We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f\left(s\right)={\sum }_{n=1}^{\infty }aₙn^{-s}$ with ${\left|\right|f\left|\right|}_{\infty }:=su{p}_{\Re s>0}\left|f\left(s\right)\right|<\infty$, then ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-2}{\le \left|\right|f\left|\right|}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }\left|aₙ\right|n^{-1/2}{\le C\left|\right|f\left|\right|}_{\infty }$, C being an absolute constant.

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an earlier result...

The aim of this paper is to obtain a generalization of W. A. Woyczyński and B. Ram results concerning integrability of power series in terms of their coefficients for the class GM of general monotonic sequences.

The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...

In the paper we obtain that, under some condition, the Rademacher-Dirichlet series or the Steinhaus-Dirichlet series on the whole plane and on the horizontal zone almost surely have the same growth.

We define Knopp-Kojima maximum modulus and the Knopp-Kojima maximum term of Dirichlet series on the right half plane by the method of Knopp-Kojima, and discuss the relation between them. Then we discuss the relation between the Knopp-Kojima coefficients of Dirichlet series and its Knopp-Kojima order defined by Knopp-Kojima maximum modulus. Finally, using the above results, we obtain a relation between the coefficients of the Dirichlet series and its Ritt order. This improves one of Yu Jia-Rong's...