#### Year of Publication

1994

#### Paper Type

Master's Thesis

#### College

College of Arts and Sciences

#### Degree Name

Master of Science in Mathematical Sciences (MS)

#### Department

Mathematics & Statistics

#### First Advisor

Dr. Jingcheng Tong

#### Second Advisor

Dr. Adel Boules

#### Third Advisor

Dr. Peter Braza

#### Abstract

Let A = {*a _{n}*}

^{∞}

_{n = 1}be a sequence of positive integers. There are two related sequences

*P*and

_{n}*Q*obtained from A by taking partial convergents out of the number [0; a

_{n}_{1}, a

_{2}, ..., a

_{n}, ...], where

*P*and

_{n}*Q*are the numerators and denominators of the finite continued fraction [0; a

_{n }_{1}, a

_{2}, ...,a

_{n}].

Let *P*(*n*) be the largest positive integer *k* , such that *P _{k}* ≤

*n*. The sequence

*Q*(

*n*) is defined similarly.

- • A known result of Barnes' Theorem states that

*P*

- (

*n*

- ) =

*o*

- (

*n*

- ) and

*Q*

- (

*n*

- ) =

*o*

- (

*n*

- ).

- • In this paper we improve this result as

*P*

- (

*n*

- ) =

*O*

- (log n) and

*Q*

- (

*n*

- ) =

*O*

- (log n), where it follows that

*P*

- (

*n*

- )=

*o*

- (

*n ^{ε}*

- ) and

*Q*

- (

*n*

- ) =

*o*

- (

*n ^{ε}*

- ) for any

*ε*

- >0.

#### Suggested Citation

Vafabakhsh, Seyed J., "Density of the Numerators or Denominators of a Continued Fraction" (1994). *UNF Theses and Dissertations*. 80.

http://digitalcommons.unf.edu/etd/80