Let $A\in(-\infty,+\infty]$ and $\Phi$ be a continuously on $[\sigma_0,A)$ function such that $\Phi(\sigma)\to+\infty$ as $\sigma\to A-0$. We establish a necessary and sufficient condition on a nonnegative sequence $\lambda=(\lambda_n)$, increasing to $+\infty$, under which the equality$$\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln M(\sigma,F)}{\Phi(\sigma)}=\overline{\lim\limits_{\sigma\uparrow A}}\frac{\ln\mu(\sigma,F)}{\Phi(\sigma)},$$holds for every Dirichlet series of the form $F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n}$, $s=\sigma+it$, absolutely convergent in the half-plane ${Re}\, s<A$, where $M(\sigma,F)=\sup\{|F(s)|:{Re}\, s=\sigma\}$ and $\mu(\sigma,F)=\max\{|a_n|e^{\sigma\lambda_n}:n\ge 0\}$ are the maximum modulus and maximal term of this series respectively.

Нехай $(\lambda_n)$ $-$ невід'ємна зростаюча до $+\infty$ послідовність, $\tau=\limsup\limits_{n\to\infty}\frac{\ln n}{\lambda_n}$, а $\rho$ $-$ додатне число. З класичної теореми Ж. Валірона випливає, що для кожного цілого ряду Діріхле вигляду $F(s)=\sum a_ne^{s\lambda_n}$ правильна оцінка$$\limsup_{\sigma\to+\infty}\frac{\ln \sup\{|F(s)|:\,\text{Re}\, s=\sigma\}}{e^{\rho\sigma}}\le e^{\rho\tau} \limsup_{n\to\infty}\frac{\lambda_n}{e\rho}|a_n|^\frac{\rho}{\lambda_n}.$$В роботі доведено точність цієї оцінки.

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UDC 517.53For the entire Dirichlet series f (z) = X 1 n=0 ane zλn , we establish necessary and sufficient conditions on the coefficients an and exponents λn under which the function f has the Paley effect, i.e., the condition lim supis satisfied, where M f (r) and T f (r) are the maximum modulus and the Nevanlinna characteristic of the function f, respectively.