Let t, b be mutually prime positive integers. We say that the residue class t mod b is basic if there exists n such that tn ≡ -1 mod b; otherwise t is not basic. In this paper we relate the basic character of t mod b to the quadratic character of t modulo the prime factors of b. If all prime factors p of b satisfy p ≡ 3 mod 4, then t is basic mod b if t is a quadratic non-residue mod p for all such p; and t is not basic mod b if t is a quadratic residue mod p for all such p. If, for all prime factors...

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.

Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1’s in the binary expansions of real algebraic numbers. A central result is that if a real $y$ has algebraic degree $D\>1$, then the number $\#\left(\right|y|,N)$ of 1-bits in the expansion of $\left|y\right|$ through bit position $N$ satisfies$$\#\left(\right|y|,N)\>C{N}^{1/D}$$for a positive number $C$ (depending on $y$) and sufficiently large $N$. This in itself establishes the transcendency of a class of reals ${\sum }_{n\ge 0}1/2^{f\left(n\right)}$ where the integer-valued...