For any k ≥ 1, we study the distribution of the difference between the number of integers n ≤ x with ω(n) = k or Ω(n) = k in two different arithmetic progressions, where ω(n) is the number of distinct prime factors of n and Ω(n) is the number of prime factors of n counted with multiplicity . Under some reasonable assumptions, we show that, if k is odd, the integers with Ω(n) = k have preference for quadratic non-residue classes; and if k is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Richard Hudson. However, the integers with ω(n) = k always have preference for quadratic residue classes. Moreover, as k increases, the biases become smaller and smaller for both of the two cases.

In this paper, we study the number of distinct distances among any N points in hyperbolic surfaces. For Y from a large class of hyperbolic surfaces, we show that any N points in Y determines ≥ c(Y )N/ log N distinct distances where c(Y ) is some constant depending only on Y . In particular, for Y being modular surface or standard regular of genus g ≥ 2, we evaluate c(Y ) explicitly. We also derive new sum-product type estimates.

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Under the Riemann Hypothesis, we connect the distribution of k-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of ζ(s). Moreover, with additional assumptions, we prove the existence of a limiting distribution of $e^{-\frac{y}{2k}}M_k(e^y)$ and study the tail of the limiting distribution, where $M_k(x)=\sum_{n\leq x}\mu_k(n)-{x}/{\zeta(k)}$ and μk(n) is the characteristic function of k-free numbers. Finally, we make a conjecture about the maximum order of Mk(x) by heuristic analysis on the tail of the limiting distribution.