We study universal Dirichlet series with respect to overconvergence, which are absolutely convergent in the right half of the complex plane. In particular we obtain estimates on the growth of their coefficients. We can then compare several classes of universal Dirichlet series.

It is an open question whether Nehari's theorem on the circle group has an analogue on the infinite-dimensional torus. In this note it is shown that if the analogue holds, then some interesting inequalities follow for certain trigonometric polynomials on the torus. We think these inequalities are false but are not able to prove that.

A well known theorem of Nehari asserts on the circle group that bilinear forms in H² can be lifted to linear functionals on H¹. We show that this result can be extended to Hankel forms in infinitely many variables of a certain type. As a corollary we find a new proof that all the $L^p$ norms on the class of Steinhaus series are equivalent.

Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying $su{p}_{z\in Dₙ}\left|fₙ\left(z\right)\right|\le C\varrho ⁿ$ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f.

Pour $q\in ℂ$, $\left|q\right|\&lt;1$, on définit la $q$-analogue de la fonction zeta de Riemann par les égalités ${\displaystyle {\zeta }_q\left(k\right)=\sum _{n\ge 1}{\sigma }_{k-1}\left(n\right)q^n=\sum _{n\ge 1}\frac{n^{k-1}{q}^n}{1-q^n}}$.Dans [8], W. Zudilin énonce deux questions à propos de ces fonctions de $q$. La première concerne l’indépendance linéaire sur $ℂ\left(q\right)$ des fonctions ${\zeta }_q\left(k\right)$, pour $k\ge 1$, et la seconde l’indépendance algébrique sur $ℂ\left(q\right)$ des fonctions ${\zeta }_q\left(2\right),{\zeta }_q\left(4\right),{\zeta }_q\left(6\right)$, et des fonctions ${\zeta }_q(2k+1)$, $k\ge 0$. Dans [5], Y. Pupyrev répond positivement à la première question, et donne des résultats partiels pour la seconde.Dans cet article, nous considérons la fonction ${\displaystyle L(x,y)=\sum _{n\ge 1}\frac{y^n{x}^n}{1-x^n}}$, et, avec ${\displaystyle \tau =y\frac{d}{dy}}$, les...

In this paper, we consider a random entire function $f(s,\omega )$ defined by a random Dirichlet series ${\sum }_{n=1}^{\infty }{X}_n\left(\omega \right){\mathrm{e}}^{-{\lambda }_{n}s}$ where $X_n$ are independent and complex valued variables, $0\le {\lambda }_n\nearrow +\infty$. We prove that under natural conditions, for some random entire functions of order $\left(R\right)$ zero $f(s,\omega )$ almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J. R. Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341–353, by relaxing condition on the distribution of $X_n$ for such function...

Under mild conditions on the weight function K we characterize lacunary series in the so-called ${}_K$ spaces.