Ukrainian Mathematical Journal, Vol. 57, No. 4, 2005 ON THE STABILITY OF THE MAXIMUM TERM OF THE ENTIRE DIRICHLET SERIES O. B. Skaskiv and O. M. Trakalo UDC 517.57 We establish necessary and sufficient conditions for the logarithms of the maximum terms of the entire Dirichlet series F ( z ) = ae n z n n λ = +∞ ∑ 0 and B ( z ) = abe n n z n n λ = +∞ ∑ 0 to be asymptotically equivalent as Re z → +∞ outside a certain set of finite measure. Let λ = ( λ n ) ⊂ R be an arbitrary sequence. By S ( λ ) we denote the class of Dirichlet series F ( z ) = ae n z n n λ = +∞ ∑ 0 (1) absolutely convergent in the entire complex plane. For σ ∈ R, let μ ( σ , F ) = max { | a n | e n σλ : n ≥ 0} denote the maximum term of series (1). Let L be the class of functions l positive and continuous on [0; + ∞ ) and such that l (x ) ↑ + ∞ , x → + ∞, i.e., the functions l monotonically increase to + ∞ on a certain interval [ x 0 ; + ∞ ). Let W denote the class of functions w ∈ L such that x w x dx - +∞ ∫ 2 1 () < + ∞. For an arbitrary complex nonzero sequence ( b n ) and a function w ∈ W, we introduce the Dirichlet series B ( z ) = abe n n z n n λ = +∞ ∑ 0 , B – ( z ) = ab e n n z n n - = +∞ ∑ 1 0 λ , B w ( z ) = ae n w z n n n ( ) λ λ + = +∞ ∑ 0 . If the sequence { b n : n ≥ 0 } ⊂ C \ { 0 } satisfies the condition lim inf ln n n n n b b →+∞ - | | | | + ( ) 1 1 λ < + ∞, (2) then F ∈ S( λ) if and only if B ∈ S( λ) and B – ∈ S( λ) , and the relations B w ∈ S( λ) and ln ( ) | | | | + - b b n n 1 ≤ w ( λ n ) , n ≥ n 1 , yield { F , B , B – } ⊂ S( λ ). Lviv National University, Lviv. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 571 – 576, April, 2005. Original article submitted July 11, 2003. 686 0041–5995/05/5704–0686 © 2005 Springer Science+Business Media, Inc.