In an earlier paper Gyarmati introduced the notion of f-correlation for families of binary pseudorandom sequences as a measure of randomness in the family. In this paper we generalize the f-correlation to families of pseudorandom sequences of k symbols and study its properties.
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Let p be a prime number, ℚp the field of p-adic numbers, and $$ \bar {\mathbb{Q}}_p $$ a fixed algebraic closure of ℚp. We provide an analytic version of the normal basis theorem which holds for normal extensions of intermediate fields ℚp ⊆ K ⊆ L ⊆ $$ \bar {\mathbb{Q}}_p $$ .
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We investigate the distribution of $Φ(n) = 1+ ∑_{i=1}ⁿ φ(i)$ (which counts the number of Farey fractions of order n) in residue classes. While numerical computations suggest that Φ(n) is equidistributed modulo q if q is odd, and is equidistributed modulo the odd residue classes modulo q when q is even, we prove that the set of integers n such that Φ(n) lies in these residue classes has a positive lower density when q = 3,4. We also provide a simple proof, based on the Selberg-Delange method, of a result of T. Dence and C. Pomerance on the distribution of φ(n) modulo 3.
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We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.
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