A Note about Iterated Arithmetic Functions

A Note on Rational Arithmetic Functions

A Law of the Iterated Logarithm for Arithmetic Functions

Let X,X1, X2, . . . be a sequence of centered iid random variables. Let f(n) be a strongly additive arithmetic function such that ∑ p<n f2(p) p → ∞ and put An = ∑ p<n f(p) p . If EX2 < ∞ and f satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm: lim sup N→∞ ∑N n=1 f(n)Xn AN √ 2N log logN a.s. = ‖X‖2. We also prove the validity of the corresponding weighted...

(1) A NOTE ABOUT THE {Ki(z)} ∞ i=1 FUNCTIONS

In the article [10] A. Petojević has considered the sequence of functions Ki(z) and he gave some statements about this sequence. In this note we give some simple proofs of Theorems 3.3. and 3.6. from the article [10], and also we give a solution of the open problem which is proposed in the same article by Question 3.7. At the end of this note we give a proof of differential transcendency of the...

A note on stable iterated function systems

Let X denote a compact metric space with metric d, and let f : X → X denote a continuous self-map on X . For any subset E of X , we let Cl(E) denote the closure of E. Following [1], we denote by { ,X} an iterated function system, or IFS, onX . That is, is a finite family { f1, . . . , fm} of continuous self-maps on X . In this paper we do not consider the case in which is an infinite family. Le...

A note on iterated galileo sequences

Galileo sequences are generalizations of a simple sequence of integers that Galileo used in early 17th century for describing his law of falling bodies. The curious property he noted happens to be exactly what is needed to quantify his observation that the acceleration of falling bodies is uniform. Among the generalizations and extensions later given are iterated Galileo sequences. We show that...