σ + it)|. It is well known that log M (σ) is an increasing convex function of σ such that lim σ→+∞ log M (σ) = +∞ and lim σ→−∞ log M (σ) = −∞. The Ritt order (R-order ) of an entire function F given by (2) is (cp. [1]) ρ R = lim σ→+∞ log log M (σ) σ .Suppose that f (z) = ∞ n=1 a n z n has finite order ρ. Execute the change z = e s . Then F (s) = f (e s ) = ∞ n=1 a n e ns is an entire function with ρ R = ρ. Consider the strip S(a, t 0 ) = {s = σ + it : |t − t 0 | ≤ a}. Put M s (σ) = max |t−t 0 |≤a |F (σ + it)|. The quantity ρ s = lim σ→+∞ log + log M s (σ) σ (a + = max(a, 0)) is called the R-order of F in S(a, t 0 ). Let lim n→∞ n λ n = D < ∞, D * = lim λ→+∞ 1 λ λ 0 D(x) dx, The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00417-a).Ufa.
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