The estimations of lower order $\left(R\right)$$\lambda$ in terms of the sequences $\left\{a_n\right\}$ and $\left\{{\lambda }_n\right\}$ for an entire Dirichlet series $f\left(s\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{s\lambda n}$, have been obtained, namely :$$\begin{array}{cc}\hfill \lambda & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{{\lambda }_{n_p}log{\lambda }_{n_{p-1}}}{log|a_{n_p}{|}^{-1}}\hfill \\ & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{({\lambda }_{n_p}-{\lambda }_{n_{p-1}})log{\lambda }_{n_{p-1}}}{log\left|a_{n_{p-1}}\right|a_{n_p}|}.\hfill \end{array}$$One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

For entire functions defined by absolutely convergent Dirichlet series, a theorem on their mean values is established which include the results of Kamthan, Juneja and Awasthi.

We study the supremum of some random Dirichlet polynomials $D_N\left(t\right)={\sum }_{n=2}^N\epsilon ₙdₙn^{-\sigma -it}$, where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials ${\sum }_{n{\in }_{\tau }}\epsilon ₙn^{-\sigma -it}$, ${}_{\tau }=2\le n\le N:P⁺\left(n\right)\le p_{\tau }$, P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, $su{p}_{t\in ℝ}\left|{\sum }_{n=2}^N\epsilon ₙn^{-\sigma -it}\right|\approx \left(N^{1-\sigma }\right)/\left(logN\right)$. The proofs are entirely based on methods of stochastic processes, in particular the metric...

L’article donne des réponses optimales ou presque optimales aux questions suivantes, qui remontent à Stieltjes, Landau et Bohr, et concernent des séries de Dirichlet $A_j=\sum _{n=1}^{\infty }a(j,n)n^{-s}$$(j=1,2...$