For some random Dirichlet series of order(R) infinite almost surely, every horizontal line is a strong Borel line of order(R) infinite and without exceptional Little functions. | Turk J Math 32 (2008) , 245 – 254. ¨ ITAK ˙ c TUB On the Distribution of Random Dirichlet Series in the Whole Plane Qiyu Jin and Daochun Sun Abstract For some random Dirichlet series of order(R) infinite almost surely, every horizontal line is a strong Borel line of order(R) infinite and without exceptional Little functions. Key Words: Random Dirichlet series, Order(R), strong Borel line, little function. 1. Preliminaries For random Dirichlet-Rademacher, Steinhaus and N series of order(R) infinite almost surely (.), it was proved that . every horizontal line is a Borel line of order(R) infinite and with a possible exceptional value [10], [11]. Later, in [12], by generalized Paley-Zygmund lemma in [8], it is proved that for more general random Dirichlet series of order(R) infinite . every horizontal line is a Borel line of order(R) infinite and without exceptional values. In this paper, we replay exceptional values by exceptional Little functions, and prove that for the random Dirichlet series of order(R) infinite ., every horizontal line is a strong Borel line of order(R) infinite and without exceptional Little functions. Our method can be applied to study some random Dirichlet series of generalized Orders(R) as, [1], [5], [11], [13]. The books [2], [3], [9] are very enlightening and helpful in the related research. 2000 AMS Mathematics Subject Classification: 30D35, 60H90 245 JIN, SUN Consider random Dirichlet series f(s, ω) = +∞ an Zn (ω)e−λn s , () n=0 and an associated Dirichlet series g(s) = +∞ an e−λn , () n=0 where {an } ⊂ C, s = σ + it ∈ C, 0 ≤ λ0 0 and ∀ϕ ∈ H ln+ n(σ, t0 , η, f(s, ω) = ϕ(s)) = +∞, σ→−∞ −σ lim where n(σ, t0 , η, f(s, ω) = ϕ(s)) = {s|f(s, ω) = ϕ(s), s ∈ B ∗ (σ, t0 , η), ϕ ∈ H}, 246 () JIN, SUN B ∗ (σ, t0 , η) = {s|Res ≥ σ} ∩ B(t0 , η), B(t0 , η) = {s||Ims − t0 | 0, consider the simple mapping z = φ1 (s) = exp[− π (s − it0 )], 2η w = φ2 (z) = z −1 . z +1 () Denote the inverse mappings by s =
