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.

If $w=(q_k{)}_{k\in \mathbf{N}}$ denotes the sequence of best approximation denominators to a real $\alpha$, and $s_{\alpha }\left(n\right)$ denotes the sum of digits of $n$ in the digit representation of $n$ to base $w$, then for all $x$ irrational, the sequence $\left(s_{\alpha }\right(n)·x{)}_{n\in \mathbf{N}}$ is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if $\alpha$ has bounded continued fraction coefficients.

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character sums.

In finite Galois extensions $k_1,...,k_r$ of $\mathbf{Q}$ with pairwise coprime discriminants the integral and the prime divisors subject to the condition $N_{k_1/\mathbf{Q}}{𝔞}_r=\cdots =N_{k_r/\mathbf{Q}}{𝔞}_r$ are equidistributed in the sense of E. Hecke.

There exist infinitely many integers $n$ such that the greatest prime factor of $n^2+1$ is at least $n^{6/5}$. The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.

The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae.

Various properties of classical Dedekind sums $S(h,q)$ have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to...