In this paper we give characterizations of those holomorphic functions in the unit disc in the complex plane that can be written as a quotient of functions in A(D), A∞(D) or Λ1(D) with a nonvanishing denominator in D. As a consequence we prove that if f ∈ Λ1(D) does not vanish in D, then there exists g ∈ Λ1(D) which has the same zero set as f in Dbar and such that fg ∈ A∞(D).

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.