We consider the behavior of the power series ${}_0\left(z\right)={\sum }_{n=1}^{\infty }{\mu }^2\left(n\right)z^n$ as z tends to $e\left(\beta \right)=e^{2\pi i\beta }$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then ${}_0\left(e\left(\beta \right)r\right)=O\left({(1-r)}^{-1/2-\epsilon }\right)$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that ${}_0\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-1+\delta }\right)$.

Consider the power series $\left(z\right)={\sum }_{n=1}^{\infty }\alpha \left(n\right)zⁿ$, where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity $e^{2\pi il/q}$. We give effective omega-estimates for $\left(e(l/p^k)r\right)$ when r → 1-. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

Let $\left(z\right)={\sum }_{n=1}^{\infty }\mu \left(n\right)z^n$. We prove that for each root of unity $e\left(\beta \right)=e^{2\pi i\beta }$ there is an a > 0 such that $\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-a}\right)$ as r → 1-. For roots of unity e(l/q) with q ≤ 100 we prove that these omega-estimates are true with a = 1/2. From omega-estimates for (z) we obtain omega-estimates for some finite sums.

We prove, in a general framework, the existence of a closed infinite-dimensional subspace consisting of universal series.

Let $E={\bigcup }_{n=1}^{\infty }{I}_n$ be the union of infinitely many disjoint closed intervals where $I_n=[a_n$, $b_n]$, $0<a_1<b_1<a_2<b_2<\cdots <b_n<\cdots$, ${lim}_{n\to \infty }{b}_n=\infty .$ Let $\alpha \left(t\right)$ be a nonnegative function and ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system $\left\{t^{{\lambda }_n}{log}^{m_n}t\right\}$ in $C_0\left(E\right)$ is obtained where $C_0\left(E\right)$ is the weighted Banach space consists of complex functions continuous on $E$ with $f\left(t\right){\mathrm{e}}^{-\alpha \left(t\right)}$ vanishing at infinity.

The paper deals with the computation of Aden functions. It gives estimates of errors for the computation of Aden functions by downward reccurence.

We investigate how the growth of an algebroid function could be affected by the distribution of the arguments of its a-points in the complex plane. We give estimates of the growth order of an algebroid function with radially distributed values, which are counterparts of results for meromorphic functions with radially distributed values.

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 ).