For the Dirichlet series F(s) = ∑1 n=0 fn exp{sλn} with nonnegative increasing +∞ exponents λn and abscissa of absolute convergence σa ∈ (−∞; +∞], the connection between the growth on (−∞; σa) of the maximum of the module M(σ; F) = sup{|F(σ + it)| : t ∈ R} and the behavior of the coefficients fn is studied. For this purpose, L denotes the class of continuous increasing up to +∞ on (x0; +∞) functions α. The membership of α in the class L0 means that α ∈ L and α((1 + o(1))x) = (1 + o(1))α(x)
as x → +∞, and α ∈ Lsi if α ∈ L and α(cx) = (1 + o(1))α(x) as x → +∞. For entire Dirichlet series (σa = +∞), for example, it is proved that if α ∈ L, β ∈ L0, then
lim
σ!+1
(exp{α(ln M(β−1(β(σ) + ln q); F))} − p exp{α(ln M(σ; F))}) = +∞ for such p >
1 and q > 1 that lim
n!1
α(λn)=β (λ− n 1 ln (1=|fn|)) > ln p= ln q. If α ∈ Lsi, β ∈ L0,
dβ−1(cα(x))
d ln x = O(1) as x → +∞ and ln n = o(λnβ−1(cα(λn))) as n → ∞ for each
c ∈ (0; +∞), α(λn+1) ∼ α(λn) as n → ∞ and ln |fn| − ln |fn+1|
λn+1 − λn ↗ +∞ for n0 ≤ n → ∞, then
lim
σ!+1
(exp{α(ln M(β−1(β(σ) + ln q); F))} − p exp{α(ln M(σ; F))}) = −∞ for such p > 1
i q > 1, which lim
n!1
α(λn)=β (λ− n 1 ln (1=|fn|)) < ln p= ln q.Similar results were obtained for Dirichlet series absolutely convergent in the half-plane {s :
Res < 0}. For example, it was proved that if σa = 0, β ∈ Lsi, α(ex) ∈ Lsi and α(x) = o(β(x)) as
x → +∞, then
lim
σ"0 (exp {α (ln M (−β−1(β(1=|1σ|) + ln q); F))} − p exp{α(ln M(σ; F))}) = +∞
for such p > 1 and q > 1 that lim
n!1
α(λn)
β(λn= ln+ |fn|) > ln p= ln q.

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