Let F (z) β R[z] be a polynomial with positive leading coefficient, and let Ξ± > 1 be an algebraic number. For r = deg F > 0, assuming that at least one coefficient of F lies outside the field Q(Ξ±) if Ξ± is a Pisot number, we prove that the difference between the largest and the smallest limit points
Arithmetic functions with linear recurrence sequences
β Scribed by Florian Luca; Igor E. Shparlinski
- Publisher
- Elsevier Science
- Year
- 2007
- Tongue
- English
- Weight
- 158 KB
- Volume
- 125
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
β¦ Synopsis
We obtain asymptotic formulas for all the moments of certain arithmetic functions with linear recurrence sequences. We also apply our results to obtain asymptotic formulas for some mean values related to average orders of elements in finite fields.
π SIMILAR VOLUMES
We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences.
Let G(k, r) denote the smallest positive integer g such that if 1=a 1 , a 2 , ..., a g is a strictly increasing sequence of integers with bounded gaps a j+1 &a j r, 1 j g&1, then [a 1 , a 2 , ..., a g ] contains a k-term arithmetic progression. It is shown that G(k, 2) > -(k & 1)Γ2 ( 43 ) (k&1)Γ2 ,
Sa rko zy and other authors have characterized the multiplicative and the additive sequences among the solutions g: N Γ C of homogeneous linear recurrence equations with complex coefficients. Their results are special cases of a much more general theorem concerning recurrent sequences g satisfying c