The number of irreducible polynomials over finite fields of characteristic 2 with given trace and subtrace

In this paper we obtained the formula for the number of irreducible polynomials with degree n over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al. (2003) [2].     

Exponential sums formed with the Möbius function

Let $\mu \left(n\right)$ be the Möbius function and $x$ real. In this paper, we investigated the best possible estimates for the sum ${\sum }_{n\le x}\mu \left(n\right)e\left(n^k\theta \right)$ under the weak Generalized Riemann Hypothesis. A similar result also holds for the Liouville function $\lambda \left(n\right)$.     

The Möbius function of permutations with an indecomposable lower bound

We show that the Möbius function of an interval in a permutation poset where the lower bound is sum (resp. skew) indecomposable depends solely on the sum (resp. skew) indecomposable permutations contained in the upper bound, and that this can simplify the calculation of the Möbius sum. For increasing oscillations, we give a recursion for the Möbius sum which only involves evaluating simple inequalities.     

Two meromorphic functions on annuli sharing some pairs of values

In this article, we prove that two admissible meromorphic functions $f$ and $g$ on an annulus must be linked by a Möbius transformation if they share a pair of values ignoring multiplicities and share other four pairs of values with multiplicities truncated by 2. We also show that two admissible meromorphic functions which share $q(q\ge 6)$ pairs of values ignoring multiplicities are linked by a Möbius transformation. Moreover, in our results, the zeros with multiplicities more than a certain number are not needed to be counted in the sharing pairs of values condition of meromorphic functions.     

The Extension of Möbius Groups

Let G be a non-elementary subgroup of SL(2,Г _n ) containing hyperbolic elements. We show that G is the extension of a subgroup of SL(2,C) if and only if that G is conjugate in SL(2,Г _n ) to a group G' with the following properties: (1) There are g ₀, h ? G', where g ₀ and h are hyperbolic, such that fix(g ₀) = {0,∞}, fix(h)∩fix(g ₀) =  and fix(h) ∩ C ≠ ; (2) tr(g) ? C for each g ? G'. As an application, we show that if G contains only hyperbolic elements and uniformly parabolic elements, then G is the extension of a subgroup of SL(2,C), which also yields the discreteness of G.     

The Möbius invariants for circle packings

Möbius invariants of general circle packings are defined in terms of cross ratios. The necessary and sufficient conditions of existence of circle packings are established by the techniques of Möbius invariants. It is shown that circle packings are uniquely determined, up to Möbius transformations, by their Möbius invariants. The rigidity of infinite circle packings with bounded degree is proved using the approach of Möbius invariants.